Computational Heat Transfer, VOL1; Mathematical Modelling_1995.pdf

Computational Heat Transfer Volume 1 MaShematScal Modslling

A.A. SAMARSKll P.N. VABISHCHEVICH

Computational Heat Transfer Volume 1 Mathematical Modelling

A. A. Samarskii

P. N. Vabishchevich Russian Academy of Sciences, Moscow

JOHN WILEY & SONS Chichester New York

Brisbane

Toronto

Singapore

Copyright

1995 by John Wiley & Sons Ltd. Baffins Lane. Chichester West Sussex PO19 IUD, England National Internotional

(01243) m 7 7 7 ( t 4 4 ) 1243 779777

All rights reserved. No part of this publication may be reproduced by any means, or transmitted, or translated into a machine language without the written permission of the publisher.

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British Library Cataloguing in Publication Data A catalogue record for this book is available fmm the British Library ISBN 0 471 95659 7 Produced from camera-ready copy supplied by the authors using TeX Printed and bound in Great Britain by Biddles Ltd, Guildford and King's Lynn This book is printed a n acid-free paper responsibly manufactured fmm sustainable forestation, far which at least two trees are planted for each one used for paper production.

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Use of Computers for Mathematical Modelling . . . . . . . . . . . . . . . Computational Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Modelling of Heat Transfer Processes . . . . . . . . . . . . . .

3 6 12

M a t h e m a t i c a l M o d e l s of Physics of H e a t . . . . . . . . . . . . . . . Heat Conduction in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Closing Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PhaseTransitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convective Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Radiation of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 16 22 30 37 45 49 55

Analytical M e t h o d s of H e a t Transfer . . . . . . . . . . . . . . . . . . Dimensionless Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analytical Solution of Linear Problems. . . . . . . . . . . . . . . . . . . . . . Exact Solutions of Nonlinear Problems . . . . . . . . . . . . . . . . . . . . . . Asymptotic Methods of Heat Transfer . . . . . . . . . . . . . . . . . . . . . . Bibliography and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 58 62 69 75 84

S t a t i o n a r y P r o b l e m s of H e a t Transfer . . . . . . . . . . . . . . . . . . Boundary Value Problems for Second-order Elliptic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Construction of Difference Schemes . . . . . . . . . . . . . . . . . . . . . . . . . Uniform Convergence of Difference Schemes . . . . . . . . . . . . . . . . . . Convergence of Difference Schemes in an Energetic Space . . . . . Direct Methods for Solving Grid Equations . . . . . . . . . . . . . . . . . . Iterative Methods of Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . Iterative Methods for Solving Grid Equations . . . . . . . . . . . . . . . .

1 1

CONTENTS

Numerical Solution of Problems in Irregular Domains . . . . . . . . . Nonlinear Problems of Stationary Heat Transfer . . . . . . . . . . . . . . Bibliography and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

188 207 218

N o n s t a t i o n a r y P r o b l e m s of H e a t Trasfer . . . . . . . . . . . . . . . Boundary Value Problems for Second-Order Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Difference Schemes for Nonstationary Problems . . . . . . . . . . . . . . Uniform Convergence of Difference Schemes for the Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theory of Stability of Difference Schemes . . . . . . . . . . . . . . . . . . . Stability and Convergence of Difference Schemes for the Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asymptotical Stability of Difference Schemes for the Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hyperbolic Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regularization of Difference Schemes . . . . . . . . . . . . . . . . . . . . . . . Nonlinear Nonstationary Problems . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Economical Difference Schemes for Nonstationary Heat Conduction Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . Implementation of Implicit Schemes on a Computer . . . . . . . . . . The Method of Alternating Directions . . . . . . . . . . . . . . . . . . . . . . Factorized Difference Schemes for the Heat Equation . . . . . . . . . Additive Difference Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Locally One-dimensional Difference Schemes . . . . . . . . . . . . . . . . . Bibliography and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

299 300 309 316 324 334 344

H e a t C o n d u c t i o n P r o b l e m s w i t h P h a s e Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variable Domain Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fixed Domain Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transformation of Dependent Variables . . . . . . . . . . . . . . . . . . . . . Quasi-Stationary Stefan Problem . . . . . . . . . . . . . . . . . . . . . . . . . . Modelling of Phase Transitions in Binary Alloys . . . . . . . . . . . . . . Bibliography and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

347 348 361 370 377 388 394

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

397

Dedicated to the memory of Nikolai Nikolaevich Govorun

Preface

This book, published in two volumes, is devoted to mathematical modelling of heat transfer problems. Recently considerable progress in scientific research has been made due to powerful computational means involving both computers and numerical methods. Conventional analytical methods of applied mathematics are currently considered only to be auxiliary tools. Computational experiment employed in modern research involves three stages, namely the model, algorithm and computer code. The authors give a comprehensive review of methods employed in modern applied mathematics and describe the technology of computational experiment applied to the physics of heat, a thoroughly developed branch of science. The book is intended for both experts in mathematical modelling and those studying the field. The book might be of interest to specialists in mathematical modelling applied to other branches of science than heat transfer. The latter problems, on the one hand, are of keen interest, and the corresponding problems are studied thoroughly. On the other hand, mathematical models employed in heat transfer problems are very diverse and, hence, can he considered as typical of a wide range of investigations. The book can be used as a textbook for graduate students whose major is applied mathematical modelling. Depending on the audience, the emphasis in the course can be put on either theoretical or applied topics related t o numerical methods. Some classes of applied problems can be comprehensively studied in optional or special courses. We hope that researchers dealing with numerical simulation will also find the subject matter useful. The reason is that a considerable portion of the material has not yet been published in monographs or textbooks. The presentation is based on conventional mathematics familiar to students specializing in applied mathematics. Elements of the operator theory in finite-dimensional Hilbert spaces comprise the core of the mathematical tools employed. We use only a minimal amount of mathematics that is necessary for a thorough and comprehensive consideration of numerical methods applied in mathematical physics. The authors think that the book can be useful for training engineering students. This is why we emphasized the algorithmic aspect of numerical methods, which affected the choice of topics covered and the style of

x

PREFACE

presentation. We describe the solution of some particular problems, report the results and include the codes in the text to make the book useful to a large audience of readers. A great number of books for engineers are devoted to numerical simulation of various heat transfer problems. The authors of many of them consider only foundations of numerical mathematics to make their texts understandable for engineers. In addition, a lot of references to original works are given without any attempt to analyse them. We tried to match the state-of-theart in the theory and applications of numerical methods rather than to cover these topics superficially. From our viewpoint, training of specialists in applied fields should be based on this, somewhat more advanced, mathematical level. Nowadays numerical simulation is intensively used for solving twwdimensional nonstationary nonlinear problems in mathematical physics. Therefore we selected as the examples tw+dimensional problems that are well studied and securely treated in practice , although in simple computational domains. Computational algorithms considered are used as a base for original investigations of more complex problems. Control problems and inverse heat exchange problems are important applied problems. This book is probably the only one available at the moment where all three basic problems of mathematical physics, namely the direct, control and inverse problems for partial differential equations, are considered in great detail. The theory of numerical methods of mathematical physics develops in several directions. First, we should mention finite difference and finite element methods. The former are considered throughout the book. The two approaches differ in the way problems are discretized, i.e. constructing systems of algebraic equations. For regular grids in regular domains studied in the book, the difference between the two, if any, is minor. Another reason in favour of difference methods is a wide experience in solving numerically nonstationary nonlinear problems for partial differential equations, in particular heatjmass transfer problems. This book is concerned with several aspects of applied mathematical research. This first volume is divided into seven chapters. In the Introduction (Chapter 1) we describe in general applied mathematical modelling and use of computational tools. In Chapter 2, we briefly describe mathematical models of physics of heat and classify the principal problems. We mainly selected the material associated with heatjmass exchange involving solids. Therefore, important applied problems of heatjmass exchange via radiation, for example, are only considered in an approximation of surface radiation. A study of an applied problem starts with conventional (analytical) methods of applied mathematics. Much attention is paid to the solution of simplified problems. Some illustrative examples of implementation of analytical methods in typical heat transfer problems are given in Chapter 3.

PREFACE

xi

Chapters 4-6 comprise the core of the book. They deal with the solution of classical heat transfer problems by finite difference methods. In Chapter 4 we describe the numerical solution of stationary problems. We first recall some fundamental results of the theory of partial differential equations. In particular, we formulate the classical maximum principle for stationary equations, which has a clear thermophysical interpretation. We construct difference equations so that they preserve the main properties of a differential problem. We discuss the construction of difference schemes for stationary problems and describe both direct and iterative methods for finding an approximate solution. The same topics are covered in Chapters 5 for nonstationary problems. We pay particular attention to general stability theory for difference schemes. In order to examine particular schemes, we verify whether necessary and sufficient conditions, formulated as simple operator inequalities, are satisfied. Economical difference schemes (schemes of alternating directions, locally onedimensional schemes, etc.) are considered in Chapter 6. Chapters 7-10 deal with numerical simulation of specific problems of heat exchange. The presentation is based on the difference methods studied for solving stationary and nonstationary problems in heat transfer. In Chapter 7 we study heat transfer problems with phase transitions involving a solid phase. We consider the classical Stefan problem and discuss modelling of melting/solidification of binary alloys. In the second volun~eof this book, starting with Chapter 8, solution techniques for complicated heat transfer problems are presented. In Chapter 8 we investigate special features of heat radiation from the surface of a solid. For example, when modelling heat exchange in a concave body, it is necessary to take into account radiation from separate parts of the boundary. Convective heat transfer problems (Chapter 9) comprise an important class of heat exchange problems. For these problems we introduce general families of monotonic difference schemes, for which the maximum principle holds. The heat/mass transfer is considered in the Boussinesq approximation. We discuss the solution of these problems both in the natural variables 'velocitypressure' and in the variables 'vortex-stream function'. We also suggest the boundary layer approximation. In Chapter 10 numerical methods are used to solve problems in thermoelasticity. For example, the thermoelastic state of a solid with a rectangular cross-section and that of a thin plate are simulated. Thermal process control problems are covered in Chapter 11. We briefly discuss gradient methods, which are the base for numerical solution of the control problems. We distinguish control problems for steady- and unsteadystate sources of heat and for various quality criteria (functionals). Problems of controlling the boundary regime constitute another important class of applications.

xii

PREFACE

The main classes of inverse problems are presented in Chapter 12. The heat conduction problems are solved approximately for the cases in which initial values, boundary regimes or coefficients in equations are recovered. Stability of the solution is studied by perturbation of either original differential (quasi-inversion methods) or difference (regularization of difference schemes) equations. Chapter 13 deals with practice of numerical modelling by the methods presented. We take actual two-dimensional problems of heat transfer as examples. First, we pose a problem and separate the dimensionless parameters of that problem. Then we construct in detail numerical algorithms and give FORTRAN codes. Finally, we describe computational results and present program source listings. Throughout the book, except for the sections Bibliography and Comments, there are no references to other works. We used more space than usual to make the presentation complete. As a result, the book became more convenient for study. No systematic analysis of existing literature on the subject is attempted. We only referred to the work that we used in the book and that can be used by the reader as well to study certain topics more carefully. There are several reasons for not presenting references to original work. In any case, the choice of reference literature from the vast variety of publications would reflect the authors' preferences, demand some research of priority, etc. In particular, the extended list of references would inevitably include more work by Russian authors. We worked on this book in the stimulating and creative atmosphere of the Chair of Computational Methods at the Department of Computational Mathematics and Cybernetics at the M. V. Lomonosov Moscow State University and the Institute of Mathematical Modelling of the Russian Academy of Sciences. We are grateful to our colleagues and disciples for their assistance at different stages of the work. The authors are particularly indebted to M. M. Makarov for his help in preparing the FORTRAN codes listed in the book. In view of the complexity of the work undertaken and inevitable omissions, the authors will appreciate all criticism and suggestions concerning the book.

Aleksander 4 . Samarskii Petr N. Vabishchevich

Introduction

We start by discussing the general problems of implementation of mathematical methods in research. Numerical modelling and computers are currently widely used and contribute substantially to theoretical investigation. Since actual experiments become available, we need to relate experiment with theoretical investigations. Therefore, we can speak of new methods of research that combine theoretical and experimental approaches and are based on up-twdate mathematical modelling. The tools of research considered here involve computational experiment that is supposed to he the most advanced stage of mathematical modelling because of the availability of computers and numerical methods for analysing mathematical models. We shall briefly describe various types of mathematical modelling, emphasizing specific features of using the conventional tools of applied mathematics in modern research and the role of computational algorithms and associated software.

1.1 Mathematical Modelling Modelling has been widely employed when solving applied problems in different branches of technology and is based on studying the properties and characteristics of various objects using natural or artificial analogues (models). Thus, modelling, in this wide context, implies constructing some models and analysing these models after they have been constructed. Any model involves simplification of an object being modelled. A model cannot embrace all properties of an object and is only intended to render those that are of interest. In other words, a model is simpler than the original object and, what is most important, the model is more convenient and easier to handle than the object.

2

COMPUTATIONAL HEAT TRANSFER

In order to investigate an object in detail, several models are employed, each of them simulating some of its characteristics, and even the same property can be studied by different models. Models vary in the extent of qualitative and quantitative adequacy as to the object studied, and modelling turns out to be successful if an adequate model or a set of models has been chosen. Logically, this choice is subjective and is based on all available experimental and theoretical knowledge of the object as well as on all the previous experience in modelling. Various models can be classified into physical and mathematical models. Mathematical models are most typical ideal (theoretical) models for investigation in natural sciences. Physical models are material (objective) models that simulate some number of properties of the object studied and are of the same nature as this object. A model of an object rather than the object itself is studied experimentally in physical modelling. Physical models have an important advantage over mathematical ones, namely, they can possess those properties of an object that, for some reason, cannot be incorporated into mathematical models. For example, this is the case if there is no mathematical model or available models are so complicated that they cannot be analysed with the required degree of completeness. Thus, physical modelling is sometimes the only way we can obtain reliable information about an object. Physical modelling is based on the theory of similarity. In addition to geometrical similarity (similarity of shape), physical similarity between the model and the object is required. This implies that the values of physical quantities for the object should be proportional to those for the model at the corresponding points of space and at the corresponding moments in time. The main idea of this type of modelling is that the values of some dimensionless similarity criteria should be the same for the model and the object. In mathematical modelling, the analysis of mathematical models of an object replaces the investigation of properties and characteristics of the object proper. Mathematical models are studied by means of applied mathematics. Currently, computers are widely used for mathematical modelling as well as methods of numerical analysis. Nowadays mathematical concepts are often used in natural sciences, humanities, and technology. Some people even use the extent to which mathematics is used in a branch of science to judge the level of development of this branch. The familiar aphorism says: 'The more mathematics is involved, the more scientific is the knowledge.' At the first stage we neglect the specific nature of a phenomenon, generalize it, and isolate its mathematical form (i.e. construct a mathematical model). At the second stage we investigate mathematical models as purely m a t h e matical abstract objects and use the general purpose mathematical tools as well as those specially constructed for the purpose. Computers and numerical

INTRODUCTION

3

methods now make it possible to study mathematical models. At the third stage we interpret a model, i.e. render a specific meaning t o mathematical abstractions. An expert in applied mathematical modelling in collaboration with experts in an applied area always recognizes their practical application. Mathematical modelling is somewhat heuristic because a mathematical rather than an actual experiment is carried out. Instead of studying an effect on the object, we study the mathematical model parametrically, i.e. how the solution depends on parameters. This experiment supplements an actual experiment, which allows us to better understand a phenomenon or a process.

1.2 Use of Computers for Mathematical Modelling The advent of computers and rapid development of numerical analysis have substantially enlarged the possibilities of mathematical modelling. With the aid of computers and numerical methods we can solve a lot of problems that could not be solved before with sufficient accuracy, and in a reasonable time period realize large projects in science and technology. As an example note that computers are used for launching and monitoring space vehicles, processing the data from seismic prospecting of minerals, numerical modelling of aerodynamical properties of an aircraft, contour computer tomography in medicine, and so on. Computers have even found application in pure mathematics; we only mention compelling computations on a computer, solution of the famous four-colour problem, etc. New branches of knowledge based on using computational tools are rapidly developing for theoretical investigation of applied problems. In this connection, we should mention computational physics, computational geometry, and computational algebra which logically include computational heat transfer. Investigation of mathematical models essentially involves qualitative analysis and derivation of an exact or an approximate solution. Computers can be used not only to find an approximate solution numerically but also to investigate a mathematical model qualitatively. Qualitative investigation starts with dimensional analysis. The problem is reduced t o a dimensionless form in order to decrease the number of parameters that define the problem. Some dimensionless parameters turn out t o he small or large, and the problem can then he substantially simplified and its specific features taken into account when developing numerical methods for its solution. The mathematical model itself can he complicated and nonlinear and often cannot be studied by conventional methods of applied mathematics. Therefore, in most cases simpler (but still relevant t o the original problem)

4


models are used for qualitative analysis. In this case, we speak of model (or simplified) problems for the main mathematical problem (models for a model). When qualitatively investigating a mathematical problem, much attention is paid to how well it is posed. First of all, the existence of a solution is studied. The corresponding strict results (existence theorems) ensure that the model is correct. Besides, constructive proofs of the existence theorem for a problem can often be used as a basis for approximate methods for solving the problem itself. Stability of solution with respect to small perturbations in initial data is important in applied mathematical modelling. Inverse problems are often unstable, and this should be taken into account when constructing approximate solutions. Nonlinear mathematical models can have multiple solutions. Qualitative analysis of mathematical models includes the study of branching points, bifurcations of solutions, problems of how to single out the desired solution, etc. Methods of qualitative analysis for different types of mathematical models have been developed to different extents. For example, qualitative methods led to the most impressive results for ordinary differential equations. Qualitative methods are also employed in the theory of partial differential equations, although to a smaller extent. An important example we should mention is the maximum principle for second-order parabolic and elliptic equations, which is related to mathematical models of heat transfer. This principle can be used to qualitatively investigate mathematical models based on partial differential equations. An exact or approximate solution is found by analytical or numerical methods. Classical examples of analytical methods include the method of separation of variables and the method of integral relations for linear problems in mathematical physics. Methods of linearization and different versions of perturbation methods are significant for nonlinear mathematical models. This approach is based on asymptotic expansions with respect to a certain small parameter. Much consideration is given to these methods when studying singularly perturbed problems. Qualitative behaviour of a nonlinear problem can be well represented by some of its specific solutions. Specific solutions are sought using automodel variables and the results of group analysis applied to equations that underlie the mathematical model. Complex nonlinear multiparameter models can be investigated on a computer by numerical methods. Unlike an analytical solution, which provides explicit dependence on some initial data of the problem, numerical methods require repetitive solution of the problem for every value of a parameter. However, an analytical solution can rarely be found for nonlinear problems,

INTRODUCTION

5

while a numerical solution of nonlinear problems has become an everyday experience. Let us pass on t o specifying the main stages of using computers for mathematical modelling. We focus on using mathematical means for the numerical solution of a problem. We should mention that computers can also be used for the qualitative investigation of a mathematical model or for seeking analytical solutions to test problems. For example, computers can be used to find self-similar solutions. If an automodel variable has been specified, the original problem for a partial differential equation is reduced, e.g. to an ordinary differential equation, whose general solution can he found using software for analytical computations (methods of computational algebra), e.g. the REDUCE code. We can single out at least two stages, or levels, of using computers for mathematical modelling. The first is to investigate simple mathematical models. At this stage, computers are used together with other (analytical, i.e. purely mathematical) methods of applied mathematics. This stage is characterized by a conventional chain 'client (theorist)executor(app1ied mathematician)'. The client formulates a problem and analyses the results, while the executor solves the problem by a computer. In this case we speak of a specific (rather narrow) problem with a definite set of initial data. The second stage (level) of using computers involves the study of complicated mathematical models, where computational means become decisive and dominating. Conventional tools of applied mathematical modelling play an auxiliary, servicing role here (qualitative investigation of heavily simplified problems (model problems), testing of computational algorithms, etc.). It is this possibility of investigating complex mathematical models by numerical methods on computers that allows us to consider the methodology of scientific research from a new viewpoint. Powerful computers, highly efficient numerical algorithms, and modern software are used together t o organize scientific research within the limits of the unique technology of computational experiment that includes both theoretical and experimental studies. Firstly, we indicate the role of mathematics in theoretical and experimental investigations and then move on to characterizing a new integrated technology of scientific research. The use of mathematical methods is typical of a theoretical level of investigating nature. As our theoretical outlook develops, the mathematical apparatus is applied more widely, with the use of new mathematical concepts and ideas. Mathematical models can he properly classified into fundamental and applied models. For each type of model, mathematical apparatus, and especially such powerful tools as computers and numerical methods, plays a different role. The theory of each branch of science starts with basic laws and statements. The science is then developed by deriving and analysing corollaries of these

6


fundamental postulates and comparing these corollaries with experimental data. Fundamental models cannot be derived from an experiment; however, they can be suggested by one. Corollaries of fundamental models, which we call applied models, should be compared with experiment. This comparison is a criterion for the suitability of the fundamental model itself. Creation of a new fundamental model leads to revision of general laws and the very theoretical ground of the science. Mathematical methods are logically used for investigating corollaries of fundamental models. This is why we usually discuss applied mathematical modelling, which implies that we deal, to a certain extent, with applied mathematical models. The modern stage of development of mechanics, physics, and technology is characterized by complex fundamental models that lead to more and more complex and more advanced mathematical models. However, possibilities of mathematical methods increase as well. Let us make some remarks on using mathematical methods in experiments. An experimenter, in the most general scheme of his or her investigation, affects the object studied, collects information about the results of this influence, and processes it. Results of each experiment are processed statistically. Empiric relationships that interpolate the experimental data to some accuracy are constructed to assess qualitatively the influence of distinct factors (parameters). In this case, we speak of using approximating mathematical models in which there are no significant mathematical models. Deriving criterion dependencies in physics of heat gives an example of approximating models. Nowadays, more and more advanced devices are used in experiment. These devices necessarily perturb a phenomenon or a process studied. In order to get rid of these perturbations, a mathematical model of a device is constructed. Experimental investigations are often carried out by integrated measuring and computing systems that can obtain, store, and process experimental data. When processing experimental data, we often deal with inverse problems. Methods for numerical solution of these problems are now rapidly developing. Thus, computational means are used more and more widely on the stage of processing and interpreting experimental data.

1.3 Computational Experiment Theoretical and experimental studies are to a great extent autonomous. If fundamental models are common and approved, we can pose the problem of a closer relationship between theoretical and experimental investigations. We mean thereby a new integrating technology of scientific research called computational ezpen'rnent.

INTRODUCTION

rH-@ TI mathematical model

I

computations

experiment

under study

investigation of the model

Fig. 1.1.

Let us first give a general scheme of computational experiment and then briefly describe its main stages. A more detailed description of computational experiment is presented in the following chapters. We think a computational experiment to be, in a narrow sense, the creation and investigation of mathematical models for an object studied by computational means. In this sense a computational experiment basically includes the triad 'modelalgorithm-code'. In a wide (methodological) sense, we consider computational experiment to be a new technology of scientific research. The main stages of computational experiment are shown in Fig. 1.1. We only deal with deterministic objects, inherent to heat transfer. For probabilistic objects, different mathematical means are used. Firstly, a mathematical model is constructed for the object studied. The model is based on conventional fundamental models. Numerical experiment actually assumes an investigation of a group of close models. Firstly, a simple model is constructed that, however, is significant and complete from the viewpoint of description of processes studied and closeness t o experimental data. This model is then refined in the computational experiment, new parameters are involved, etc. Hence, we can (moreover, must) speak of a set, more exactly an ordered set (hierarchy), of mathematical models, each describing reality with a certain accuracy. We should attain agreement with experiment within the framework of the simplest model. As soon the mathematical model is constructed, it is preliminarily investigated by the standard means of applied mathematics. The essence of computational experiment is to investigate mathematical models on computers by numerical methods. However, we now discuss only the p r e liminary investigation of the mathematical model. At this stage, the wellposedness of the problem is studied as completely as possible t o the level of accuracy accepted in mathematics. The essence of the preliminary investigation is to single out test (simplified) problems and study them thoroughly, because the complete mathematical

8


model is very complicated. Test mathematical problems are constructed in computational experiment for two purposes: firstly, to qualitatively investigate the complete problem (and, hence, the object under study) and, secondly, to test numerical algorithms for approximate solution of the complete problem. When qualitatively investigating test (simplified) problems, multiplicity of solutions, their stability, etc. are studied. Specific solutions of essentially nonlinear problems, asymptotic solutions, etc. are important as well. Thus, conventional mathematical tools are used at this stage. The next stage of computational experiment includes construction of a discrete problem and a numerical method for its solution. The mathematical model usually involves partial differential equations (the kernel of the mathematical model) or systems of differential and algebraic equations. Numerical algorithms are constructed and studied by means of numerical analysis. There are two trends in scientific research here. According to traditions (paradigm) of pure mathematics, some investigators study discrete models and numerical methods for their analysis without any reference to applied modelling, realization on a computer, or solving an applied problem. They rigorously prove the existence of solutions for discrete problems, derive theoretical estimates of the discrepancy of an approximate solution, and study the convergence of the iterative process. Scientists that belong to the applied direction in numerical analysis use a somewhat different ('physical') level of accuracy, which operates with such nonstrict concepts as 'practical convergence', 'real grids', etc. Requirements for complete severity in applied mathematical modelling do not lead to good results. Numerical experiment has two characteristics, which should be taken into account when creating adequate software. The first one is multivariant computation within the framework of a fixed mathematical model, while the second is availability of several models. It is impossible to get by with one computer code because it is necessary that this code be easily transformed to solve similar problems (for a set of models). Software for computational experiment includes applications software and applications packages. Applications software is designed for solving similar problems (with respect to their mathematical nature) from the same branch. It includes libraries of program modules, more or less independent, that comprise applications programs. In applications software, programs are constructed from modules by hand. In applications packages, system software is used to construct programs, thus this process is essentially automatic. Software packages can be treated as a technology for solving problems within the framework of computational experiment. They can be effectively used to accumulate and make the best use of available software and increase the productivity

INTRODUCTION

Then, in the cycle of a computational experiment, a series of test computations is performed on a computer for different values of parameters of the problem. Applied specialists then analyse and interpret the data obtained. The results are processed taking account of theoretical views and experimental data. This is done, to a large extent, using traditions of classical actual experiment. The experimental data are represented as tables, graphs, photographs of display, films, etc. It should he kept in mind that the amount of information to process and detail the results obtained in computational experiment is incomparably larger than in the actual experiment. Problems associated with storing and processing information in computational experiment become more and more important. When analysing the results, one resolves whether the mathematical model and its numerical realization have been chosen successfully. If necessary, models and numerical methods are refined, and the whole cycle of computational experiment repeated. To characterize computational experiment as a whole, it is very important to note that it is universal. Therefore this technology can easily be extended to include investigation of other objects. This is a characteristic of mathematical modelling, which is due to the fact that many phenomena and processes have the same mathematical models. Since computational experiment is general-purpose, it allows researchers to use the accumulated experience of mathematical modelling and banks of numerical algorithms and software to solve new problems quickly and effectively. The second characteristic of computational experiment as a technology of scientific research is its interdisciplinary character. We constantly emphasize this when saying that an applied mathematician has actually combined a theorist and an experimenter to reach the common goal faster. Numerical experiment can be considered as a convenient form of cooperation in mental activity t o increase its productivity. Theorist, experimenter, applied mathematician, and programmer work together in the united cycle of computational experiment. We can mention the following advantages of computational experiment a s compared with actual experiment. Firstly, computational experiment is carried out even when actual experiment is impossible. As an example, we can investigate heat processes with thermonuclear parameters (explosion of a nuclear bomb is now the only way to attain these ~ararneters). Secondly, computational experiment sharply decreases the costs of research and saves time. This is due to the fact that computations are multivariant and mathematical models for simulating real conditions are easy t o modify. Development of new products and technologies is associated with hard, expensive, and long procedure. It is this stage where computational means substantially economize on time and money.

10


Data of experimental investigations are used for calibration of mathematical models and accuracy control of approximate solutions to the problem. Following a common practice of numerical investigation, we affect the mathematical model and process the results (this is why we speak of experiment, although computational). It is rare that we control the accuracy of our 'device' by comparing it with a standard. Following a trend of theoretical investigation, in computational experiment we deal with a mathematical model rather than with the object itself. These common features are considered as additional arguments in favour of treating computational experiment in a wide (methodological) sense as an integrating technology of scientific research. Computational experiment should be considered as a new technology of scientific research in future, as a trend, as a logic in the development of research organization. Currently it is generally realized in a narrow sense, along the chain 'customer-applied mathematician'. A closer relationship between theoretical and experimental investigations in a unique technology of scientific research is now a clearly defined trend. It is remarkable that computational experiment is the major link in this methodology. Let us now briefly describe the main branches of science and technology in which computational experiment is employed. We pay much attention to classification of types of experiment with respect to applications and types of mathematical models involved. This self-consistent classification helps a researcher to use adequate mathematical means for investigating mathematical models. Mathematical modelling has been traditionally developed within the context of fundamental sciences, namely in mechanics and physics. These sciences are characterized by the highest level of theoretical investigations (in other words, by the highest level of using mathematics). Modern mathematical methods, including numerical methods, have been successfully introduced in these sciences. For example, there are conventional mathematical models employed in heat physics as well as the collection of basic problems. Therefore, primary attention is paid there to construction of computational algorithms and creation of versatile software. The mathematical arsenal of an engineer or technologist is substantially poorer. Up to now, scientific knowledge in technology is introduced indirectly. New ideas first appear in fundamental sciences, then they are transformed in an applied branch, and only after that are they employed in specific technology projects and developments. This refers first of all to implementation of modern mathematical methods of theoretical investigation, mathematical modelling, and computational experiment. This approach to transforming ideas into a specific scientific and technology solution or a new technology is unwarrantedly long and wasteful.

In modern conditions, mathematical methods should be universally introduced in science and technology. Mathematical modelling of technological processes is expected to he of great benefit and actually leads t o a new qualitative level of technology. Technology and industry are the fertile ground for mathematical modelling. We specify three main types of computational experiment, namely searching, optimizing, and diagnostic experiments. This classification is made, on the one hand, with respect to application fields and, on the other, with respect to the nature of mathematical models employed. In order to explain this, let us consider each type in more detail. The usual approach to studying a new process or phenomenon is to construct a mathematical model and to perform computations while varying parameters of the problem. This type of computational experiment is called a searching experiment. If the mathematical model is based on partial differential equations, a direct problem of mathematical physics is investigated and solved within the cycle of computational experiment. As a result, we describe observable phenomena, predict the behaviour of the object studied under different, probably unreal, conditions. This type of computational experiment is typical of investigations in fundamental sciences. On the other hand, an optimizing computational experiment can be taken as the main one when mathematically simulating technological processes. It includes solving an optimization problem to reduce expenses, lighten construction, etc. The corresponding optimal control or optimization problem is posed for the mathematical problem formulated. A typical example is provided by optimal control problems for equations of mathematical physics, e.g. for boundary control, where boundary conditions are chosen so as to minimize the appropriate functional (the quality functional). In this case, multivariant computations are carried out t o adjust control parameters, which results in an optimal, in a sense, solution. When processing data of actual experiments, a diagnostic computational experiment is employed. Additional indirect measurements are used t o draw a conclusion about internal links of a phenomenon or process. If the structure of a mathematical model for the process studied is specified, an identification problem is posed, e.g. coefficients of the model are determined. Diagnostic computational experiment usually corresponds to an inverse problem of mathematical physics. We often encounter a situation in which there is no mathematical model for the process or phenomenon studied, and it is impossible to construct such a model. This situation is typical, in particular, for processing data of an actual experiment. Then processing is made in a 'black-box' regime and we deal with approximating models.

12


Thus, direct problems are correspondent with computational experiment, optimization problems with optimizing experiment, and inverse problems with diagnostic experiment.

1.4 Numerical Modelling of Heat Transfer Processes Let us use the simplest model of heat transfer, namely the onedimensional heat equation, to specify types of computational experiment and describe main classes of problems of applied modelling. Let us consider a thin cylindrical rod with thermally insulated lattice surface. We assume that one end is thermally insulated, while a specific thermal regime is specified at the other. The temperature T(x, t) is determined by the heat equation

In equation (1) c is specific heat capacity, p is density, and k is thermal conductivity. Additionally, we pose the boundary conditions

and the initial conditions T(x, 0) = To(x).

(4)

We study the influence of nonlinear thermal coefficients c = c(T), p = p (t), and k = k(T) on the process of heat propagation along the rod, i.e. we consider a direct initial boundary value problem (1)-(4). Even in this simple problem one can observe striking phenomena because the mathematical model is nonlinear. In particular, a heat wave propagates at a finite speed. This investigation can he related to a searching computational experiment. Let us consider an optimization problem within the framework of the same equations governing the process. We assume that it is necessary to maintain the temperature at the point x = 1 close to a given function g(t) controlling the temperature at the other end (a boundary control problem). In terms of mathematics, this problem is formulated as follows. Find a control v(t) such that the quality functional

INTRODUCTION

13

where t,, is the time of observation, is minimal. Here T ( x ,t ) is specified as a solution of equation (1) equipped with conditions (2)-(4). Such a minimization problem is typical of an optimization computational experiment. Of course, this is the simplest formulation of the prohlem. The actual situation can be aggravated by more complex equations, quality functionals, which can be nonunique, etc. In such complicated situations one often has to limit oneself to the simplest search methods for solving optimization prohlems. Finally, let us consider one example to illustrate a diagnostic experiment. We pose a problem to find a boundary (thermal) regime a t the left end of the rod (with x = 0) which is inaccessible for direct measurements. In addition, the temperature (i.e. the function g(t)) at the other end (with x = 1 ) is measured with an error. In terms of mathematics, the problem is formulated in the same fashion as the above optimal control problem. We should find a function u(t) for which the functional in (5) is minimal, given additional restrictions (1)-(4). In this case, we have a typical inverse prohlem in an extremal formulation. In the theory of inverse problems this prohlem is considered as an inverse boundary value problem of heat conduction. It is absolutely necessary to demonstrate specific features of optimization (synthesis) problems as opposed to inverse problems. We have seen that both these problems have the same mathematical form (i.e. they are posed identically). However, there are principal differences between them. First of all, note that in both cases we are dealing with ill-posed problems. A problem is said to be well-posed (in the classical sense) if a solution of the problem exists, is unique, and continuously depends on initial data. If any of these conditions is not satisfied, the problem is ill-posed. We do not discuss here the existence of a solution. Instead, let us study the consequences resulting from the breakdown of the other conditions of well-posedness in optimization and inverse problems. Assume a solution t o be nonunique. In optimization problems this is favourable, because we get several (and, probably, many) solutions to the problem. We thus have a good choice and can, in addition, optimize our solution with respect to some other parameters or characteristics. In the case of inverse problems, the situation is completely different. Nonuniqueness of solution to the inverse problem implies that we cannot solve the problem in the formulation chosen. We should refine (complete the formulation), and in the case of processing data of an actual experiment additional data are needed. A similar situation occurs if the solution is unstable with respect to small perturbations of initial data (small perturbations of the function g(t) in our illustrative example). Large perturbations of v(t) lead t o small perturbations for the optimization and inverse problems. This is favourable for optimization problems, in which we should find a control u(t), because we can substantially vary the control and only slightly affect the quality functional. However, in

14


the inverse problems this implies that we can only restore the boundary regime with a large error. Thus, in this aspect, optimization and inverse problems are essentially different. These differences should he emphasized because identical mathematical formulations of these problems can lead to wrong methodological views.

1

Mathematical Models of Physics of Heat

We study temperature field and heat transfer from some parts of a rigid body to its other parts. The temperature field at a time moment t is specified by the temperature distribution in the body, i.e. by a function T = T(x, y,z, t), where (x, y, z ) are Cartesian coordinates. In the simplest case heat flow is directed from the higher-temperature region to a region in which temperature is lower. There are three main types of heat transfer, namely (a) heat conduction, that is heat transfer due to interaction between microparticles of bodies that are in contact; (b) convection, that is heat transfer, caused by movement of the substance. This type of heat transfer is observed in moving media (fluids or gases); (c) mdantaon, that is a process in which energy is transmitted as electromagnetic waves. In numerous applied problems heat is transferred by two or three of these mechanisms concurrently (complex heat transfer). When describing convection it is necessary to take into account heat conduction between different parts of a continuous medium (heat and mass transfer). Radiation can be accompanied by heat conduction, convection, etc. Such complicated processes occur, for instance, in phase transitions or chemical reactions. In the case of heat conduction in solids, the thermal effects can be significant. In this chapter we present reference material and give basic mathematical models of heat transfer. The main heat equation is written in several orthogonal coordinate systems. Much attention is paid to boundary conditions and conditions at the contact between different media (junction conditions).

16


The classical Stefan problem is considered to be basic for modelling melting and solidification. In more general situations more complex models are needed. As an example, we shall describe some special features of modelling binary alloys. Considerable difficulties arise when modelling radiative heat transfer. Because of the large dimension of the problem, it is very difficult to use complete models. Therefore, simpler models become important, e.g. diffusion approximation and radiative heat transfer approximation. In this book, we consider the simplest models of radiation from the surface of a rigid body as basic ones. Describing convection is associated with modelling the motion of a medium. The Navier-Stokes equation in the Boussinesq approximation for this purpose is most frequently used in the theory of heat and mass transfer. Further simplification involves the use of parabolized equations, boundary layer approximation, etc. Deformation of rigid bodies subjected to heat is described by equations of therm~elasticit~. The mathematical model is based on the heat condition equation supplemented by thermoelasticity equations. We should specifically mention simpler models used, for example, for describing the stressed state of thin plates, shells, etc.

2.1 Heat Conduction in Solids

2

. THE HEAT EQUATION

The basic assumption of the heat conduction theory, known as the Fourier law, is that heat flow is proportional to temperature gradient in a homogeneous immobile medium, i.e. q = -kgradT, where k is thennal conductivity. To derive the heat transfer equation, let us rewrite the energy conservation law

where f represents the capacity of internal heat sources, c is specific heat capacity, and p is the density of the medium. If we substitute the expression for the heat flow into ( I ) , we can write the basic differential heat equation

aT cp- = div(k grad T ) at

+f.

(2)

The coefficients and the right-hand side in this equation can vary in space (inhomogeneous medium). In this case we have c = c ( x ,y, z ) , p = p(x, y, 2 ) .

17

MATHEMATICAL MODELS OF PHYSICS OF HEAT

k = k(x,y, z ) , and f = f ( x , y , z ) , and the heat equation becomes a secoudorder linear parabolic equation. The thermal properties of a medium usually depend on temperature, i.e. c = c(T), p = p(T), k = k(T), and f = f(T). Thus, we come to a quasi-linear heat equation. If the thermal properties of the medium do not depend on space variables (homogeneous medium), the heat equation (2) can be written as

Here a = k/(cp) is the t h e m a l daflusivity and A = divgrad is the Laplace operator. Another special case of the heat equation (2) corresponds to steady-state temperature fields. The stationary heat equation looks like (aT/at = 0 in (2))

and is a secoud-order elliptic equation. The heat equation (2) is appropriately modified for modelling heat transfer in a moving medium (in a moving coordinate system). Namely, the partial derivative with respect to time a / a t is replaced by the complete derivative with respect to time (substantive derivative)

d dt

-=

a + v grad, at

where v is the local velocity of the medium. Therefore, the heat equation for a moving medium is written as = div(k grad 2')

+f.

(5)

Here the term with v g r a d T defines temperature change due to convection. The heat equation in a usual (Cartesian) coordinate system (x,y,z) is derived from the above invariant form (equations (2)-(5)) provided that g

-1,

a a a ax' a y ' az

-

Hence, (2) can be rewritten as

aa, aa, diva=-+-+-

ax

ay

aa,

az

18


Other equations are rewritten similarly. For example, to rewrite the Laplace equation (3) we should take into account that the Laplacim operator A in the Cartesian coordinates looks like

2.1.2 CURVILINEAR COORDINATE SYSTEMS

Now let us write the heat equation in the curvilinear orthogonal coordinate systems widely used in computational practice. Let functions x1 = xl(x, y,z), 5 2 = XZ(X, y,z), m d x3 = xs(x, y, t) determine the coordinates of a point (x,y,z) in a curvilinear orthogonal coordinate system (xl, x2, x3) and let this transformation be nondegenerate. The elementary length and volume in a curvilinear coordinate system are given by the formulae ds = (gfdxt +gzdxi + gidxi)1'2 and dv = g1gzg3dxldxzdx3, where g l , g2, and g3 are metric coefficients. Let il, iz, and i3 be the unit vectors of the local basis. Then for an arbitrary scalar ip we get 1 aip. gradip = - -11 g1 8x1

1 a9. 1 aip. +-12 +g2 ax2

-13

g3 ax3

The basic operations of vector analysis are performed over a vector a = (al, az,a3) in the curvilinear coordinate system as follows:

Among repeated operations, let us mention the Laplacian operator

19


In the above (see (6)) we wrote the heat equation in the Cartesian coordinates, for which X I = x, xz = y , x3 = 2, gl = 1, g* = 1, and g3 = 1. For a cylindrical coordinate system we obtain X I = T , xz = (o, x3 = t , g~ = 1, gz = T , and g3 = 1. According to (8) and (9), the heat equation (2) becomes

Similarly, for a spherical coordinate system we get X I = T , xz = 0, x3 = p, g, = 1, gz = T, and g3 = r sin0. Therefore the heat equation is written a s cp-=---

aT at

a

a (a ~ g )a~ + (t& mf g) 1

k 2

TZ a T

+--6--)+f.alp T

1 sin o sip

1 sm 0

(13)

Formulae (10) and (11) can also be used to obtain the corresponding expressions for the curl and Laplacian operators in cylindrical and spherical coordinates. 2.1.3 ANISOTROPIC MEDIA We presented the heat equation for the case in which the medium is isotropic. Thermal characteristics of many solids (e.g. crystals or composite materials) are anisotropic, i.e. their properties are different in different directions. In this case the heat conduction is a second-order tensor rather than a scalar. For the heat conductivity tensor k we have

and this tensor is symmetrical (k,, = kyr, k,, = k,,, etc.). The heat equation preserves its form (2) when (14) is taken into account. In coordinate-wise form we obtain cp-aT =-a (kzc

at

ax

+ -aay

g+

+k (k , , -E

+a ( k,, a~ instead of (6)

aT kzy -

E+ -

"1 "1at

a~ + kzz at

yy

aT -+ky,By

aT k,, - k,, ay

+

")

+f

(15)

b-,,hkq


In the case of an anisotropic medium and constant thermal characteristics, anisotropic heat equation (14) is simplified by taking into account the symmetry of heat conductivity tensor k and becomes

This equation contains mixed derivatives; we can get rid of them by passing t o a new Cartesian coordinate system (x',y', z') in which (16) has the simplest form, namely

The coordinate axes (z',y',zi) are said to be the principal axes of heat conduction, while the coefficients k,, , k,,, and k,, are said to be the principal heat conductivities. 2.1.4 HYPERBOLIC HEAT EQUATION

The Fourier law is based on the assumption that heat propagates in a medium at infinite speed (the gradient and density of heat flow correspond to each other at each moment in time). For high-intensity unsteady heat processes it is necessary to take into account that heat propagates with a finite speed (the inertia) and to add by the Fourier law an extra term

where 7, = const is the relaxation time of heat stresses. In order to derive the heat equation, we substitute (18) into (1) to obtain

Equation (19) is hyperbolic and is said to be the hyperbolic heat equation. 2.1.5 PROBLEMS

1. Write the heat equation in an orthogonal coordinate system for a moving homogeneous medium.

MATHEMATICAL MODELS OF PHYSICS O F HEAT

21

Solution. Consider the heat equation

in an arbitrary orthogonal curvilinear coordinate system. Taking into account (8) and ( l l ) , we obtain

2. Transform the heat equation for a moving homogeneous medium to a self-adjoint form, given that curlv = 0 (irrotational potential motions). Solution. We again consider (20). It is not self-adjoint because of the convective term vgradT. In order to eliminate this term, let us try to represent the solution as

and choose an appropriate function R(x, y, 2 , t). If curl v = 0, any velocity field can be represented as v = grad O, where O is the velocity potential. We take account of (21) and rewrite the heat equation as

Let us select the function R so that Rgrad O = 2agradR, e.g. R = exp(Of(2a)). Then we obtain the desired equation for determining the unknown u

This equation no longer contains terms with first derivatives with respect to space variables.

22


2.2 Closing Relations

2.2.1 INITIAL AND BOUNDARY CONDITIONS The temperature distribution in a medium at different moments in time is determined by a partial differential equation (the heat equation). In order to specify the temperature field T(x, y, z, t ) uniquely, it is necessary to formulate additional (closing) relations because a solution of a partial differential equation is defined to within an arbitrary function. To eliminate this arbitrariness, we formulate some additional relations, i.e. we specify the solution itself, some of its directional derivatives, etc. at some points. The temperature field is always calculated in a certain space domain. For simplicity, we only consider the case in which the computational domain R, where we search for a solution of the heat equation, remains unchanged. For definiteness, we assume that the process of heat transfer is studied > 0. We thus search starting from the time t = 0 till a time t = t,, for a solution of heat equation (2) of Section 2.1 in the cylinder Q = {(x,y,z,t)l ( x , Y , ~E) 0, 0 < t < t,,,), i.e.

aT

cp - = div(kgradT) at

+f,

(x, y, z, t ) E Q.

(1)

This equation involves partial derivatives with respect to both space and time variables. Therefore, additional conditions have to be specified on some i.e. on some sets sets of the spatial domain fl and the time interval (0, t,,,), of the points from cylinder Q. Boundary value problems are usually posed for the heat equation. In this case, additional conditions are specified on the boundary of Q. Conditions on the lateral surface of the cylinder Q correspond to those with respect to space variables (on the boundary of the spatial domain), hence they are called boundary conditions. Conditions on the bottom base of Q are said to be initial conditions. More complicated conditions can be specified as well. For instance, instead of initial conditions for t = 0, we can specify additional conditions on another section of the cylinder Q, i.e. for some t = t * . In other words, the set of points on which additional conditions are given can also lie inside Q. Some of these possibilities are discussed in the following, where we specify the main types of problems for the heat equation. It is generally assumed that the temperature field is specified at the initial moment in time, i.e.

MATHEMATICAL MODELS O F PHYSICS O F HEAT

23

When considering high-intensity processes described by hyperbolic heat equation (19) of Section 2.1, it is necessary to specify two conditions with respect to time. For example, the temperature and the rate a t which it changes with time are given a t the initial moment. This allows us to specify the condition

along with (2). Specifying conditions like (2) in applied modelling requires direct measurements a t a fixed moment in time. Such measurements are not always possible. Hence, other approaches can be used. For example, conditions at the final moment in time may be appropriate for (I), that is we specify the condition

= Trn(x,y,z), ( x , Y , ~E) 0 , (4) T(x,y,z,t,.,) instead of (2). In this case, the temperature field must be restored a t a previous given condition (4). We have thus formulated a moment in time t < t,,,, retrospective problem for the heat equation. General boundary conditions for the heat equation are divided into three kinds. The simplest situation occurs when the temperature field is given on the boundary an (first-kind boundary conditions), i.e. ) r, (5) T ( x , Y , z , ~= ) g ( ~ , ~ , x , t ) , ( x , Y , z , ~E where I' is the lateral surface of 0, r = {(x, y, z , t)I (x, y, z ) E 8 0 , 0 < t <},. First-kind conditions (5) are also called the Dirichlet conditions. Second-kind boundary conditions (the Neumann conditions) correspond to specifying the heat flow on the boundary. This condition for the heat equation (1)in an isotropic medium is written as

where a l a n denotes the external, with respect to the domain 0,normal to the boundary 8 0 . A more complicated situation occurs when we pose second-kind boundary conditions for the heat equation in an anisotropic medium (see equation (15) of Section 2.1). Let cos(n,x), cos(n, y), and cos(n,z) be the direction cosines of the external normal. The flow is specified by the equation

24


which corresponds to differentiation along the normal. The second-kind boundary condition is written as

and generalizes condition (6) (specifying a / a v = k a l a n ) for the case of anisotropic media. A third-kind boundary condition simulates convective heat transfer between the surface of a rigid body and the environment which has the temperature T,.It is usually assumed that the heat flow is proportional to the temperature difference between the surface and environment, consequently, for an isotropic medium we get

where a is the heat transfer coeficient. The appropriate condition in the case of anisotropic media [see (a)] is formulated similarly. The third-kind boundary conditions can be treated as the most general of the above conditions. These conditions can be written as

Then (10) yields second-kind condition (6) as a condition (5) as a + co.

-+

0 and, conversely, first-kind

2.2.2 JUNCTION CONDITIONS

Conditions on the interface between two media with different thermal properties, i.e. junction conditions, are worth considering separately. We first focus on the question as to what junction conditions are natural for the heat equation, i.e. follow from this equation itself, and hence should not be formulated explicitly. In this case, contact conditions are determined only by discontinuities of thermal characteristics observed when passing the interface between the media. Suppose that the plane z = 0 is the interface berween two homogeneous media. We mark thermal characteristics of the medium that fills the half-space z > 0 by plus and those for the second medium (with z < 0) by minus. Let us consider the heat equation in the form


25

in which the coefficients are discontinuous at x = 0, namely

Taking into account the discontinuity of the coefficients of ( l l ) , it is logical to expect that the solution of the equation itself (temperature T ) is continuous, while the first derivatives of the solution are discontinuous. We thus write the continuity condition for temperature at the interface between two media

where the brackets denote the jump at the interface. In our case [TI = T(x 0, y, z ) - T(x - 0, y, 2). Now we only have to formulate the junction conditions for the first derivatives of temperature. In order to derive realistic (natural) junction conditions for the heat equation (11) with discontinuous coefficients (12) we can separate a bounded region 6s on the interface and integrate the original heat equation (11) over the domainof width ZE, namely 0, = {(x, y,z)1 - E < x < E , (y,z) E 65). We take account of the continuity of temperature (condition (13)) and finiteness of discontinuities as E -+ 0 and obtain

+

Since the element 6S is arbitrary, we find the junction condition in the form

The junction condition (14) reflects the continuity of heat flow. Natural junction conditions are written in the same fashion (see (13) and (14)) on an arbitrary interface S between two media, which is an internal boundary of the domain a. They reflect the continuity of temperature and heat flow and are written as

The junction conditions (15) and (16) are the conditions of ideal contact. We emphasize once again that (15) and (16) are natural conditions for (11) with discontinuous coefficients (12). This implies that they are embedded into the

26


equation itself so that there is no need to formulate them separately every time. In the case of an anisotropic medium, the condition of continuity of heat flow in the form (x,y,z) E S,

(17)

where the notation of (7) is used, is employed instead of (16). If the ideal contact conditions (continuity of temperature and heat flow) are not satisfied on the interface between two media, then some junction conditions needed for closure of the heat equation in each medium should be formulated specifically. These junction conditions reflect special features of thermal processes on the contact boundary and peculiarities of behaviour of the solution to the problem when passing the interface and may not be included in the heat equation. Let us consider several options. Suppose there exists a surface heat source with capacity qs on the interface S . Then the temperature continuity assumption holds, i.e. (15) is satisfied, while the heat flow has a discontinuity. Instead of homogeneous junction condition (16) we should write a nonhomogeneous condition

The conjunction conditions (15) can be included in the heat equation written in the whole computational domain without separating the contact boundary S. The surface heat source is taken into account by introducing an additional term in ( l l ) , namely

In equation (19) ds is a surface delta-function specified such that

J

n

asP(x, y, z) dxdy dz =

J

P(x, y, t)ds

s

Other types of junction conditions are also used in computational heat physics, e.g. the condition of concentrated heat capacity, in which temperature is continuous, while the discontinuity of the heat flow is determined by the

where cs is the lumped heat capacity of the contact. The junction conditions (15) and (20), in which the surface delta-function is used, are included in the heat equation similar to (19).

MATHEMATICAL MODELS O F PHYSICS OF HEAT

27

Conditions of nonideal contact deserve much attention in applied research. They are realized, for example, if two rough rigid bodies are in contact with each other but not tightly enough together. The heat flow is continuous for a nonideal contact, which results from the law of conservation of energy. We thus have one junction condition, namely condition (16). Temperature has a discontinuity when passing through the boundary of the nonideal contact, with the height of the step being proportional t o the heat flow, i.e.

where the coefficient of contact heat transfer a is associated with the contact conditions. 2.2.3 DIRECT AND INVERSE PROBLEMS

FOR THE HEAT EQUATION Let us specify basic classes of problems for the heat equation. Firstly, we consider boundary value problems, in which the corresponding boundary and initial conditions are given. For instance, we can pose a first-kind boundary value problem when the equation is supplemented with initial condition (2) and first-kind boundary condition (5). Similarly, we can pose a second-kind problem, in which (6) is used instead of (5), or a third-order problem, in which condition (9) is employed. We can specially separate the case in which boundary couditions of one kind are given on a part of the boundary aO1 and boundary conditions of another kind on the remaining part a02 (80, = a0 \ aOl). For example, boundary couditions for (1) can be

where r, = {(x, y, z , t) I(x, y, z) E a%, 0 < t < t,,,), a = 1,2. We thus have mixed boundary conditions and a mixed boundary value problem. The class of boundary value problems that we have just described is characterized by additional conditions that are set on the boundary a R (the lateral surface of r) and for t = 0 (initial conditions). This is an important class of problems for the heat equation, which is thoroughly studied in the theory of partial differential equations. These boundary value problems belong to the class of Hadamard well-posed problems. Recall that a problem for a partial differential equation is said to he a well-posed problem if the following three conditions are satisfied: (1) the problem has a solution;

28


(2) this solution is unique; (3) the solution continuously depends on coefficients of the equation and additional (boundary and initial) conditions. If at least one of these conditions is not satisfied, the problem is ill-posed. Condition (3), which says that the solution should be stable with respect t o small perturbations in the initial parameters, is most frequently violated. Considering boundary value problems from the viewpoint of cause-andeffect relations allows us to classify boundary value problems a s direct problems for the heat equation. Breakdown of cause-and-effect relations in inverse problems often manifests itself in the ill-posedness of an inverse problem. Inverse problems for the heat equation involve incomplete boundary or initial conditions and/or an incomplete equation (e.g. the coefficients, the right-hand side, or the computational domain are not determined). Instead, some additional information is given about the solution, the equation, the domain, etc. It should be kept in mind that there are many quite different ways to give this additional information. Below we discuss some possibilities. In the simplest inverse problem of heat transfer conditions (4) for the time t,,, are specified instead of initial conditions (2) for t = 0 (a retrospective problem or a problem with inverse time). Boundary value problems for the heat equation in which boundary conditions are incomplete are important for practice. For instance, suppose that two conditions are given on a part 8% of the boundary and no conditions are given on the remaining part 802, i.e.

an,

This situation occurs when the part is for some reason inaccessible for direct measurements of temperature and heat flow.

I 2.2.4 OPTIMIZATION PROBLEMS

Additional information in inverse heat transfer problems can he merely information about temperature and/or heat flow at internal points of the computational domain R and/or on its boundary, like in direct heat transfer problems. We separate a class of problems in which unknowns are determined by searching for a minimum of one or several functionals. These problems are actually problems of conditional minimization, in which the restrictions are such that the minimum is searched for among the solutions of a boundary value problem for the heat equation. We thus deal with optimization problems for the heat equation. For example, the inverse problem in which it is necessary to restore the boundary regime in (I), (2), (22), and (23) can also be formulated as an optimization problem. We denote


29

where u is the sought-for function. We denote the solution of heat equation (I), supplied with initial condition (2) and boundary conditions (22) and (24), by T = T(u;x, y, 2, t). We find the boundary regime on rz (the function w) from the condition (see (23)) J(w) = min J(u),

"

(25)

where

Optimal control problems for the heat equation are formulated in the same way. Minimization problem (25), (26) on the set of restrictions (I), (2), (22), (24) is interpreted in this case as follows. Find a boundary thermal regime (24) (optimal regime) that would provide necessary quality (functional (26)). We thus separated three main classes of problems for the heat equation. The first includes standard boundary value problems, i.e. direct problems of heat transfer. The second class of applied problems includes inverse problems of heat transfer. The third class includes optimization and optimal control problems. 2.2.5 PROBLEMS

1. S u p p o s e we have a n isotropic medium, in which t h e r e is a homogeneous inclusion with t h e b o u n d a r y S. Let t h e h e a t conductivity of t h e inclusion b e m u c h greater t h a n t h a t of t h e medium. T h e inclusion supplies t h e m e d i u m w i t h a n a m o u n t of h e a t Qs(t). Consider t h e heat transfer in t h e m e d i u m a n d pose necessary b o u n d a r y conditions o n t h e b o u n d a r y of t h e inclusion. Solution. Since the heat conductivity of the inclusion is much greater than that of the medium, we may assume that the temperature on the boundary S is constant, i.e.

This temperature is defined by an additional integral relation. By the law of conservation of energy, we have

Conditions (27) and (28) are desired nonlocal conditions on the boundary S of the inclusion.

30


We can take into account the heat capacity of the inclusion to refine condition (28)

where M is the mass of the inclusion and c is its specific heat. 2. Formulate t h e conditions of ideal contact for t h e h e a t equation i n moving media. Solution. We consider heat equation (5) of Section 2.1, which, for convenience, can be rewritten for each distinct medium as

Reasoning as in the above, we arrive at the following junction conditions:

where u, is the normal velocity component.

2.3 Phase Transitions

2.3.1 THE CLASSICAL STEFAN PROBLEM

We consider solid-liquid phase transitions, the examples of these being solidification and melting in metallurgy. The mathematical models of these processes contain moving surfaces of phase transition that are unknown beforehand. That is, we deal with problems with free (unknown) boundaries. The main assumption that underlies the modelling of solid-liquid phase transitions is that the phase transition occurs at a given, fixed temperature of the phase transition T*. Let us denote the interface between two phases by S(t). This interface divides the computational domain R into two subdomains: the domain W ( t ) = {(x, y, z) E R, T ( x , y,z,t ) > T ' }filled with the liquid, in which the temperature is greater than the phase transition temperature, and the domain W ( t ) = {(x,y,z)l (x, y,z) E 0, T(x, y , z , t ) < T'} filled with the solid phase (see Fig. 2.1). We use similar notation for thermal characteristics of each phase.


31

Let us write the corresponding heat equation. In the solid phase we have

where Q- = { ( x ,y, z, t)l ( x ,y, z ) E R-, 0 < t < t,.,). In the liquid phase we additionally take into account convective heat transfer to get

.

.

We are interested in boundary conditions on the interface S of the phase transition. Firstly, the continuity of temperature is assumed on this boundary of contact between two media, i.e.

(X,Y,~)ES.

(3)

The phase transition is accompanied by a release/absorption of a certain amount of heat. Therefore the heat flow is discontinuous on the phase transition boundary and is defined by the equation

Here L is the phase transition enthalpy and V. is the normal velocity of the phase transition boundary. As mentioned above, we assume that the phase transition occurs at a constant temperature T'.Hence the interface is determined a t each moment in time as follows: S = S ( t ) = { ( x ,y , t) E 0,T ( x ,y, z , t ) = T ' ) or, in other words, the first-kind conditions

hold on the interface

Fig. 2.1.

32


Conditions (1)-(3) are called the Stefan conditions and the corresponding problem for equations (1) and (2) is referred to as the Stefan problem. In this problem, processes in both phases are studied; and the problem is also called the two-phase Stefan problem. The extreme case of this problem is when the temperature field in one of the phases is known (the temperature equals the phase transition temperature). We thus should find the temperature field only for one phase, i.e. we deal with the monophase Stefan problem. In this case, the unknown phase transition boundary S is external rather than internal. For example, assume that the domain 0- is filled with the solid phase at temperature T*. Then in order to find the temperature in the liquid phase, we use equation (2) in a variable domain R+(t) supplied with the following conditions on S:

I

I

Conditions like (6) and (7) are typical for the monophase Stefan problem. In some cases the Stefan approximation is inappropriate. Various improvements of mathematical models for phase transitions are actively discussed in the literature. Without going into details, we mention the main point. The Stefan condition in (5) is based on the assumption that the temperature instantly levels to the phase transition temperature, which actually corresponds to the assumption that the velocity of the phase transition is unbounded. This assumption is not valid in some cases, e.g. in the problem of describing high-intensity thermal processes (the hyperbolic heat equation). To avoid this, we can use a more general third-kind condition instead of the first-kind condition in (5). To derive this condition we should describe the kinetics of the phase transition. For example, the condition

aTt an

k+ -+ a* (T" - T*)= 0,

(2,y, z ) E

S(t),

(8)

which bounds heat flows toward the phase transition boundary, can be used instead of (6) in the monophase problem. Condition (8) is used along with ( 7 ) , which expresses the law of conservation of energy for any motion of the interface. 2.9.8 GENERALIZED FORMULATION OF THE STEFAN PROBLEM

The formulation of the Stefan problem can be generalized so that conditions (3)-(5) are included in the heat equation itself. This is very important from the viewpoint of constructing efficient computational algorithms. We have already discussed how to include nonhomogeneous junction conditions on a given

33


interface in the equation itself (see (19) in Section 2.2). The Stefan problem is complicated because the unknown phase interface S itself is unknown. Let us explain how we can pass from equations (1) and (2) equipped with conditions (3)-(5) to a single heat equation. According to (18) and (19) in Section 2.2, we can rewrite (1) and (2) as the single equation cp

(g+

v grad T

1

= div(k grad T ) - 6sLV.

+ f,

( x y z t) E Q.

(9)

Near the boundary of the phase transition we introduce local orthogonal coordinates (x', y', 2') with unit metric coefficients. The surface delta-function hs in these coordinates becomes 6s = 6(x1 - xb), where the equation z' = xb defines the boundary S . Similarly, the velocity of the free surface is V, = dx'ldt. The Stefan conditions in (5) correspond to T = T(x', t) and T(xb, t) = T* in the new coordinates. Taking this into account, we obtain 6sVn = 6 (x' - xb)

dx' = 6 ( T - T') dt

dT dt

-

Substitution of (10) into (9) yields the desired equation (cp

+ L6(T - T*))

= div(kgradT) + f , ( x , ~ , z , t E)

Q.

(11)

Equation (10) is remarkable because it does not include the unknown boundary of the phase transition explicitly. Taking into account the heat of phase transition amounts to specifying the effective heat capacity ceR = c + p-'L6(T - T*). Effective numerical procedures for approximate solution of the Stefan problem are based on quasi-linear heat equation (11). 3.3.3 THE QUASI-STATIONARY STEFAN PROBLEM

We single out the problem with phase transitions. Consider a stationary temperature field in a moving medium. Let the medium move a t a constant velocity vo along the vector s. Then v = vo = uos in equation (11). The stationary temperature field T = T(x, y, z) with the phase transition taken into account is defined by the equation aT vo(cp L6(T - T*)) - = div(k gradT)

+

as

+ f,

(x, y, z) E 0.

(12)

Thus, it is typical of the quasi-stationary Stefan problem that the temperature field in each subdomain (in R+ and W ) is described by the second-order elliptic equation (12) and that the interface is immovable but unknown.

34


2.3.4 PHASE TRANSITIONS I N MULTICOMPONENT SYSTEMS The Stefan problem above refers to the case in which a pure substance undergoes a phase transition. However, the solidification/melting of impure substances and multicomponent systems is very important in practice. Let us dwell on the case of an alloy of two substances. If we consider hardening, one of the substances changes state at one temperature and the other at another, different temperature (i.e. we have two phase transition temperatures). Therefore there exists a temperature interval in which one part of the substance is in liquid state while the other is in solid state (two-phase zone). Let the subscript 0 denote the first substance and the subscript 1 the second and let C be the concentration of the second substance. The processes that occur in this situation are schematically shown in the diagram of phase state in Fig. 2.2. For C = 0 and C = 1 (only one substance is present) the phase transition occurs at a given phase transition temperature (To* and Ti, respectively). For an intermediate concentration C', solidification starts at a certain temperature Zi,(Ct) (liquidus temperature) and is completely finished at a temperature TSoI(C1)(solidus temperature). Thus, the simplest model for a binary alloy can be based on specifying this characteristic temperature interval. The computational domain now consists of three subdomains (Fig. 2.3), namely Q+(t), filled with the liquid phase, R- filled with the solid phase, and Ri(t), where solidification occurs and two phases coexist, i.e. R*(t) = {(x, y,z)l (x, y,z) E R, T,,I < T < Zi,}. Let *(T) denote the part of the solid phase at temperature T. Then the heat equation becomes

0

C'

Fig. 2.2.

I

C

Fig. 2.3.

35


Evidently, equation (13) yields equation (11) for a pure substance, where we have

and T' = Tli, = T.,,. Equation (13) is the basis for the simplest quasiequilibrium model the two-phase zone. In special cases, the width of the two-phase zone may turn out to be much smaller than a typical linear dimension in the problem. Then we can restrict our analysis to the simplest assumption of a given temperature of the phase transition for every concentration (T* = Tli, = T.,I):

When modelling alloys, it is often necessary to consider disproportionation of an impurity due to phase transitions (especially in the liquid phase). Without going into details we will consider a representative problem of a narrow two-phase zone. The temperature field is described by equations (1) and (2) closed by conditions (3) and (4) on the phase transition boundary S, which is determined by (14). The problem of determining the concentration is formulated similarly. In particular, the diffusion equations are in this case

a'- - - div (D-grad C - ) , at

act + div (vC') at

= div (D' grad C')

,

(x, y, z, t) E Q-,

(15)

(x, y, z, t) E Q',

(16)

where D t and D- are diffusivities in liquid and solid phases, respectively. Let us formulate the simplest conditions on the phase transition boundary for concentration. The concentration of impurity is discontinuous on the phase transition boundary, and it is usually assumed that approximately we have C- = k c f ,

(x, y, z) E S,

(17)

where k is a coefficient that describes the distribution of impurity. For definiteness we assume that condition (14), which defines the phase transition boundary, is written for C + . The law of conservation of mass can be used to write the junction condition for flows ( v = 0 on S in equation (16))

36


We take into account (17) and rewrite this condition as follows:

The junction conditions (17) and (18) complete diffusion equations (15) and (16). Of course, we can set different conditions on the boundary of the computational domain, but these will not he discussed here. 2.3.5 PROBLEMS

1. Formulate conditions for t h e velocity in t h e liquid phase o n t h e phase transition b o u n d a r y taking into account t h e density j u m p . Solution. If the density is continuous on the phase transition boundary, we obtain a natural condition v = 0, (x, y,z) E S , for the velocity in the liquid phase. Changes in density (the density typically decreases) when passing from the solid into the liquid phase lead to a nonzero velocity along the normal to the phase transition boundary (expansion of substance). At the same time, the tangent component remains zero, i.e.

The law of conservation of mass gives the following relationship between u, and the velocity V. of the phase transition boundary:

This implies u,,

= v n = (I

-

5)vn,

(x,g,z) E S.

(20)

Conditions (19) and (20) completely define the velocity of the liquid phase on the phase transition boundary. 2. I n t h e quasi-stationary Stefan problem for a homogeneous medium, transform t h e heat equation s o as t o g e t rid of convective t e r m s ( t e r m s including directional derivatives along 1). Solution. The stationary heat equation

holds in separate subdomains 0+ and 0-. Let us consider the expression a R

- div(R grad T) = aAT

+ a (grad R, grad T)

37


We compare it with (21) and choose a function R(x, y,z) so that a gradR = vo. R Equation (22) holds for any constant vector vo if

where r = xi + yj desired equation

+ zk.

div(RgradT)

(22)

Substitution of (22) and (23) into (21) yields the

f = 0, +R acp

(x, y, z)

E

RC U 0-.

Thus, we have obtained a self-adjoint equation in each separate subdomain.

2.4 C o n v e c t i v e Heat T r a n s f e r

2.4.1 THE NAVIER-STOKES EQUATIONS

Convective heat transfer is due to motions of the medium itself. It is very important when considering heat transfer in a fluid (melt). Mathematical models of heat transfer in this case should be supplemented by models of motion of the medium itself and by equations of continuum mechanics. Let us present typical models of motion at a heat-conducting medium which are widely employed in practical research. We consider heat transfer in a fluid (a gas). The law of conservation of mass yields the continuity equation ""

In many cases (if medium motions are slow as compared with the speed of sound) the density of the fluid can be assumed to be constant for the whole medium a t any moment in time (the model of an incompressible fluid). In this case equation (1) becomes (2) divv = 0. We use the notation v = (u,v, w) for the velocity components. Equations of motion for a viscous medium (the Nauier-Stokes equations) are written as

38


In (3), f = (f,, fy, f z ) is the vector of mass forces and a, are components of the symmetric tensor of viscous tensions. For an incompressible Newtonian fluid we get

where q is the viscosity. Substitution of (4) into (3) yields the desired equations of motion for an incompressible viscous fluid. The Navier-Stokes equations for a homogeneous medium, in which all characteristics are constant, reduce to

If we take into account compressibility of the medium, we obtain an extra term in equation ( 5 ) , namely

where ( is the second viscosity. It is usually simply assumed that E = 0. The continuity equation and the Navier-Stokes equations describe motion of the medium. We only have to write the heat equation in the fluid. This equation has the simplest form for an incompressible medium cPp

(g+

+ n@.

v grad T = div(k grad T)

(7)

Here c, is the specific heat at constant pressure and q@ is the term that describes the dissipation of energy to beat due to viscosity or viscous friction. The quantity @ can be expressed as

The allowance for the compressibility of the medium refines the dissipative function @ and brings about an additional term to heat equation (7). The system of equations (2), (5), and (7) is basic for modelling convective motions of viscous homogeneous media. The unknowns are temperature T,

39


velocity v , and pressure p. The density is assumed to be given. This system is closed by the appropriate boundary and initial conditions. For example, the homogeneous conditions

can be posed on the boundary of the computational domain, i.e. the uwpenetration ( v n = 0) and no-slip ( v x n = 0) conditions of rigid wall are satisfied. The system of (2), (5), and (7) is peculiar in the sense that there is no equation for pressure while there are, actually, two equations (vector equation (5) and scalar equation (2)) for the velocity. The same partially refers to boundary and initial conditions (see e.g. (9)). We can eliminate pressure from the equation of motion (5). For this purpose we use the relation vgradv =

1

gradvZ - v x curlv.

(10)

We substitute (10) into (5) assuming that p and 7 are constant and apply the curl operation to both sides to obtain

a

-- curl(v x curlv) = vA curlv

at

+ curlf,

(11)

where u = q l p is the kinematic viscosity. We thus arrive at the system of equations (2) and (11) to determine the velocity. 8.4.2 TWO-DIMENSIONAL FLOWS Let us consider in more detail the case in which thermal and hydrodynamic characteristics of a moving fluid do not depend on one of the coordinates, e.g. on z . We thus come to a two-dimensional (in space) problem of heat and mass transfer. Such planar flows are usually simulated using 'vorticity' and 'stream function' coordinates rather than physical (natural) coordinates, that is velocity and pressure. The continuity equation for an incompressible fluid (2) becomes

Equation (12) implies that the velocity components u and u can be expressed in terms of the stream function li, (the lines li, = const are said to be streamlines), namely

It is essential that continuity equation (12) holds in this representation automatically.

40


The vorticity for two-dimensional flows is introduced as follows:

We take into account (13) to get

We take equations of motion in the form (13) and rewrite the convective term in (15) in somewhat different form:

We take into account (13) and rewrite the convective term in (15) in a somewhat different form:

in terms of the Jacobian of the mapping (w,$) --, (x, y). The system of equations (14) and (15) is the desired system that governs motions of a viscous incompressible fluid in the vortex and stream function variables. We substitute (14) into (15) to eliminate the vorticity and obtain a single quasi-parabolic fourth-order equation for the stream function (the $-equation)

The heat equation is transformed similarly

where the dissipative function Q is

Passing to the vortex-stream function variables simplifies the original equations. In particular, instead of three equations (the continuity equation and two equations for the velocity components) we get two standard equations of mathematical physics, namely elliptic equation (14) for the stream function

41


and parabolic equation (15) for the vorticity. However, it is difficult to pose boundary conditions in these new variables. For example, the uonleakage and attachment conditions (see (9)) yield two conditions for the stream function

aIj,-- o, -

Ij, = const,

an

an.

( 5 , ~E )

(17)

We thus have two conditions for equation (16) but no condition for the vorticity. Besides, the constant in (17) is mast likely to be unknown (e.g. for flows in multiply connected domains) and to be determined from additional conditions (see Problem 2). 2.4.3 FREE CONVECTION

Much attention in theoretical and applied investigations of heat and mass transfer is paid to free convection, i.e. the motion of a fluid in the gravity field, accounted for by an inhomogeneity of the temperature field. The fluid is usually assumed to be incompressible and changes in density are only taken into account when specifying the gravity forces. The temperature and density are reckoned from steady-state equilibrium values To = const and po = const. Therefore T = To T' and p = po + p ' , where T' and p' are small. The behaviour of density versus temperature is described by the following equation for p': p' = -poPT', where P is the coeficient of thermal expansion. Navier-Stokes equations (5) are linearized provided that f = pg, where g is the vector of free fall acceleration. In a linear approximation (the Boussinesq approximation) we get

+

The equation for temperature is taken as

The system of equations (2), (IS), and (19) is a base for describing freeconvection ruotions of a fluid when equipped with appropriate boundary and initial conditions.

2.4.4 OTHER MODELS Steady flow of a homogeneous fluid in a tube of arbitrary cross-section is an important applied model of heat and mass transfer. We assume that a constant temperature regime is maintained on the boundary. Let the axis of the

42


tube be directed along the z-axis. The velocity of the fluid is directed along the z-axis and is a function of the coordinates x and y. Similarly, T = T(z, y). Under these assumptions the continuity equation (2) is an identity, while (5) yields

Equations (20) and (21) imply that pressure is constant on a section of the tube, and since u = u(x, y), it follows from (20) that d p l d z = const = 6 p / l , where 6 p is the known pressure drop across the tube and 1 is the length of the tube. The Poisson's equation (20) for longitudinal velocity is considered on a section of the tube 0 and is augmented by the attachment boundary condition = 0, (z, Y) E an. (23) Similarly, using (7) and (8) we derive the equation for the temperature in a homogeneous fluid

The boundary condition is

T = To = const,

(x,y) E an.

(25)

Equation (24) together with boundary condition (25) specifies the temperature field, provided that the velocity field is already calculated from equation (20) equipped with condition (23). In both cases we deal with the classical Dirichlet problem for the Poisson equation. Many simplified models are based on the assumption that changes in velocity and temperature in the longitudinal direction are small, e.g. a thin layer model and a model of film flows. The most widespread model of this kind is the boundary layer model. Consider, a s an example, a flow of a viscous heat-conducting incompressible fluid over a thin plate (Fig. 2.4). We direct the x-axis along the plate and the y-axis across the plate. There exists a thin layer near the boundary in which the flow undergoes substantial changes (the fluid slows down). This thin region


X

Fig. 2.4.

Fig. 2.5.

adjacent to the boundary of the body is called a boundary layer. Similar changes are observed in the flow at the entrance of a thin slot (Fig. 2.5). We assume the flow to be steady and planar (two-dimensional) and write the boundary layer equations for the flow over the plate as follows:

Equations (26) and (27) are supplemented by the corresponding boundary conditions. It follows (see equation (26)) that the boundary layer approximation is essentially based on neglecting second derivatives in the longitudinal direction. A heat boundary layer is considered similarly. We assume that the incoming flow has a certain constant temperature. Then the stationary heat equation in this moving homogeneous medium becomes

where a is the thermal diffusivity. Reduced Navier-Stokes equations, similar to boundary layer equations (26)-(28), underlie many applied investigations. In particular, many classes of free-convection flows can be studied within the boundary layer approximation.

44

COMPUTATIONAL HEAT T R A N S F E R

1 . Derive t h e equation for pressure for modelling planar flows of a n incompressible fluid in t h e s t r e a m function a n d vorticity variables. Solution. We apply the operator div to the equation of motion (5), take into account the incompressibility constraint (2) to obtain the Poisson equation for pressure. In the planar case, equations of motion are

We differentiate (29) with respect to x and (30) with respect to y and take into account (12) to derive

Substitution of (13) into (31) yields

2. Formulate conditions o n a rigid boundary t o uniquely specify t h e s t r e a m function w h e n modelling planar flows in a multiply connected domain. Solution. Suppose that a doubly connected domain Cl has an internal boundary y and an external boundary r. The stream function is constant on both parts of the boundary (see (17)). The velocity components are specified by the first derivatives of the stream function in (13). We thus can determine the stream function, neglecting a constant. We put

(x, Y) E r, (x, y) E y.

11, = 0, $ = const,

(33) (34)

To find an unknown constant in (34), let us use the following relation on . the boundary y -= E 7, ~

ap

atas

as

aw

T%'

where is the longitudinal derivative. Equation (35) follows from equations of motion (29) and (30) and the attachment and nonleakage conditions on the boundary y.


45

Pressure should be uniquely defined on y, therefore (35) implies the condition

which completes boundary conditions (33) and (34). An additional condition like (36) can also be formulated on the external boundary r. Note that condition (35) as well as the similar condition

can be considered as boundary conditions for determining pressure (see equation (32)) provided that the velocity field and, hence, the vorticity w , is given.

2.5 Thermal Radiation of Solids

2.5.1 MAIN POINTS OF THE THEORY OF RADIATIVE HEAT

EXCHANGE Heated bodies radiate energy into the environment in the form of electromagnetic waves. The quanta of energy propagate at the speed c and are characterized by the wavelength X or frequency v, where c = Xu. The energy carried by a quantum is hv, where h is the Planck constant. Let E denote the integral density for the flux of energy of surface radiation passing through a unit surface area. This quantity for a unit frequency interval is called the spectral (monochromatic) density and denoted by E,,

Radiation incident on a body is partially reflected, partially absorbed, and partially passes through the body. When radiation is absorbed completely, we deal with an absolutely black body. Mathematical models of radiative heat transfer are complicated and difficult for numerical modelling mainly because multidimensional transport equations need t o be used. In some cases we can restrict ourselves to simpler models, e.g. a multigroup diffusion approximation and approximations of radiative heat transfer and an optically thin layer. We mainly deal with heat processes in solids, which are opaque for heat rays unless very thin. Therefore we can assume that only the surfaces of solids are

46


involved in heat exchange processes. This substantially simplifies models of radiative heat transfer. The spectrum of black-body radiation with temperature T is determined by the Planck law 1 hv3 E,, = 2 ~ c2 exp(mv/T) - 1' where m is a constant. According to (1) and (2) the integral density of the energy flux is

E=

7

E, dv = uT4.

(3)

O

Equation (3) expresses Stefan-Boltzmann law and u is a constant known as the Stefan-Boltzmann constant. Of course, bodies never absorb radiation completely, and a coefficient of 'blackness' E (E < 1) is introduced in the Planck and Stefan-Boltzmann laws to describe the fraction of radiation that is absorbed by a hody. For example, the equation E =E U T ~ (4) is used for the flux density instead of (3). The coefficient of blackness is usually taken to be a constant, although in a more general case, it may vary with temperature, i.e. E = E(T). 2.5.2 BOUNDARY VALUE PROBLEMS O F HEAT EXCHANGE TAKING ACCOUNT O F RADIATION

The simplest description of radiative heat exchange, based on the StefanBoltzmann law in the form (3) (or (4)), can be used to pose conjugate problems in which heat is transferred due to heat conduction and radiation concurrently. We use the standard heat equation inside the rigid hody and refine the boundary conditions so as to take into account radiation and absorption. Let us first consider an isolated single rigid convex body R with the boundary Xl.Inside the body heat is transferred is due to thermal conduction; therefore (see (2.1)) aT cp= div(kgradT) f , (x,y , z ) E R. (5) at We take into account only the radiation of the rigid hody itself and convective heat exchange with the environment and formulate the third-kind boundary conditions on the boundary a R

+

aT kan

+ a ( T - T,) + E U T ~= 0,

(x,y,Z) E dR.

(6)


47

Boundary condition (6) has to be refined if radiation of the environment, external radiation, etc. are present. A refinement is also needed for a nonconvex boundary a n , when it is necessary to take into account absorption from some parts of the boundary (self-irradiation). We see that mathematical models of heat transfer in rigid bodies with allowance for radiation are not much more complicated than those in which heat is transferred due to thermal conduction solely. We should only mention that boundary condition (6) is essentially nonlinear. With the Stefan-Boltzmann law taken into account, the boundary condition in the approximation of an absolutely black body and an environment is written as follows:

This condition can be rewritten as an ordinary condition of convective heat transfer aT k-+a,(T-T,)=O, (x,y,z)EaO (7) an but thermal conductivity is then nonlinear, namely

+

a , = a + O(T- T,) (T2 T:) .

(8)

The boundary value problem for (5) with conditions (7) and (8) is closest to standard heat transfer problems discussed in Section 2.2. 2.5.3 HEAT EXCHANGE BETWEEN BODIES We have already discussed how to describe heat transfer for an isolated body taking into account its own radiation. In the theory of radiative heat transfer, much attention is paid to the case in which we have many bodies separated by a medium which is transparent for radiation. When formulating boundary conditions in this case, it is essential to take into account not only internal radiation but also radiation from the other bodies. This situation is similar to the case of a nonconvex rigid body 0, for which the self-absorption of internal radiation has to be taken into account. For example, assume that the bodies are absolutely black and the radiation is isotropic, i.e. its intensity does not depend on the direction. The StefanBoltzmann law (3) gives an expression for the density of flux of radiation from the surface of a rigid body along all the directions in a half-space. The flux of radiation from a body in some direction is proportional t o the flux in the normal direction and to the cosine of the angle between the normal to the surface and this direction (the Lambert law).

48


Fig. 2.6. Let n(M) denote the normal to the surface aR at a point M and r(M, P ) denote the distance between the points M and P on the surface (Fig. 2.6). The resulting radiation flux is comprised of the flux of the proper radiation and the absorbed flux. Therefore, by the law of conservation of energy, we write

where cos(n(M),r) is the cosine of the angle between the normal and the line segment that joins the points M and P . The integration in (9) is over aW = aW(M), i.e. over a part of the boundary 0 that can be seen from the point M. We have thus obtained equation (9) to determine the density of flux of radiation. The temperature field inside the bodies is defined by heat eqnation (5). The boundary condition becomes

The model of ( 5 ) , (9), and (10) is simplified if we consider rigid bodies with isothermal surfaces. In this case, the radiation flux on the surface of each distinct body is constant. Consequently, (9) yields algebraic relations that determine the fluxes. 2.5.4 PROBLEMS

1. Consider two limiting cases for t h e Planck law, namely w h e n mu << 1 a n d mu >> 1. Solution. In the first case, in which mu << 1, we use the expansion

The Rayleigh-Jeans law says


49

In the case of mu >> 1 we neglect the unity in the denominator of the Planck formula to obtain

This formula is known as the Wien displacement law. 2. Formulate junction conditions for a nonideal thermal contact taking into account radiation into a gap. Solution. We consider the gap to be a transparent medium. The law of conservation of energy implies the continuity of heat fluxes (see (6) in Section 2.2), expressed as

The contact heat exchange condition remains the same as usual (see (21) in Section 2.2):

If the bodies in contact are not absolutely black, the conditions are written similarly (see the Stefan-Boltzmann law (4)).

2.6 Thermoelasticity

2.6.1 BASIC EQUATIONS OF THERMOELASTICITY Being subjected to heat, rigid bodies extend. This effect of linear expansion of rigid bodies is described within the framework of linear elasticity theory. We present the basic equations of thermoelasticity, which include an equation for displacements in rigid body and equations for temperature distribution. Let u = (u,v,w) denote the displacement of particles in a rigid body with respect to the equilibrium position. Equations of motion for the rigid body are

50


The components of the tensor of elastic tensions are written as

where X and p are Lame' coefficients which describe elastic properties of the medium. The terms involving temperature govern motions of the elastic medium due to thermal effects. In addition, we have y = (3X + 2p)a. Here cu is the coefficient of linear expansion, responsible for temperature effects. In equation (I), as usual, p is the density of the medium and f is the vector of volume forces, known beforehand. We assume the medium to be homogeneous, substitute (2) into (I), and obtain the Lam6 equations

The system of hyperbolic second-order equations (3) is supplemented by the appropriate boundary and initial conditions. For instance, it is logical to specify the initial displacement and the velocity:

To set the simplest boundary condition for the Lam6 system of equations would be to specify displacements on the boundary of the computational domain, e.g. (6) u(x,y,z,t)=o, (x,y,z)€a~. Thus, given a temperature field in the medium, the stressed state is determined by the boundary value problem like that in (3)-(6). A heat equation for a rigid homogeneous body that takes into account compressibility is somewhat different from usual and looks like (neglecting internal sources of heat)


51

The second term in the left-hand side of (7) is small for rigid bodies and is often neglected. The heat equation (7) with appropriate boundary conditions can be used to determine the temperature field, given deformations. Equations (3) and (7) comprise the system of equations of thermoelasticity. It is logical t o distinguish stationary thermoelasticity problems as a separate class. For the case of stationary problems the system of (3) and (7) is significantly simpler:

In the elasticity theory (as well as in problems of thermoelasticity) it is often important to reformulate the original problem for displacements in terms of new variables. In the following, we suggest some possibilities of this kind. 8.s.e A SPECIAL REPRESENTATION OF THE LAME EQUATIONS

The representation in terms of solutions to ordinary second-order equations describing oscillations is the most important among possible representations of solutions to thermoelasticity problems. We seek a solution to (3) with f = 0 in the form

u = grad ip + curl Q.

(10)

We use the identity

A = grad div - curl curl, to rewrite (3) with f = 0 as

aZu

p - = (A + 2p) grad div u - p curl curl u - y grad T . at2

Substitution of (10) into (11) yields

This implies that

ip

can be found from the equation

We thus have a vector equation for Q, namely

(11)

52


Thus, the dynamical problem of calculating thermoelastic state is reduced to solving second-order hyperbolic equations (12) and (13) and parabolic equation (7). Equation (12) describes the propagation of longitudinal waves (expansion waves) at speed cl = ((A z ~ ) / ~ ) ' / ' .Similarly, equation (13) describes distribution of transverse waves (displacement waves) a t speed cz = ( w I P ) ' ~ ~ .

+

2.6.3 PLANAR PROBLEMS

Let us discuss planar deformations in more detail. In this case, we have w = 0, and u and u depend only on x and y. We consider the state of equilibrium in which there are no mass forces, i.e. f = 0 in (8). Then we can introduce the Airy stress function 11, such that

We obtain a biharmonic equation for the stress function, namely

+

p)) 5 112. We In (14) u denotes the Poisson coeficient, u = A/(2(A take into account the simplest stationary heat equation in (7) and derive the homogeneous biharmonic equation for the stress function

This biharmonic equation is widely used in the elasticity theory; therefore, it can be considered as a basic model (equation (14)). Note that in the planar case we have

a4+

a4+ +-a4+ ax2ay2 ay4

AA+ = -+ 2ax4

'

It should be noted that a hyperbolic second-order equation (the equation of oscillations in (12) and (13)) and systems of hyperbolic equations (the Lam6 equations in (3)) are other basic models of the elasticity theory. 2.6.4 THIN PLATES

In the elasticity theory, much attention is paid to the simulation of deformation of thin plates and shells and membranes. Let us formulate the equations that describe the thermoelastic state of a thin plate. We consider an elastic cylinder of a small height h subjected to stress. We choose the Cartesian coordinate system so that r = 0 is the medium plane


53

of the cylinder and consider small deflections of this cylindrical plate under normal loads and the combined action of heat. The constant deflections of a fixed-width plate are governed by the Sofi Jermen equation DAAw = q - A M T , (15) where D is the cylindrical rigidity, q = q(x, y) is the load, and MT is the bending moment due to thermal actions. The value MT can be represented as

The temperature field is determined by the corresponding heat equation (stationary heat equation (9)). In the simplest case, in which the temperature is constant over any section, we get MT = 0, i.e. such thermal actions do not lead to displacements along the normal to the plate. The effect of inertial forces reduces to the following refinement of (15):

Equation (17) is supplemented by nonstationary heat equation (7). 2.6.5 PROBLEMS

1. Derive an expression for the deformation of an infinite elastic medium in which the temperature field is T = T(x, y,z) # To = const, ( x , Y , E~ a,and T ( x , Y , ~=) T O , ( X , Y , ~$)0 . Solution. A steady thermoelastic state is described by the equation (see (11)) (A + 2p) grad div u - p curl curl u = y grad T. This equation has a particular solution

The velocity u specified by (18) and (19) can be represented as

54


where r = ((x - x ' ) ~+ (y - y')Z+ ( Z - z ' ) ~ ) ~NOW / ~ .we take into account (19) to obtain

because the temperature is constant outside the domain R. 2. Derive a s t a t i o n a r y equation for a stressed t h i n p l a t e i n t h e case of a nonuniform t e m p e r a t u r e field i n t h e transverse direction, neglecting h e a t transfer i n t h e longitudinal direction. Solution. Under the posed ideal conditions, the heat equation (9) reduces to

The integration of (20) and the substitution of the result into (16) yields

This expression is used in equation (15). Thus, temperature gradients in the transverse direction lead to a deflection of the thin plate.

2.7 B i b l i o g r a p h y and Comments

2.7.1 GENERAL NOTES

2.1 Mathematical models of heat conduction are considered in classical manuals [3, 5, 7, 11, 121. 2.2 Formulations of boundary value problems for heat equations are discussed in textbooks on equations of mathematical physics (see e.g. 1161). Inverse problems of heat transfer are most completely covered in [I, 21. 2.3 Problems in which phase changes occur are briefly discussed in textbooks [3, 111. More detailed bibliography is presented in Chapter 7. 2.4 General models of heat and mass transfer are discussed in [6, 8, 10, 131 (see also Chapter 9). 2.5 Models of radiation, in much more general terms (not only radiation from the surface of rigid bodies), are considered in [5, 14, 151. 2.6 Among all manuals on the elasticity theory we mention 14, 9, 171. Papers specifically devoted to thermoelasticity problems are listed in Chapter 8.


55

2.7.2 LITERATURE 1. Alifanov 0 .M. (1988) Inverse Problems of Heat Transfer [in Russian]. Mashinostroenie, Moscow. 2. Beck J . V., Blackwell B. & St. Clair C., Jr. (1985) Inverse Heat Conduction, Ill-Posed Problems. John Wiley & Sons, New York. 3. Carslow H. S. & Jaeger J. C. (1959) Conduction of Heot i n Solids, 2nd edn. Oxford Univ. Press, London. 4. Green A. E. & Zerna W. (1968) Theoretical Elasticity. University Press, Oxford. 5. Isachenko V. P., Osipova V. A. & Sukomel A. S. (1981) Heot Transfer [in Russian]. ~ n e r ~ o i z d aMoscow. t, 6. Jaluria Y. (1980) Natural Convection. Heat and Moss Ransfer. Pergamon Press, Oxford. 7. Kutateladze S. S. (1979) Basics of Heat 7 h n s f e r [in Russian]. Atomizdat, Moscow. 8. Landau L. D. & Lifshitz E. M. (1987) Fluid Mechanics, 2nd edn. Pergamon Press, Oxford. 9. Landau L. D. & Lifshitz E. M. (1984) Theory of Elasticity. Pergamon Press, Oxford. 10. Loitsyanskii L. G. (1973) Mechanics of Fluids and Solids [in Russian]. Nauka, Moscow. 11. Luikov A. V. (1968) Analytical Heot Difusion Theory. Academic Press, New York. 12. Luikov A. V. (1972) Heat and Moss linnsfer [in Russian]. Onergiya, Moscow. 13. Schlichting H. (1968) Boundary-layer Theory, 6th edn. Translated by J. Kestin. McGraw-Hill, New York. 14. Siege1 R. & Howell J. (1981) Thermal Radiative Heat Tmnsfer. McGraw-Hill, New York. 15. Sparow E. M. & Sess R. D. (1971) Radiative Heat Ronsfer [Russian translation]. ~ n e r ~ iLeningrad. ~a, 16. Tikhonov A . N. & Samarskii A. A. (1977) Equations of Mathematical Physics [in Russian]. Nauka, Moscow. 17. Timoshenko S. P. & Goodier J. P. (1970) Theory of Elasticity. McGraw-Hill, New York.

Analytical Methods of Heat Transfer

Heat transfer problems are studied, in simple cases, by classical analytical methods of applied mathematics. Computers and numerical methods are employed to solve more complex problems. Analytical methods are also used in the course of numerical solution to preliminarily examine a mathematical model and test computational algorithms. To investigate an applied mathematical model, it is necessary, first, t o pass to dimensionless variables. This would allow us to analyse the problem, i.e. t o single out small or large parameters. In dimensionless variables the number of parameters of the problem is less than in the original variables, which is extremely important for multiparameter investigation of an applied model. For example, if we deal with an optimization problem, we can significantly reduce the dimension of the minimization problem. Similarity criteria can be used t o compare results of different experiments and clear up similarity of processes. The most advanced analytical methods of heat transfer are developed for solving linear problems. These methods with multiple parameters comprise thus far most of the space in textbooks on theoretical heat transfer. For example, the method of separation of variables and methods of integral transforms are commonly used for solving boundary value problems. Exact solutions of nonlinear problems are worthy of notice. These particular solutions allow us to analyse special features of a problem due to its nonlinearity. Some exact solutions of nonlinear problems in beat transfer can be found when searching for self-similar solutions. The group analysis can be involved in a more general situation. We should also mention some functional transforms, which can be used to pass from certain nonlinear problems to

58


linear ones. The problem is then reduced to finding general solutions of these linear problems. Among approximate methods of applied mathematics we should mention perturbation methods. An approximate analytical solution can in certain cases be constructed by separating a small parameter. The most important results in this way are obtained by homogenization theory. This theory can be nsed to compute effective characteristics of composite media with small inclusions. Asymptotic methods are widely nsed in the theory of heat transfer. As an example, we present the investigation of a regular thermal regime corresponding to a developed stage of the process.

3.1 Dimensionless Analysis

3.1.1 GENERAL CONCEPTS AND A MODEL PROBLEM It is logical that in practice we desire to use a unique system of units. Such a system for mathematical modelling and computations is a dimensionless system of units. This choice of a system of units is meaningful and is not simply made for convenience of unification. Rounding-off errors determined by the hardware always occur when solving a problem on a computer. In order to reduce a contribution of rounding-off errors into an approximate solution, the problem is scaled. The modulus of the desired solution must not be too large. For this reason, desired quantities are multiplied by the corresponding scaling factors, i.e. the problem is transformed. By passing to dimensionless variables the problem is scaled so that dimensionless quantities vary approximately from -1 to 1. The second reason in favour of dimensionless variables is that they allow us to separate small (large) parameters of the problem. It is in dimensionless variables that we can compare terms with one another. The small (large) parameters are used to construct simplified mathematical models and analyse the problem asymptotically (i.e. find an approximate solution). Finally, introducing dimensionless parameters, we reduce the total number of parameters of the mathematical model, which is extremely important for numerical modelling. Therefore we can investigate the influence of a group of parameters on the solution by studying the influence of a smaller group of parameters on this solution. Consider the simplest model of heating a cylinder rod of the length 1 with a thermally insulated lateral surface. For simplicity, we only consider the case in which the thermal characteristics of the rod (as well as the other parameten of the problem) are constant. Heat transfer is described by a one-dimensional heat equation, namely

59

ANALYTICAL METHODS OF HEAT TRANSFER

where we use the standard notation (see Chapter 2). We assume that the base of the cylinder is subjected to heat exchange with a medium at the temperature T, -k

aT

ax + a ( T - T,) = 0,

x = 0.

(2)

Let the heat flow be specified on the other end of the rod, i.e

In addition, we assume that the initial temperature of the rod is other than that of the environment, e.g.

These are all the necessary boundary and initial conditions for (1) 3.1.2 MAKING A PROBLEM DIMENSIONLESS

The problem in (1)-(4) is characterized by the following set of parameters: I, c, p, k , f, a,T, and q. The solution to even this simple problem depends on eight parameters, namely T = T(x, t ;1,c, p, k,f,a, T., q). Thus, it is not always possible to study the influence of each of the parameters. Usually most of the parameters are fixed and only some of them can vary in a small range. Therefore, it is often sufficient to study the influence of one parameter while the others are fixed. We should first choose characteristic quantities (values) for scaling in order to pass to a dimensionless problem. This choice is not always obvious and depends on the specific problem. It is reasonable to use common techniques of scaling for a certain branch of investigation. In the problem considered we can take the length of the rod 1 for scaling the spatial variable x. We use the same letters for dimensionless quantities but supply them with primes. Then x = lx', where x' is a dimensionless variable. If the rod is not heated strongly, we can take the temperature of the medium T, for scaling temperature, i.e. T = T,T', where T' is dimensionless temperature. Similarly, let t = tot', where a typical time interval to is not yet defined. We substitute these formulm into heat equation (1) and obtain the equation cp12 aT' --=-

kto atf

a2T'

+ Os,

0

< 2' < 1,

t'

> 0,

(5)

60


in the dimensionless variables. Equation (5) contains a dimensionless fraction (dimensionless parameter) 0 s = 12f/(kT,), that is called the Ostrogradskii number. Let us now define the time scale to = cp12/k. Then the heat equation is additionally simplified aT' - a2T' Os, at1 (a~')~

+

0

< sf < 1, t' > 0.

to include only one parameter (the Ostrogradskii number). Let us now derive dimensionless boundary and initial conditions. According to (2)

where Bi = a l l k is the Biot number. Similarly, condition (3) becomes

where Kr = ql/(kT,) is the Kirpichev number. The initial condition is written

The problem in the dimensionless variables (5)-(9) is characterized by three dimensionless parameters, namely Os, Bi and K r . Note that we have only three rather than eight parameters for the problem, i.e. T' = T'(xf, t'; 0 s ,Bi , Kr ). Logically, the problem in the dimensionless variables is much simpler for investigation. In order to transfer to dimensional quantities we should actually multiply dimensionless solutions by dimensional factors. It is dimensionless variables that reveal similarity of different problems. Indeed, let us have two problems with different linear sizes and thermal parameters. Assume that they are written identically in dimensionless variables. Then the problems are similar, i.e. they differ only in scale factors. Therefore it is easy to pass from one problem to the other. 3.1.3 PARAMETRIC ANALYSIS O F A PROBLEM

Rewriting a problem in dimensionless variables, we can separate small (large) parameters and, hence, simplify the problem. In the simplest case of (1)-(4), for example, third-kind boundary conditions are realized for x = 0. The question arises as to under what conditions of cooling can we pass from these conditions to first- or second-kind boundary conditions. The boundary regime in dimensionless variables is only specified by the Biot number. If


61

Bi >> 1 in ( 7 ) , we can put T' = 1 for x' = 0 (a first-kind boundary condition). Conversely, if Bi << 1, we get a second-kind boundary condition aT1/az' = 0 for x' = 0. A more precise answer to the question of the boundary regime can be given according t o the experience of modelling. For instance, if Bi < 0.01, the simplified boundary condition holds to within several per cent. The second example of parametric analysis of (1)-(4) is associated with the question as to when we can pass from nonstationary equation (1) to a stationary equation. Assume that a thermal process is considered in the time Then it is sufficient t o compare the typical time of interval from 0 to t,,,. If t,, >> to, we can use the stationary heat transfer to = cp12/k with t,,,. equation instead of (1). In this case we can take t,,, for scaling time. As a result we get the equation

with the small parameter E = tO/tmax instead of ( 6 ) . Evidently, the nonstationary character of (10) manifests itself for small time intervals, namely the small parameter at the time derivative makes the equation singularly perturbed. We can similarly investigate the influence of the source in (1). If 0 s << 1, it can be neglected. In the opposite case of 0 s >> 1 we come across a singularly perturbed problem, and if we are not interested in boundary effects, we can neglect heat conduction in the original equation (1). We can thus simplify the original mathematical problem and investigate the qualitative behaviour of its solution by parametric investigation of the problem in dimensionless variables. We should emphasize that we only use minimal mathematical tools for the analysis. 3.1.4 PROBLEMS

1. S e p a r a t e a dimensionless p a r a m e t e r responsible for convective h e a t transfer. Solution. The model equation is

where u is the velocity of the medium. The corresponding dimensionless equation (see (6)) is

62


The Pe'clet number Pe = ulcp/k is the desired dimensionless parameter. The typical time can be taken as to = 1/11. Then the equation in dimensionless variables appears in a somewhat different way, namely

2. Find the conditions under which the heat of phase change can be neglected while modelling heat transfer. Solution. Let us reduce the appropriate one-dimensional monophase Stefan problem to a dimensionless form. The latent heat L is taken into account by the condition (see Section 2.3)

where x = q(t) is the interface. We reduce (11) to a dimensionless form and separate a dimensionless number Ste = L/(cp), which is called the Stefan number. If Ste << 1, the heat release due to a phase transition can be neglected.

3.2 Analytical Solution of Linear Problems

3.2.1 THE METHOD O F SEPARATION O F VARIABLES Henceforth the linear heat equation written in the form

au

c(x) - = div (k(x) gradu) at

+ f(x, t ) ,

x E 0, t

>0

(1)

is the basic mathematical model. We consider general 3-D problems as well as problems of smaller dimensions; therefore x = ( X I ,22,. . . ,x,), m = 1,2,3. Equation (1) describes (see Section 2.1) heat transfer via conduction in a nonhomogeneous isotropic medium. Equation (1) is supplemented by the initial condition

and a homogeneous boundary condition, e.g. by the first-kind condition

It is important for the method of separation of variables that the boundary condition is homogeneous. Therefore, if we deal with a problem with generic

63

ANALYTICAL METHODS O F HEAT TRANSFER

boundary conditions, we should first pass to the problem with homogeneous conditions. The essence of the method of separation of variables (the Fourier method) is the construction of particular solutions of (1) that can be represented as a product u(x, t) = B(t)u(x), (4) where each factor depends on its own variable. Let us first consider the case of a homogeneous equation (i.e. f (2, t) = 0 in (1)). We substitute (4) into (1) and derive the equations for B(t) and v(x) div (k(xgradv)) + Xc(x)u = 0,

x E 0,

(5)

According to (3), equation (5) is supplemented by the boundary condition U(X)= 0,

E an.

(7)

The problem of (5) and (7) has nontrivial solutions only for some X and is referred to as a spectral problem (the Sturm-Lioville problem). The corresponding values of X are said t o be eigenvalues and the corresponding solutions u(x) are called eigenfunctions. Let us number the eigenvalues of the problem of (5) and (7) in ascending order so that X 1 < X 2 < . . . A,<

...,

lim X , + m

n-cx

All the eigenvalues are positive, i.e. X1 > 0. We denote the corresponding eigenfunctions by u,(x), n = 1 , 2 , .. . . Notice the main properties of the eigenfunctions vn(x). Let us consider a Hilbert space 71 = Lz(12) equipped with the scalar product

and the norm llyll = (y,y)'/2. Similarly, for positive c(x) we define a weight Hilbert space 71, such that

64


The eigenfunctions of the problem of (5) and (7) are orthonormalized in 71,. i.e. (v,,~.), =6,,, where

is Cronecker's delta. Given a solution of the spectral problem of (5) and (7), we can determine the general solution of (6) as

Let us now represent the solution of (1)-(3) with f ( x , t ) = 0 as a superposition of constructed particular solutions

The coefficients c, are determined by the initial condition (2), namely c, = (uo,u,), are the coefficients in the expansion of the function uo(x) in the eigenfunctions v,(x), (the Fourier coefficients). Thus, we derived the solution of problem (1)-(3) with f ( x , t ) = 0 in the form w u(x,t) = ~ ( ~ ~ , ~ ~ ) , e x p ( - ~ , t ) ~ , ( x ) .

(9)

n=1

In the case of nonhomogeneous equation ( I ) , representation (9) in the method of separation of variables includes an additional term, namely

We thus obtained the general solution (10) of the heat transfer problem (1)-(3). The cases of the first- or second-kind boundary conditions, mixed boundary conditions, etc. are proceeded with similarly. Since the solution is represented as an infinite series, it is often necessary to simplify the original problem to get a simpler solution. The general solution is constructed given the solution of the spectral problem in (5) and (7). Note that the solution of this problem is known only

65


in a few cases, and most textbooks on heat transfer do not present these solutions. The same is true for stationary problems of heat transfer. For example, let us find the solution of the simplest one-dimensional problem in which R = (0,l). We consider the heat equation

with initial and boundary conditions in (2) and (3), respectively. The corresponding eigenvalue problem (see (5) and (7)) becomes

The eigenvalue problem in (12) and (13) has the solution

The solution of (11) is thus presented according to (lo), (14) and the conditions in (2) and (3). 3.2.2 THE METHOD OF THE GREEN FUNCTIONS The method of the Green functions is a widespread method for solving boundary-value problems in mathematical physics. Consider the problem of (1) and (2) with a nonhomogeneous first-kind boundary condition U(Z,t) = g(z, t),

E an.

(15)

The Green function (a source function) for this problem is defined as the solution of the equation aG C(Z)- = div (k(x) grad G) at

+ 6(z - z', t - t'),

(16)

with homogeneous initial and boundary conditions (3). Here 6(z, t) is the delta function (see Section 2.3). Thus, the Green function G(z,zf;t , t') is the temperature field due to the instantaneous (at time t') source of heat located at the point 2'. The solution of the general problem in (I), (2), and (5) is uniquely represented in terms of the Green function. The problem is thus reduced to searching for the Green function. This function can be constructed in some specific problems of heat transfer (in particular, when stationary problems are considered in this way).

66


The solution u ( x ,t ) is written as

The first, second and third terms in the right-hand side of (17) are responsible for nonhomogeneous initial temperature, nonhomogeneous boundary condition and nonhomogeneous equation ( I ) , respectively. We separate an important case in which heat transfer is considered in an unlimited isotropic medium. In this case, Cl is the whole space Rm and the bounded solution is defined by equation (1) and initial condition (2) (an initial value problem). Under these conditions G ( x ,x'; t , t') is the fundamental solution and the corresponding integral representation of the solution is

u ( x ,t ) =

I

u o ( x l ) G ( x ,x'; t , 0 ) dx'.

Rm

For a nonhomogeneous medium we get

(

G ( x ,x'; t , 0 ) = ( 2 ( a a t )1,2 ) -3 exp --T ~ ? ; ~ X ' ) ) , where T ~ ( x , x '= ) C z 1 ( x ,- x : ) ~ . Note that the Green function can be constructed by the method of separation of variables. For this purpose we set the right-hand side f ( x , t ) of ( 1 0 ) to be the delta function, which yields m

c(x')u,(x')u,(x) exp ( - X,(t - t'))

G ( x ,x'; t , t ' ) = n=1

In particular, we can conclude from this equation that the Green function is symmetrical with respect to the spatial variables for c ( x ) = const.

67


3.2.3 INTEGRAL TRANSFORMS

An integral transform of a function (an original) is said to be the function

where K(p, t ) is called the kernel of the integral transform. Considerable attention is aid to integral transforms in the analytical theory of heat transfer. Transforms with both infinite and finite ( a and b) limits of integration are studied. The most conventional transforms are the cos- and sin-Fourier transforms, the Mellin transform, the Hankel transform, etc. The most typical example of practical employment of integral transforms for solving heat-transfer problems is provided by the Laplace transform. The integral m

F(P) = /exp(-pt)f(t)

<

p = +in

dl,

0

>

exists for ( &, given a continuous function f (t) of a real argument t such ). the image F(p), we can define the inverse that /f(t)l 5 c o n s t e ~ p ( < ~ tGiven Laplace transform

There is a lot of reference material about the Laplace transform and some other integral transforms. Without going into detail we mention that the Laplace transform can be used to find exact solutions of heat transfer problems. Let us consider the problem (1)-(3). We multiply (1) by the kernel exp(-pt) of the Laplace transform and integrate the resultant with respect to t from 0 to m. Let

be the image of the solution. Direct calculation yields c ( x ) ( p - uo) = div (k(x) gradv)

+F(x,p),

x E 0,

(18)

where F ( x , p ) is the image of the source term in the right-hand side of (1). Equation (18) is equipped with the boundary conditions U(X,P)= 0, which result from (3)

E

an,

(19)

68


We thus come to solving the Dirichlet problem in (18) and (19) for a second-order elliptic equation that includes a parameter p. If this problem can be solved, the next step is to determine the inverse Laplace transform. In this case we use the Laplace transform with respect to time. Of course, in certain problems it is more convenient to apply the Laplace transform with respect to some of the spatial variables. For example, this is the case for stationary and nonstationary heat transfer in space. Obviously, there are a lot of other examples.

3.2.4 PROBLEMS

1. A n isotropic b o d y with a fixed initial t e m p e r a t u r e exchanges h e a t w i t h t h e environment whose t e m p e r a t u r e varies i n time. R e d u c e t h e solution of t h i s problem t o t h e solution of a problem with fixed boundary conditions. Solution. We need to find the solution to the equation au c(x) - = div (k(x) gradu), at

x E 0, t

> 0,

(20)

that satisfies the conditions x E 0, u(x,o) = uo = const, au x E an, k ( ~ ) + a ( u - g(t)) = 0,

an

(21) (22)

where g(t) is the temperature of the environment. We seek the solution (by Duhamel's method) in the form

We substitute (23) into (20)-(22) and obtain the following problem of determining the function u(x, t)

au

c(x) - = div (k(x) gradu), at = 0, E n, U(X,O)

au + a(u an

I~(x)

- 1) = 0,

x E R,

t

> 0,

E an.

We thus come to the problem with fixed boundary conditions.

69


2. F i n d t h e general solution of t h e hyperbolic h e a t e q u a t i o n b y t h e m e t h o d of s e p a r a t i o n of variables neglecting i n t e r n a l sources of heat. Solution. Under these hypotheses, the heat equation becomes (see Section 2.1) c )

(

+

)

= div ( ( x ) g r a d )

r s R,

t >o

In addition to conditions (2) and (3), we pose an extra initial condition

We apply the method of separation of variables and derive the second-order equation

instead of (6). Solving this equation with the initial conditions in (2) and (24) taken into account results in the sought-after solution of the hyperbolic heat equation written as

where y = (1 - 4Xnrr)/(2rr). It is undoubtedly interesting to compare this solution with that of the conventional heat transfer problem (see (9)) as the relaxation time rr vanishes.

3.3 Exact Solutions of Nonlinear Problems

3.3.1 FUNCTIONAL TRANSFORMS OF NONLINEAR PROBLEMS

Many mathematical models of heat transfer are nonlinear; for instance, models that describe heat transfer in bodies whose thermal characteristics are variable and depend on temperature. Processes with phase changes and adjoint problems of heat and mass transfer are essentially nonlinear a s well. We cannot construct general analytical solutions of nonlinear heat transfer problems and can only give some particular exact solutions.

70


Solutions of nonlinear problems allow us to discover new effects and prw perties of mathematical models that are not inherent to linear ones. Therefore exact solutions to nonlinear equations are important for theoretical research. In numerical investigations particular solutions to nonlinear equations are used to test the accuracy of numerical algorithms as well as the other properties of the latter. As a typical example, we consider a quasilinear heat equation that looks like au c(u) - = div (k(u) grad u) . (1) at When constructing analytical solutions to this equation, it is meaningless t o formulate either initial or boundary conditions and separate a computational domain. Given a particular solution to a nonlinear equation, we can formulate the necessary closure conditions based on the solution itself. To find a solution to ( I ) , we can try to change dependent and/or independent variables to reduce the problem to a linear one. The simplest example is given by functional transforms of the form

Although transforms such as (2) rarely allow us to reduce the problem to a linear one, the problem can often be partially linearized. A well-known transform employed in theoretical heat transfer is the Kirchhoff transform

"

v=

1

k(s) ds.

(3)

0

Substitution of (3) into (1) yields qv)

ava t = Av,

where E(v) = c(u(v))/k(u(v)). Equation (4) corresponds to moving to a problem with linear thermal conduction. It is very convenient to use the Kirchhoff transform in stationary problems, in which we obtain the linear equation Av = 0 for the new independent variable v. Note that under this transform first-kind boundary conditions are preserved as linear, while second-order conditions become linearized. Heat equation (1) is also partially linearized by the Goodman tmnsform

71


In this case ( 1 ) becomes

au

- = div ( a ( u )grad u ) ,

(6)

at

where a ( u ) = IZ(u(u))/c(u(u)), i.e. we come to a problem with constant specific heat capacity. Quasilinear parabolic equations (4) and ( 6 ) can be considered as model equations of nonlinear heat transfer. When describing convective heat transfer, dependent variables in a model equation of continuum mechanics are changed in a more complicated way. An important example of such an equation is the Burgers equation

which has quadratic nonlinearity and the coefficient v simulating viscosity. We consider an initial value problem for (7), i.e. supplement it by the initial conditions u(2,O) = uo(x), -00 < x < 00. (8) The Cole-Hopf transform

is used for finding the general solution of problem ( 7 ) , (8). Substitution of ( 9 ) into ( 7 ) yields the linear heat equation

The initial condition for ( l o ) ,with ( 8 ) and ( 9 ) taken into account, is written

where uo is an arbitrary constant and

The general solution for heat equation ( l o ) , (11) is well-known (see Section 3.2) and can be represented as

u ( x ,t ) = vo

i

G ( x - c , t )exp

(--

dc.

72

I


where G ( z ,t ) = ( 4 ? i ~ t ) - ' e/ ~x p ( - x 2 / ( 4 v t ) ) . The solution of the problem for the Burgers equation ( 7 ) , (8) is obtained from ( 1 2 ) according to ( 9 ) . 3.3.2 TRANSFORMS OF INDEPENDENT VARIABLES

An original nonlinear equation can be simplified by changing dependent variables. Using self-similar variables, we can reduce the original partial differential equation to an ordinary differential equation. Let us offer some well-known examples. Consider a one-dimensional heat equation written as (see ( 6 ) )

We seek solutions to ( 1 3 ) that only depend on one variable ( = ((x, t ) and call the corresponding solutions u ( x , t ) = v ( ( ) the self-similar solutions. We can take x <=(14) 2tw as a self-similar variable for equation ( 1 3 ) .Taking ( 1 4 ) into account, we express the first-order derivatives as

au -=--- E at

dv 2t d ~ '

au - I ax z t l l z

dv d<'

Equation ( 1 3 ) rewritten in terms of the self-similar variable becomes

Thus, in order to construct the exact solution of the nonlinear heat equation ( 1 3 ) , we need to solve the ordinary differential equation ( 1 5 ) . In many important cases the latter can be solved analytically. Let us construct a travelling-wave self-similar solution of equation ( 1 3 ) :

where D = const is the velocity at which the heat wave moves. Substitution of ( 1 6 ) into ( 1 3 ) yields an ordinary differential equation for 4 0 , namely

This equation immediately yields


We set the constant to be zero and obtain

This i m ~ l i e s

If the coefficient k ( u ) is given (see Problem 2), further computations can be made using (17). The travelling-wave self-similar solution for the heat equation with a nonlinear source

au azu + f ( u ) , at axz

-= -

which is called the Kolmogorov-Petrovskii-Piskunov equation, is studied in great detail.

3.3.3 GENERAL TRANSFORMS In order to obtain exact solutions to nonlinear equations, both dependent and independent variables can be transformed. Nowadays group analysis of the equations is employed to search for these transforms. It is necessary t o find transforms under which the original equation remains unchanged, i.e. invariant. Knowledge of a group of these transforms is helpful for determining particular solutions. Thus, we search for transforms of both dependent and independent variables of model equation (13) in the form

where a l ,az, . . . , a , are parameters of the group of transforms. For example, group analysis can be used to find the transform like (18) linearizing equation (13) with k(u) = u - ~ The . corresponding transform is

The equation

74


immediately follows from (13) and (19). Hence, under transform (19) every solution of the linear equation

goes into a solution of the quasilinear equation (13) 3.3.4 PROBLEMS

1. F i n d t h e solution of t h e two-phase Stefan p r o b l e m i n a homogeneous m e d i u m filling a half-space, given first-kind b o u n d a r y conditions. Solution. We search for the solution of the equation

in separate phases that satisfies the boundary condition u ( 0 ,t ) = U = const.

(20)

The initial condition of constant temperature V is written as u ( x , 0 ) = v. The phase change occurs at the point x = q ( t ) temperature. Then the junction conditions

(21)

>

0 , say, at the zero

are satisfied. We seek the solution that depends on the self-similar variable We determine u(E) from equation (15) with k(u) = 1, namely

< (see (14)).

where the parameter cu = q / ( ~ t ' /defines ~) the location of the interface and

is the error function


The constants

..

az =

-

V erf ( a ) 1 - erf ( a )'

bz =

v

1 - erf ( a )

are determined by substituting ( 2 4 ) into (20)-(22). We thus obtain from ( 2 3 ) the eauation

to determine the interface a . The solution t o the monophase problem with V = 0 can be considered on the basis of the solution constructed. 2. Consider the propagation of a travelling heat wave in a medium at zero temperature, provided that the relation between the thermal conductivity and temperature is power-like. Solution. Let us consider the self-similar solution (16) to equation (13) with

Equation (17) thus yields

We set the constant of integration to be zero and find v Equation ( 2 6 ) yields

>

0 for

< < 0.

This solution is characterized by a finite propagation velocity (the parameter D ) of heat disturbances.

3.4 Asymptotic Methods of Heat Transfer

3.4.1 THE REGULAR REGIME OF HEAT TRANSFER There are three stages of describing heat transfer in a solid. The first stage involves the regular regime that is realized over a long time. Let us consider

76


in more detail the asymptotic behaviour of the solution to the heat equation over time. Let the temperature be defined by the equation

au

c(z) - = div ( k ( ~ ) ~ r a d u ) , z E $2, at

t > 0,

(1)

equipped with the initial condition u(x, 0) = uo(x),

5

E fl,

(2)

and the stationary third-kind boundary conditions

The solution of problem (1)-(3) tends to a stationary solution denoted by um(x) as t + oo. This solution is determined by the equation div (k(x) gradu,)

= 0,

x E fl,

(4)

and the boundary conditions

Let us represent the solution to (1)-(3) as u(x, t ) = u,(x) + w(x, t). We derive the system of equations for the difference between the solution to (1)-(3) and the stationary solution

aw

c(x) - = div (k(x) grad w), x E fl, at w(x,O) = uo(x), x E a, aw x E afl. k(x) - aw(x, t ) = 0, an

t > 0,

+

(6) (7) (8)

We use the method of separation of variables (see Section 3.2) to find the asymptotic solution to (6)-(8). This solution can be expanded as

where A, and ui(x), i = 1 , 2 , . . ., are, respectively, the eigenvalues and eigenfunctions of the problem div (k(x)gradv)

+ Ac(x)v = 0,

av + au(x) = 0,

k(x) an

x E 80.

x E 0,

(10) (11)


77

To study the behaviour of the temperature over short times we should take into account all harmonics in expansion (9),and the solution substantially depends on the initial distribution. The regular regime of heat transfer occurs at the developed stage of the process for sufficiently large times. In this case, since the eigenvalues A, grow fast with the number of harmonic i, the contribution of higher harmonics is negligible; therefore it is typical of this stage that w(x, t ) = ~ ( xt), = ( u ~ , u ~ ) , e x p ( - A ~ t ) ~ ~ ( x ) , (12) i.e. the behaviour of the solution is defined only by the first harmonic. At this stage the influence of the initial conditions and the distribution of the initial temperature are of secondary importance. Finally, the third stage of the process corresponds to the stationary regime with w(x, t) = 0. For the regular regime

-1 -au z -A1 at

= const

Hence, the relative velocity of temperature variation is constant for the regular regime (i.e. independent of either coordinates or time). In the theory of regular heat transfer much attention is paid to the dependence of the cooling rate A 1 on boundary regimes and to thermal properties of the medium. 3.4.2 PERTURBATION METHODS

In the analytical theory of heat transfer considerable attention is paid to perturbation methods. The latter involve separation of a small parameter (parameters) in equations and boundary conditions and investigation of the solution as a function of these parameters. Although small changes in a small parameter lead to small changes in the solution to a regularly perturbed problem, this is not the case for singularly perturbed problems. In a portion of the computational domain (like boundary or internal layers) small changes in a small parameter may result in significant changes in the solution. For brevity, we only consider simple model stationary problems of heat transfer. Let uE(x) be the solution to the perturbed problem and let uo(x) be that for the unperturbed problem (for which E = 0). We use the homogeneous norm I l 4 ~ ) l l c ( n= ) :E%lu(~)l to estimate the difference between the perturbed and unperturbed solutions. For regularly perturbed problems we get

78


Condition (13) is not satisfied for singularly perturbed problems, and typically

IIue(x) - uO(x)llC ( D ) 3 0 ,

if

E 3

0 while D

c R,

i.e. the perturbed and unperturbed solutions are only close in a portion D of the domain R rather than in the whole R. The essence of asymptotic analysis is to construct a function U ( X , E such ) that 0. IIu,(x) - U ( x , ~ ) l l3~ 0~, ~ , if E +

The asymptotic approximation U ( X , E is ) usually constructed as a series in powers of the small parameter, namely

u(X,E ) =

Ekuk(x,E ) .

(I4)

k=l

Let us give simple examples to illustrate typical characteristics of singularly perturbed problems. Consider the following boundary value problem for a second-order ordinary differential equation

d2u, + E - =du, O, dx2

dx

O
The boundary value problem (15), (16) describes the temperature field taking into account convective heat transfer. Our aim is to understand special features of problems with dominating convection. Evidently, the exact solution of (15), (16) has the form

For the degenerate problem

(E

= 0 ) we obtain uo(x)= a.

Equations (17) and (18) immediately yield llu,(x) - U ~ ( X ) ~ ~=~O( ( ~E )J ,i.e. ) the problem in (15),(16) with dominating conduction ( E is a small parameter) is regularly perturbed. The corresponding asymptotic expansion is easily written. We substitute (14) into the original equation (15) and equate the terms with equal powers of E . We arrive at the problem

to find u t


79

The problem

is an example of a singularly perturbed problem. It differs from (15), (16) only in that the small parameter multiplies the second (leading) derivative (the small parameter E is replaced by the large parameter I / € ) . In problems of heat and mass transfer this corresponds to domination of convection over heat conduction. The exact solution of the singularly perturbed problem (19), (20) is

Equation (19) implies that uo(x) is a constant. The constant uo(x) is defined by the boundary conditions, however it is not clear a priori which of the conditions (20) should be chosen. The term proportional to E governs the heat diffusion due to heat conduction, and the second is responsible for convective transfer due to motion of the medium itself. For the convective term it is natural to pose boundary conditions where the medium moves inside the computational domain. Consequently, we obtain

for the solution to problem (19), (20). Obviously, this representation also results from (21). Even for small values of the perturbation parameter the solution of the perturbed problem is not identical to that of the degenerate problem in a certain region (namely, in a vicinity of the point x = 0). This is because of the additional boundary conditions posed for the perturbed problem. This phenomenon is typical of problems with small parameters at the leading derivatives. If a boundary layer is present, asymptotic approximations should be constructed in a different way. These approximations, along with regular terms, contain singular terms (boundary functions) that take into account the behaviour of the solution inside the boundary layers. This situation is also typical of more general heat transfer problems, namely for nonstationary, multidimensional and nonlinear problems. The methods of a small parameter are used to construct approximate solutions of nonlinear heat transfer problems. In this case, each term of the asymptotic series is a solution of a linear problem, which is simpler than the original problem.

80


3.4.3 PROPAGATION OF HEAT IN THIN BODIES Asymptotic models are widely employed for constructing approximate models, i.e. those in which only first terms of asymptotic expansions are taken. A well-known model of this kind describes propagation of heat in thin cylindrical bodies. Let us consider heat propagation in a thin homogeneous cylinder of constant cross-section D. Let 1 be the length of the cylinder, d its diameter, and let the axis of the cylinder be identical to the x3-axis. The stationary process inside the rod is described by the equation

We assume that the cylinder exchanges heat with the environment according to the law au k - + a u ( x ) = 0, (XI,x2) E aD. (24) an In addition, let some boundary conditions be given on the faces, e.g

When we construct the asymptotic solution with respect to the small parameter E = d l l , we obtain boundary layers near the faces due to conditions (25) and (26). We do not pay attention to this here. The parameter E appears when we pass to a dimensionless equation. We choose the typical length to be the length of the rod, do not change the notation, and rewrite (23) in a dimensionless form as follows:

The boundary condition (24) yields

. in We assume the Biot number to be small, so that Bi = G E ~Seemingly, order to derive an approximate solution it is sufficient to put E = 0 in (27) and (28) and get an ordinary differential equation along the cylinder axis. However, the truncated equation is more complicated. We search for the asymptotic solution in the form


81

We obtain the following problem for the first term in the asymptotic expansion (29):

Similarly, we find for the second term

An arbitrary function of the longitudinal coordinate uo = uo(x3) is the solution of (30),(31).The problem in (32),(33)is a Neumann problem, which is solvable if the condition

is satisfied. This identity with (32) and (33) taken into account yields the desired equation for the temperature distribution in the thin rod

S,

/SD

ds dxldxz is the ratio of the perimeter to the cross-section. where x = Nonstationary problems, those for slightly curved rods or rods with variable cross-section, etc. are considered similarly. 3.4.4 HEAT CONDUCTION IN COMPOSITE MATERIALS Asymptotic methods are widely used for describing heat conduction in composite materials. We consider nonhomogeneous media with periodical structure with small inclusions. Various averaging procedures are used to describe heat transfer in these media. The asymptotic expansion with respect to a small parameter that characterizes the size of the inclusion is used to pose a problem for averaged characteristics of the process. We present a typical example of homogenization. We consider a composite material that occupies a domain R. It has a periodic structure with size 6, i.e. it consists of the cubes D". Although

82


properties of the material are nonhomogeneous over a cube, they are identical for each cell of the composite medium. In the simplest case, the composite medium is a periodic structure with inclusions of another material. We consider the heat equation

in the domain 0. It is equipped with the following boundary and initial conditions:

The heat capacity and thermal conductivity of the material are periodic fnnctions cC(x/&)and kE(x/&)with the period E in each direction. The asymptotic solution of problem (34)-(36) is constructed, provided that the size of inclusions is small as compared with that of the domain R. Let us denote y = Z/E. We construct a two-scale expansion for the solution ue(x)

where the functions uk(x,y) are periodic with respect to y. The zeroth term in expansion (37) is of the most practical importance. It can be interpreted as an ordinary medium with effective thermal characteristics. The corresponding heat equation becomes

where

is the heat capacity averaged over an elementary cell. The heat conductivities are determined by averaging the solutions of special problems in the cells. Let wyY) be the periodic solution of the problem

akE

div (kE(y)gradwi) = --a y i

-

satisfying w1(y) = 0. Then we get

83


Thus, we should first solve problems for a single cell in order to find effective characteristics of the composite material. We now consider heat exchange through a fine-grained boundary. Let QE have a fine-grained (wavy) boundary aRL with a typical wave size &. Let the body exchange heat with the environment through this boundary according to the law auE ) Eav. k - u ~ u ~ (=x 0, (39) an When averaging the boundary, we consider an effective third-kind condition with a different heat exchange coefficient

+

Asymptotic analysis yields u = aE,y for the effective heat exchange coefficient, where ,y stands for the local ratio of the length of the boundary of 80' to the length of the averaged boundary 8 0 . 3.4.5 PROBLEMS

1. Find an expression for the rate of cooling XI as a measure of nonuniformity of the temperature over the surface of a body and over its volume. Solution. According to (12) we should find an expression for the first eigenvalue of problem (lo), (11). Equation (10) yields XI = -

=-

div (k(x) grad ul)

1

c(.)a(x)

div (k grad ul) dx

R

R

Using ( l l ) , we can rewrite the first integral as div (k grad ul) dx = -a R

an

Therefore (40) and (41) yield

The latter relation is referred to as Kondrat'ev's theorem

84


2. Using a n averaging procedure derive a heat equation for t h i n homogeneous media. Solution. For a thin homogeneous body with a cross-section D we write the heat equation

( x ~ , x zE) D , Let

0 < x3

< 1,

au +CY(U(X) - uc(x3,t)) = 0,

k-

an

t > 0.

(42)

(x1,x2)E aD.

(43)

We integrate (42) over the cross-section D.Let

be the temperature averaged over the cross-section. We thus have

We take into account the boundary condition (43)and rewrite the last term

Since the body is thin, we can assume that the temperature is constant over a cross-section. Consequently, substitution of (45)into (44)yields the desired heat equation

This equation for the stationary case was discussed from the viewpoint of the asymptotic analysis.


3.5.1

GENERAL NOTES

3.1.Passage to dimensionless problems of heat transfer is discussed in [2]. Basics of the similarity theory for heat transfer problems are given in more detail in [5].


85

3.2. Classical methods for solving boundary value problems in mathematical physics a r e described in [lo]. Application of these t o heat transfer problems are discussed in [2, 71. 3.3. General methods to search for exact solutions t o nonlinear equations a r e suggested in [6, 81. 3.4. T h e regular regime of heat transfer is investigated in [3]. Methods of a small parameter a s applied t o various problems in applied mathematics a r e presented in [4, 111. T h e homogenization theory is most completely described in [I, 91.

3.5.2. LITERATURE 1. Bakhvalov N. S. & Panasenko G. P. (1984) Averaging of Processes i n Periodic Media [in Russian]. Nauka, Moscow. 2. Belyaev N. M. & Ryadno A. A . (1982) Methods of the Heat Transfer Theory [in

Russian], in two vols. Vysshaya Shkola, Moscow. 3. Cherdakov P. V. (1975) The Theory of Regular Regime [in Russian]. Energiya,

Moscow. 4. Colle J. 119681 Perturbation Methods i n Applied .. Mathematics. Blaisdell Publishinrcornpan;, w i t h a m (USA). 5. Gukhman A . A . (1973) Introduction to the Similarity Theory [in Russian].

Vysshaya Shkola, Moscow. 6. Ibragimov N. Kh. (1983) Groups of Transforms i n Mathematical Physics [in

Russian]. Nauka, Moscow. 7. Kartashov E. M. (1979) Annlyticol Methods i n the Heat Transfer i n Solids [in

Russian]. Vysshaya Shkola, Moscow. 8. Ovsyannikov L. V. (1978) Group Analysis of Ordinary Differential Equations [in

Russian]. Nauka, Moscow. 9. Sanchez-Palencia E. (1980) Non-homogeneous Media and Vibmtion Theory.

Springer, Heidelberg. 10. Tikhonov A. N. & Samarskii A . A . (1972) Equations of Mathematical Physics

[in Russian]. Nauka, Moscow. 11. Vasil'eva A. B. & Butuzov V. F. (1990) Asymptotic Methods i n the Theory of Singular Perturbations [in Russian]. Vysshaya Shkola, Moscow.

Stationary Problems of Heat Transfer

We start the presentation of numerical methods employed in the physics of heat with the simplest problem of calculating the stationary temperature field. Heat is transferred due to heat conduction, and the temperature is described by a second-order elliptic equation equipped with appropriate boundary conditions. We discuss the main topics of the theory of difference schemes, namely how to pass to a discrete problem and then investigate and solve it. First we present the basic facts of the theory of boundary value problems for elliptic equations. A grid problem should necessarily inherit properties of the original differential problem. For example, the maximum principle is such a property. To study boundary value problems, a priori estimates are derived. A similar technique is involved to prove that the difference solution converges to the exact one. We consider differential problems from the viewpoint of the operator theory in Hilbert spaces. The first question arises as to how the difference problem should be constructed for solving an applied problem. We consider various methods of constructing grid problems, taking as an example a simple one-dimensional problem. In particular, we use the integro-interpolation method, which has a clear physical interpretation. The finite element method can also be applied to construct difference schemes. The main theoretical question is to estimate the accuracy of an approximate solution. We formulate the maximum principle for difference problems and use it to prove convergence (in the homogeneous norm) of the approximate solution to the exact solution for simplest stationary problems of heat transfer written in dimensionless form. We use grid a priori estimates to prove convergence of difference solution in Sobolev spaces.

88


We consider basic methods for solving difference elliptic problems. In simple cases in which variables can be separated, faster direct methods can be involved. In a more general situation, iterative methods are used. We present the general theory of iterative methods in Hilbert spaces. Triangular variation iterative methods lead to the most considerable advances. It is somewhat difficult to solve elliptic problems in irregular domains. We briefly consider the main approaches to the solution of these problems. Methods involving irregular computational grids are conventionally used, e.g. the method of fictitious domains. Recently, much attention has been paid to methods of decomposing of complex domains into simple subdomains. We briefly discuss the approximate solution of nonlinear stationary problems of heat transfer. We pay specific attention to the Newton-Kantorovich method.

4.1 Boundary Value Problems for Second-order Elliptic

Problems

4 . 1 . 1 LINEAR STATIONARY HEAT EQUATION

The heat equation for a generic anisotropic medium occupying a bounded domain R becomes (see Section 2.2)

Here kmp = kOa, a, = 1,2,. . . , m , because the heat conductivity tensor is symmetrical. It is logical to assume that the condition of uniform ellipticity is satisfied, i.e.

em,

for any a = 1,2,. . . ,m,. We take into account the motion of the medium by introducing additional terms with first derivatives in equation (I), namely

where b,(x) stand for the corresponding velocity components.

89

STATIONARY PROBLEMS O F HEAT TRANSFER

Simpler models involve the heat equation in the form -

a

"

(

k

au )

f

x E ~ .

a=1

which governs heat transfer in an isotropic medium. For a homogeneous medium (k(x) = const) we come to the Poisson equation

(we replaced f (x) by f (x)/k). Equations (1) and (3)-(5) can be considered as basic equations for modelling stationary heat transfer. They are supplied by appropriate boundary conditions. We mainly use the Dirichlet condition

when a given temperature is maintained on the boundary. The Nenmann condition (the heat flux is given) for heat equations (1) and (3) is written as

au

av = g(x),

a: E a n ,

(7)

where

is the derivative along the conormal and cos(n,x,) are the direction cosines of the external normal n. In the case of equations (4) and (5), condition (7) is simplified because

The Neumann problem (I), (7) is soluble neglecting a constant, provided that

Convective heat exchange with the surrounding media is simulated by the third-kind boundary condition

au av + U ( X ) U = s(z),

E

an,

where u 2 0 is the coefficient of convective heat exchange.

90


If it is not stated to the contrary, we everywhere assume that coefficients in equations and boundary conditions, the boundary of the computational domain and the solution u ( x ) of the problem are smooth. For example, we assume the function u ( x ) to be twice continuously differentiable in the domain, i.e. u ( x ) E C 2 ( R ) ,and so on.

4.1.2 THE MAXIMUM PRINCIPLE We consider the heat equation in the most general form (3). A simple conclusion that the highest temperature in a body is attained only at its boundary, given no heat sources, leads to the maximum principle for secondorder elliptic equations. T h e o r e m 1. Let L u 5 0 (Lu 2 0) in a bounded domain fl. Then the function u ( x ) achieves its maximum (minimum) on the boundary of the domain, i.e. max u ( x ) = max u ( x ) Z E ~ =€an

( 2 E u ( x ) = min u ( x ) =can

(10)

The maximum principle implies that the Dirichlet problem ( 3 ) , ( 6 ) is uniquely soluble. The following comparison theorem is an important corollary of the maximum principle. T h e o r e m 2. Let the inequalaties

hold for the functions u ( x ) and u ( x ) . Then u ( x ) 5 u ( x ) in the whole domain R. The maximum principle can be applied to deriving simple a priori bounds for Dirichlet problem (3), ( 6 ) in the uniform norm. T h e o r e m 3. The solution of the Dirichlet problem (3), (6) can be estimated as

Ilu(x)llc(n)5 l l d ~ ) I l c ( a n+) M l l f ( ~ ) l l c c n ) ,

(12)

where the constant M depends on the diameter of the domain R and on the coeficients of (3). The a priori bound in (12) reflects that the solution of the Dirichlet problem continuously depends on the right-hand side and boundary conditions. The solution to the third-kind boundary value problem ( 3 ) , (9) can be similarly estimated, given that o ( x ) is separated from zero.

91

STATIONARY PROBLEMS OF HEAT TRANSFER

Theorem 4. Given a(x) 2 rro estimated as

> 0, the solution of problem

(3), (9) can be

where the constant M depends on R, L, and the constant a. Instead of (3) we consider the equation Lu

+ C(X)U= f (x),

x E R.

(14)

The maximum principle yields the following conclusion.

Theorem 5. estimated as

Given c(x)

>

0 , the solution of problem (Id), (6) can be

Simple a priori bounds in (12), (13) and (15) can be extended in different ways (see literature on partial differential equations).

4.1.3 PROBLEMS OF STATIONARY HEAT CONDUCTION IN HILBERT SPACES Let us formulate problems of stationary heat conduction in the space of square-integrable functions on 0 , i.e. in the simplest Hilbert space 'K = Lz(0). The scalar product in 'K is given by

and the corresponding norm is JlyJI= (y,y)1/2. We consider heat conduction problems with homogeneous boundary conditions, i.e. g(x) = 0 for x E 8 0 in (6), ( 7 ) and (9). The corresponding differential operator is defined on this set. For instance, let a differential operator A be defined by the formula

on the set of functions U(X)= 0,

Ea ~ .

(17)

92


The operator A is self-adjoint and positive definite in R ' (A = A' and A 2 6E, where E is the identity operator), i.e. ( A U , ~=) (u,Au), (Au,u) 2 611~[1~, 6 > 0.

(18) (19)

We prove ( 1 8 ) using the formula of integration by parts for a function of many variables

in (20) and derive

We consider the operator A (see (17)) on the set of functions vanishing on the boundary. Therefore (21) immediately yields

Since k,p = kp,, we find condition ( l a ) ,which says that the operator A is self-adjoint. We also obtain (22) for the operator A defined on the set of functions satisfying the condition

Similar computations for the third-kind conditions

93


result in

Thus, the operator of the third-kind boundary value problem is also selfadjoint. Now let us check that the operator A is positive definite on the set of functions in (17), i.e. let us show that ( 1 9 ) is satisfied. According to ( 2 2 ) and the ellipticity condition ( 2 ) , we find the inequality

Consider all functions u ( x ) defined in a bounded domain R. Then the Fnedrichs inequality

holds, where the positive constants cl and c2 depend only on the domain. For the functions u ( x ) that vanish on the boundary a R inequality ( 2 7 ) becomes

If R is a rectangle, namely R = {XI x = ( x l , x z ) , 0 < x , < l,, a = 1,2}, we can obtain the expression cl = n-2/(1,2 + 1;') for c1 in ( 2 8 ) . We combine ( 2 6 ) and ( 2 8 ) and derive the desired condition of positive definiteness

The third-kind boundary conditions ( 2 4 ) with u ( x ) with (25) give rise to the inequality

> a0

>

0 together

94


We take into account the Friedrichs inequality (27) and get the conditions of positive definiteness of the operator A for the third-kind boundary value problem m 00 ( A u , ~2) min (-, -)11~11~. (30) c1 c2 Consideration of the operator of the Neumann problem (16), (23) is essentially based on the Poincari inequality

where the constants c3 and c4 do not depend on u(x). The solution of the second-kind boundary value problem is defined neglecting a constant. Let us separate the unique solution u(z) such that

In this subspace we derive from (22) and (31) the inequality

for the operator A. Consequently the operator A is also positive definite on the set of functions satisfying (23) and (32). More complex problems of stationary heat conduction are considered similarly. In particular, we can mention problems with mixed boundary conditions in which conditions of different kinds are specified on different portions of the boundary. Let us consider the case in which the source term includes a term proportional to the temperature, i.e. let us instead of (1) consider the equation

The corresponding operator A' is self-adjoint. Given c(x) is of a fixed sign if the inequality

2 co, this operator

is satisfied. In the simplest case co 2 0 both operators A and A' are positive definite.

95


4.1.4 A PRIORI ESTIMATION IN HILBERT SPACES

Let us find simple bounds of solutions to problems of stationary heat conduction for equation ( 1 ) equipped with conditions ( 6 ) , ( 7 ) and ( 9 ) . The norm in the Hilbert space W ; ( R ) is defined as

The solutions of boundary value problems ( I ) , ( 6 ) and ( I ) , ( 9 ) are estimated as follows:

The bound in (34) also holds for the solution of the Neumann problem ( I ) , ( 7 ) ,given the normalization condition (32). We can derive a similar bound in the case in which the original equation has a divergent term in the right-hand side. Instead of ( 1 ) we consider the equation

The corresponding a priori bounds for equation (35) supplied by conditions ( 6 ) , ( 7 ) and ( 9 ) look like

Bounds in other norms can be derived using embedding theorems from the bounds of the solution in W t ( n ) . For instance, bounds (36) and (35) yield bounds in the homogeneous norm in the one-dimensional case (m = 1).

4.1.5 PROBLEMS 1. Show t h a t o n e c a n t a k e t h e constant M in t h e b o u n d in (12) for t h e solution of t h e Dirichlet problem (3), (6) t o b e [ l ~ ( z ) l l ~ ~ ~ ~ , w h e r e t h e function v ( x ) satisfies t h e conditions

LW 2 I, 2 0,

W(X)

x E E

n, an.

96


Solution. Let us consider the function

+

4 5 ) = u ( ~ ) l l f ( ~ ) l l c ( Inl)d ~ ) l l c ( a n )

We take (38) into account to derive w ( x ) domain, (39) immediately yields

4%).

(39)

> 0 on the boundary a R . Inside the

We take into account (37) and finding Lw 2 0. Because of the maximum principle, w ( x ) 2 0 in the whole domain 0 . Since w ( x ) is nonnegative, we derive from (39) the bound

i.e. M = IIu(x)llccn, in (12). We thus reduced the problem to finding an a priori bound to the problem of searching for a majorant function u ( x ) satisfying (37), (38). The latter is simple in some special cases. 2. Prove that the operator

is positive definite on the set of functions satisfying the homogeneous first-kind boundary conditions if

i.e. for convective heat transfer in an incompressible medium. Solution. Let us represent the operator A' in the form

where A is the operator defined by (16).The properties of the operator A are studied below. In particular, we studied the condition of positive definiteness (19) (see (29)).According to (41),we rewrite the second term in (42) as

au x

( x )( ''a

)x = -

i

u 2 ( r )div b d x = 0

n

Therefore ( A 1 u , u )= ( A u , u ) 2 ( n l / c l ) l l u ~ l 2where , nl and cl are constants in (2) and (28).

STATIONARYPROBLEMS OFHEATTRANSFER

4.2 Construction of Difference Schemes

4.2.1 APPROXIMATE SOLUTION O F BOUNDARY VALUE PROBLEMS

We solve problems of the physics of heat in an approximate manner by numerical methods. We first discuss the appearance of an approximate solution itself. Suppose we search for the approximate solution of a stationary problem of heat transfer. We denote the exact solution by u(x),x E $2. For simplicity, for the moment we consider a one-dimensional problem, i.e. n = (0,l). We mention the two main classes of numerical methods, namely grid and projective methods. In grid (difference) methods we replace functions of a continuous variable by functions of a discrete variable. In our case, we introduce nodes x, E (0, I), i = 0,1,. . . , N , xo = 0, X N = 1 in the closed interval [0,1] that form the grid Gh = wh awh, where wh is the set of internal nodes (i = 1,2,. . . ,N - 1) and a w h is the set of boundary nodes (a = 0 and i = N). Let h be a parameter of this grid (e.g. the density of nodes or the distance between the nodes). We seek the approximate solution of the continuous problem a t the grid nodes and denote it by yh, = yh(x,), i = 0,1,. . . ,N. In order to determine this function we formulate a difference problem. Let the linear boundary value problem be written as

+

Lu = f (x), LU = g(x),

x E R, E

an

We relate to the continuous problem ( I ) , (2) the difference problem

where Lh and 11, are some difference operators that approximate the operators L and I of the differential problem (I), (3). In projective methods for solving problems of mathematical physics functions of a. continuous argument are also approximated by functions of a continuous argument, which is the principal difference of these methods as compared with the grid methods. We pass to a finite-dimensional problem in the following way. Let the solution of the problem u(x) belong to a space X.In the latter space we separate a subspace X M which is a linear span of the elements

98


wk(x), k = 1 , 2 , . . . , M . We thus search for the approximate solution in the form M

Then we use some reasoning to pose the problem of determining the coefficients ak, k = 1,2,. . . ,M , in expansion (5). Hence, a projective method is characterized by the choice of a basis (the functions wk, k = 1,2,. . . ,M ) and by the way of determining the coefficients in the expansion. Let us mention some approaches to determining ak, k = 1 , 2 , . . . , M . For simplicity, we consider the boundary value problem (I), (2) with the homogeneous boundary conditions (g(x) = 0). We assume that the functions wk(x), k = 1 , 2 , . . . , M , satisfy this condition, i.e. lwk = 0, x E 80. The problem is thus reduced to determining the coefficients ah, k = 1 , 2 , . . . , M so that equation (1)is satisfied approximately. We introduce the discrepancy

that appears when we substitute approximate solution (5) into equation (8). Coefficients of expansion are specified in different ways to provide minimal discrepancy (6). Let us consider the problem in a common Hilbert space 71 = Lz(R). We specify some weight functions yk, k = 1,2,. . . , M by the M conditions

Let us give some methods of choosing the weight functions yk(x), k = 1 , 2 , .. . ,M. In the least-square method we find the minimum of the norm of the discrepancy, i.e. (RM,R M ) + inf. This yields

In the collocation method the original equation is satisfied exactly only at M selected points (collocatiou points xk, k = 1,2,. . . ,M ) . In a more general formulation the collocation method corresponds to the choice yk(x) = 6(x - xk), k = 1.2,. . . , M , where 6(x) is the 6-function. The class of Galerkin projective methods is associated with the choice

The Ritz method is close to the latter (for self-adjoint methods they are simply identical). Let the solution of problem ( I ) ,(2) with the homogeneous boundary conditions be equivalent to the solution of the minimization problem for the

99

STATIONARYPROBLEMS O F H E A T T R A N S F E R

functional J ( u ) = (Lu, u) - 2(f,u). Then it is logical to find the approximate solution UM(Z)of the form (5) as the minimum of this functional. Using conditions (7) we can write the corresponding system of linear algebraic equations for determining ah, k = 1 , 2 , . . . ,M. For example, for the Galerkin method (8) we have M

~(~w,,w,)a,=(f,uri),

i = 1 , 2 ,..., M .

(9)

j=1

Note that in the main projective methods (except for the collocation method) we integrate systems of linear eqnations to determine unknown coefficients in the expansion (see e.g. (9)). Projective-ged methods (finite element method) are now widely used for practical computations. In these methods special elements wk, k = 1,2,. . . , 11.1, (finite elements) are chosen. These functions have local supports, i.e. they vanish everywhere except for a small neighbourhood. In several cases the coefficients a k , k = 1 , 2 , .. . ,M, of (5) can be associated with the approximate solution at some points of the domain 0.Therefore projective methods can be considered as methods for constructing difference problems. The above approaches can also he used to determine the coefficients in the finite element method.

4.2.2 BASIC CONCEPTS IN THE THEORY OF DIFFERENCE SCHEMES In the course of the numerical solution of applied problems by difference methods we encounter the following problems related with one another. The problem is to obtain the approximate solution with a given accuracy for appropriate computational expense. For this reason we pass from the differential t o a discrete problem. When we construct discrete problems (approximate equations and boundary conditions), it is desirable that the difference solution contains the most important qualitative characteristics of the sought-for solution. As an example, we mention conservativity, i.e. conservation laws must be satisfied for the difference solution as well. The difference solution should converge to the exact solution as the grid becomes finer. It is desirable to estimate the discrepancy between the difference and exact solutions versus the parameter h. The theory of convergence of difference schemes studies the discrepancy and stability of a difference solution with respect to small perturbations of the right-hand side of the equation and boundary conditions. In order to derive the approximate solution, we should solve the corresponding system of linear eqnations. For this purpose we develop various methods for solving grid problems which most take into account the specific problem.

100


Let us introduce some basic concepts of the theory of difference schemes. We again solve problem (I), (2) (not necessarily onedimensional) reducing it to a difference problem (3), (4). Choosing different grids CJh we obtain a set of difference solutions {yh) depending on the parameter h. We thus consider the family of difference problems (3), (4), which is referred to as a difference scheme. We choose the grid C7h ( h ( ~ )is a parameter) according to the desired accuracy of the approximate solution E . Let Hh be the space of grid functions We and uh(x) be the values of the exact solution u(x) at the grid nodes of 31,. use the notation zh = yh - uh, x E Zh for the discrepancy of the difference solution. The main problem of theoretical research is to estimate the discrepancy of the solution a priori. Let us formulate the problem for zh. We substitute yh = zh + uh into (3) and (4) and obtain

Here Qh = p h - Lhuh is the residual in the equation for the discrepancy, which is called the error of approximation of equation (1) by difference equation (3). Respectively, uh = x h - lhuh is the residual in the boundary condition for the discrepancy, which is called the e n o r of approximation of the boundary condition (2) by the difference boundary conditions (4). We stress that these errors of approximation are considered on the solutions of the original problem ( I ) >(2). The solution of difference problem (3), (4) converges to the solution of problem (I), (2) if llzhll~h= llYh - uhlllh -+ 0 as h -+ 0 in some grid norm I/ . llIh. The difference scheme converges at the rate O(hk)),k > 0, (i.e. it has the kth order of accuracy) if for any h 5 ho, however small, l l ~ h l l l h5 M h k , where the positive constant M is independent of h. Difference scheme (3), (4) approximates problem ( I ) , (2) if ll$h1(2h -+ 0 as h -+ 0 and IIVh113h -+ 0 as h -* 0, where 11. l l ~ hand II.113h are some grid norms. Stability of the solution with respect to small perturbations of the righthand side and boundary conditions is an important characteristic of a difference problem. The stability of the difference scheme for linear problem (3), (4) follows from the relation

This relation yields a similar relation for the discrepancy

Consequently, if the scheme is stable and approximates the original problem, it converges.

101


Investigation of convergence for linear difference schemes is thus reduced to studying the errors in approximating the original equation and the boundary conditions and deriving a priori bounds for the discrepancy of the difference solution. 4.2.3 SIMPLE DIFFERENCE OPERATORS Let us consider approximation for the simplest operators of the first and second derivatives. We denote the uniform grid J (the subscript h is omitted) on the interval = [0,1] with the step h (w is the set of internal nodes), namely G = {XI x = x, = ih, i = 0,1,. . . , N , N h = I). Let us examine different approximations of the differential operator of the duldx on the grid introduced. We consider smooth first derivative Lu functions u(x) E d k ) ( 0 ) ,k 2 2. We relate to difference operator Lh the differential operator L. We call a stencilthe set of points involved in the construction of the operator Lh. Let us define the error of approximation of the operator L by the difference operator Lh at the ith node by the formula $i = Lhui - (Lu),. The Taylor series in a neighbourhood of the internal node x = x, is written

=

Therefore the left-hand difference derivative is (we omit the subscript i )

Thus the operator Lh approximates the operator L with the first order ($ = O(h) at each internal node) for u(x) E C(')(0). Similarly, the right-hand side difference derivative looks like

If we use a three-point stencil (involving the nodes use the central difference derivative

xi and x,+l), we can

which approximates the derivative with the second order if u(x) E C(3)(0). Similar algebra for the second derivative Lu = d2u/dx2 yields

102


The latter operator approximates the second derivative with the second order if u(x) E C(4)(R). The approximation error is separately estimated at each node. In order to evaluate the approximation error on the grid w, we should use grid norms. For example, we can use the norms in C(w) and L z ( w ) ,namely

In the above examples (14)-(17) the approximation error had the same order in both norms. Of course, this is not always the case. The approximation order can be different in different norms. As an example, let us consider an approximation of the operator of the second derivative on a nonuniform grid yj = {XIx = xi, i = 1 , 2 , . . . , N , xo = 0, X N = 1) with the steps hi = xi-xi-1. We introduce the difference operator

on the three-point stencil, where hi = 0.5(hi+l +hi). Direct computations yield the approximation error

Therefore IIvllc = O(ho) and II$II = O(ho), where ho = maxh,, i.e. we only rEu get the first-order approximation. Investigation of the structure of the approximation error shows that operator (18) still has second-order approximation in an appropriate norm. For instance, let

We take into account

and find

STATlONARYPROBLEMS O F H E A T T R A N S F E R

103

for the principal term of the error. According to (19) we represent the approximation error in the form $, = $: $f, where

+

We thus obtain

II$PII-l

= O(hi). Therefore II$II-l 5 We take into account (20) to find 5 Mh;, i.e. operator (18) approximates the second 1$:-1 11$:ll-1 derivative with the second order on each nonuniform grid. This is because of a special (divergent) form of the error and is associated with the choice of the norm. The above difference operators are obtained by the Taylor formula (operators of the first derivative) and by repetitive differentiation (difference operators (17) and (18) for the second derivative). Alternatively, difference operators can be found from the definition of a derivative as a solution to a corresponding integral equation applying some quadrature formulae. Let d k u / d x k = f ( x ) , then

+

We thus can define the kth derivative as a solution of integral equation (21), given u(x). For example, let us construct difference operators that approximate the first derivative (k = 1 in (21)). On a uniform grid w we have

/

T.+1

u

-1

=

j ( t ) dt.

(22)

Ti-,

We apply the rectangle quadrature formula to estimate the integral in (22) and derive the central difference derivative u,+1 - ui-1 = ji 0(h2). 2h If we use quadrature formulae with several internal nodes, we obtain the socalled compact difference opemtors. For example, the Simpson formula applied to (22) leads to

+

104


In this case we can increase the order of approximation without formal expansion of the stencil. However, we then have to invert a tridiagonal matrix to evaluate the derivative.

11

4.2.4 THE METHOD OF DIRECT APPROXIMATION

Difference schemes for problems with smooth coefficients in regular domains can be constructed by passing from differential to difference operators. Let us illustrate this approach by a model one-dimensional boundary value problem of stationary heat transfer. We consider the equation

for k(x) 2 n i > 0 equipped with the mixed boundary conditions

Let us construct a difference scheme for the problem (23)-(25) on the uniform grid 3 with the step h. Let us consider the difference relation

According to (14) and (15) we get

At the internal nodes of the grid w we replace differential equation (23) by the difference equation

where ip, = f(x,),i = 1 , 2 , . . . , N- 1. In order to determine a,,let us compare (261 and


105

If a, are chosen so that

we approximate the original differential equation (23) by difference equation (27) with the accuracy O ( h Z ) . In particular, equations (28) are satisfied if we determine a; as

Difference equation (27) is supplemented by the condition

(see (25)). It is of interest how to approximate the third-kind boundary condition (24). We replace the first derivative for x = 0 by the right-hand difference derivative and come to the following approximation of boundary condition (24): (31) -k(O)y,,o YO = g ~ . The boundary condition is approximated on the two-point stencil with the first-order accuracy. To improve the accuracy, let us consider the difference expression a l y Z s oon solutions of problem (23)-(25).Similar algebra yields

We take into account equation (23) and boundary condition (24) and derive the following second-order approximation:

We can obtain a higher order of accuracy by approximating the boundary condition on the solutions of the original differential problem. This technique

106


Fig. 4.1.

is widely used when constructing difference schemes with increased accuracy. Below we discuss certain examples. Boundary value problems for elliptic equations are considered similarly. For simplicity, we only study two-dimensional problems. Usually there is no problem about passing to a problem of a higher dimension, therefore we do not cover them in the book. We assume that the computational domain is the rectangle 0 = 1x1 x = ( x l , x z ) ,O < x, < l,, a = 1 , 2 ) . We introduce a grid w uniform in both directions with the steps hl and h z , namely i = O , l , .. . , NI, = (ihl,jhz), j = O , l , . . . ,N z , N , h , = l , , u=l,2},

a = { x l x = x,j

and let w be the set of internal and aw the set of external nodes (Fig. 4.1). We can approximate elliptic equations using the above approximations for ordinary differential equations. For example, we have a problem of stationary heat transfer in a nonhomogeneous medium with a given temperature regime on the boundary, i.e. u ( x ) is defined as the solution of the following Dirichlet problem:

We relate to the boundary value problem (33)-(35) the difference problem


Fig. 4.2.

Taking into account (27) and (35) we put 2

Ay =

1Amy,

LY= - ( a m ~ s ) s. m

(38)

m=1

According t o ( 2 9 ) , we set the coefficients of difference scheme ( 3 6 ) , ( 3 8 ) to be e.g. a l ( x l , x z ) = k(x1 - 0 . 5 h l r x 2 ) and a z ( z 1 , x z ) = k(x1,xz - 0.5hz). Difference problem (36)-(38) approximates the differential problem with the second order if the coefficients and solutions of problem (33)-(35)are smooth. Approximation of heat transfer problems in anisotropic media, in which the heat equation contains mixed derivatives (see Section 2.1), are worthy of notice. We consider the Dirichlet problem ( 3 3 ) , ( 3 4 ) , where

with appropriate restrictions on the coefficients k,p. For problem ( 3 3 ) , (34), (39) we consider difference problem ( 3 6 ) , (37) provided that 2

Ay =

Lpy. (40) o.p=1 Each of the operators Lop is approximated by a difference operator, namely

Difference scheme (36), (37) is considered, in this case, on the nine-point a h l , x ~ phz), a,P = 0 , f l . stencil (Fig. 4.2) involving the nodes ( X I In particular, if the coefficients of equations (33) and (40) are constant, approximations ( 4 0 ) and (41) yield

+

+

108


Let us represent the operator A,p in the form

It is easy to check that the operators A;p and A& have the first-order and Amp has the second-order approximation. For cu = we have

Let us now consider the operator with the mixed derivative

1\12 Y = -$ ( k 1 2 ~ , ) , , . Taylor's formula yields (see (14) and (15))

We put v = k12uz2 and find

Similar computations lead to

whence we conclude that A12u - L12u = 0 ( J h J 2 )


109

Difference scheme (36), (37), (40) is considered similarly if we use the approximation

instead of (41). Boundary conditions for the problems considered in the rectangle are approximated as in the one-dimensional case. We do not discuss this in more detail. Problems in irregular domains are discussed specifically.

4.2.5 CONSERVATIVE SCHEMES Basic differential equations are derived by applying conservation laws (in the integral form) to elementary volumes and then tending these volumes to zero. This passage to the limit results in the differential form of the conservation laws. The method of finite differences is actually the inverse transition from the differential to the difference model. It is logical to require that the conservation laws be satisfied after this transition. Difference schemes that express the conservation laws on a grid are called conse~uatiuedzffeerence schemes. The conservation laws for the whole grid should be an algebraic conclusion of the difference equations. Let us give an example of a nonconservative scheme that diverges if the thermal conductivity is discontinuous. We consider the problem

Let us perform differentiation in the left-hand side of (42) and rewrite it as

According to the principle of direct approximation, we relate to problem (43), (44) the difference scheme

Let us show that scheme (45), (46) diverges in the class of piecewise constant thermal conductivities

110

I


We assume that the conditions of ideal contact

=

[u] u(E

+ 0 ) - u(€- 0 ) = 0 ,

kg]

=O

are satisfied at the discontinuity point x = E. Under these conditions problem (41), (42) has a piecewise linear solution

The constants a0 and Po are defined by the junction conditions a0 = ( X + ( 1 - x)E)-', PO= xao and x = k ~ l k z . Let us now solve the difference problem (45), (46). Let I = x , +Oh, where x, = n h , 0 < 0 < 1. Since the thermal conductivity is piecewise constant, we obtain yz,,, = 0 for all a # n and a # n + 1. Therefore

In order to determine the coefficients cu and and x = x,+l. We thus find

- x,+1) - ( 1 -ax,))

b,(P(l b,+lPh

a, let us rewrite (45) for x = x , + anah = 0 ,

+ a,+~(P(l- %,+I) - ( 1 - ax,))

+

(48)

= 0,

+

where a, = (5k1 - k2)/4, a,+' = (kl 3k2)/4, b, = ( 3 k ~ k2)/4 and b,+l = (5k2 - k 1 ) / 4 . We take into account that x , = - Oh and x,+l = E ( 1 - 8 ) h and derive from (48)

Passage to the limit as h

-3

+

0 yields

We use linear interpolation to complete the definition of the grid functions in the whole interval 0 5 x 1 and tending h -t 0 obtain the limiting function

<


111

We compare (49) with (47) and find that the limiting function %(x) is identical to the exact solution u(x) of the problem only for Eo = a0 and = Do, which is only possible if x = 1, i.e. kl = kz. Therefore, the difference scheme (45), (46) diverges if kl # kz. It can easily be seen that the function E(x) is the solution of problem (42), (43) with the piecewice constant thermal conductivity that satisfies the nonhomogeneous junction conditions

Po

The capacity q of the concentrated source of heat varies in a wide range depending on X. In particular, 0 + fco as y, -+ 5 f 0. Thus, there is a physical reason for divergence of scheme (45), (46). Namely, au additional source (outlet) of heat appears at the point x = x which breaks the balance of heat (conservation law). Therefore, it reasonable to construct conservative difference schemes. 4.2.6 INTEGRO-INTERPOLATION METHOD When constructing conservative difference schemes, it is logical t o start with conservation (balance) laws for separate cells of a difference grid. This method of constructing conservative difference schemes is called the integrointerpolation method (balance method). This approach is close to the method of a control uolume, in which the differential formulation of the problem is not used, while the conservation laws are taken into account for separate cells in the medium. Let us show how the integreinterpolation method is applied for constructing a difference scheme for model one-dimensional problem (23)-(25). Let us denote the heat flux by q(x) = -k(x)du/dx. We construct the difference scheme on the grid 9. We integrate heat equation (23) over the interval X , - I / ~ 2 i xi+1/2 to get

<

The balance relation (50) reflects the law of conservation of the amount of ~ . quantity qi-l/2 is the amount heat for the interval xi-1/2 x x , + ~ / The of heat incoming through the cross-section and qitl/z is the amount of heat outgoing through the section X,+~,Z.The unbalance of these fluxes is due to distributed sources (the right-hand side of (50)). Although in the integreinterpolation method we started from equation (23), in the method of control volume the same relation (50) is derived immediately.

< <

112


In order to derive a difference equation from the balance relation (50), it is necessary to complete the definition of grid functions. Let us express the fluxes at half-integer nodes in terms of the values of the function u(x) at integer nodes. For this purpose we integrate the expression duldx = -q(x)/k(x) over x 5 xi and get the interval xi-]

<

The quantity a, in (52) has the sense of a mean thermal resistance of the x xi+lj2. According to (50) and (53) we write difference interval x,-lj2 scheme (27) with the right-hand side

< <

for the internal nodes. The first-kind boundary condition (25) is approximated by (30). In order to approximate (24) we again use the integr&interpolation method. Let us x xljz. The corresponding integrate equation (23) over the interval 0 boundary condition becomes

< <

~ . determine qo we use boundary As in the above, we obtain 4112 = a ~ u , , To condition (24), namely qo = gl - fluo. This allows us to rewrite (32) as

Coefficients of the difference scheme are calculated using quadrature formulae. For example, for smooth coefficients the first formula in (29)


113

corresponds to the use of the rectangle formula for computing the integral ( 5 2 ) and the third formula to the trapezoid formula. Comparing with the scheme ( 2 7 ) , (30), (32), we can conclude that the conservative difference scheme has the second-order approximation for smooth coefficients and solutions. This scheme belongs to the class of homogeneous differenceschemes (the coefficients of the difference equation and boundary conditions are computed by the same formulae for every node of the grid). The difference scheme for the model heat transfer is similarly constructed by the balance method in the case of a nonuniform grid. Let = xi-hi12 and x , + ~ = / ~x, + hi12 for the nonuniform grid. Then we obtain the difference equation

for the internal nodes of the grid. The coefficients in (55) are computed by the same formulae ( 5 2 ) ,(54) as in the case of the uniform grid. The boundary conditions are of the form ( 3 0 ) , (32). Let us introduce the Steklou averaging operators by the relations

In the notation of (56) the coefficients of the conservative scheme are defined as a ( x ) = ( S - ( ~ - ' ( X ) ) ) -and ~ ~ ( x=) S f ( % ) .The balance equation (50) is obtained if we act on the original equation ( 2 2 ) by the averaging operator S , namely

We can introduce the operator of repetitive averaging

and construct a difference scheme by applying the operator T to the original equation. We take into account (57) and derive

114


for the averaging operator T . By this version of the integrwinterpolation method we obtain the difference scheme (27) with the coefficients

for equation ( 2 3 ) . Difference schemes for multidimensional problems are also simply constructed by the integrointerpolation method. Certain more complicated cases can be considered separately. We now dwell on the stationary problem (33)-(35) of heat transfer in a homogeneous medium. Similarly to ( 5 6 ) , we specify averaging operators along certain directions, e.g.

Similarly to ( 5 6 ) , ( 5 7 ) and ( 5 9 ) , we define the operators S z and T,, a = 1 , 2 . It is logical to define the averaging operator in the plane as the product of the corresponding one-dimensional averaging operators, namely S = SlS2 and T = TlT2. Let us construct a difference scheme on a uniform grid for problem (33)-(35) by the standard version of the integrointerpolation method. For this purpose we consider the balance equation for a rectangle

n,, = { X I X = ( X I , X Z ) x,

~

X2.j-112

.

5~x1 -5 ~ ~ ~

<

22

5~2,~+1/2).

This procedure corresponds to the action of the operator S on the original equation, namely

SLu = S f ( x ) ,

x E w.


As a result we derive difference scheme (36)-(38) with

The version in which the averaging operator T ((27) and (58) in the onedimensional case) acts on the original equation is considered similarly. The same technique can he used for considering difference schemes for the general stationary heat equation (33), (39), including problems with thirdkind boundary conditions.

4.2.7 DIFFERENCE SCHEMES OF THE FINITE ELEMENT METHOD Difference schemes can be constructed by the finite element method. We again consider the model onedimensional equation of stationary heat transfer (23). For simplicity we consider the boundary conditions

We use the uniform grid Gi with the step h. We employ the Ritz method to construct a projective-difference scheme. The problem (23), (60) is equivalent to minimization of the functional

We search for the approximate solution of the problem to minimize the functional (61) as an expansion (see (5) with M = N - 1) N-1

UN-I(X) =

a.tw.t(w). k=1

@I

116


Fig. 4.3.

We choose the functions wk(z), k = 1,2,. . . ,N - 1, in the form (Fig. 4.3)

w~(x= )

(x - xi-1 )/h,

xi-1 5 x 5 xi, xi 5 x 5 Xi+l. x > Xi+l.

(63)

The coefficients of the expansion are defined by the system of linear equations (9). Note that if the functions wk, k = 1 , 2 , . . . , N - 1, are chosen according to (62) and (63), we have yk = a h , k = 1,2,. . . ,N - 1, and we come to the equation for determining the grid function yk, k = 1 , 2 , . . . ,N - 1. Direct computations yield

The derived tridiagonal system for determining the coefficients as, k = 1,2,. . . , N - 1, can be rewritten as the difference equation (27). Relations (62) can be rewritten in the form (58). Hence, the above scheme of finite elements is identical to that obtained by the integrointerpolation method. If the basic functions are taken to be piecewise polynomials of higher degree (quadratic, cubic, etc.), corresponding difference schemes are similarly constructed.


Fig. 4.4.

We illustrate the construction of finite element schemes in a multidimensional case by a simple problem of stationary heat transfer in a homogeneous medium, in which (see Section 4.1) heat transfer is described by equations (33), (35) with k(x) = 1 and (34), where we set g(x) = 0. This problem is equivalent to minimization of the functional

To construct a finite-dimensional subspace, let us divide the computational domain Cl into elementary cells. In the tw~dimensionalcase it is reasonable to choose triangles as such cells. The approximate solution is a linear function in each of these cells. We introduce the uniform rectangular grid with the steps hl and hz and divide a neighbourhood of the node (XI,,x2j) into right triangles by diagonals that pass, for instance, through the nodes (xli,xzj) and (XI,,-I,xz,,-I).We define the basic functions associated with the internal nodes (xl,,xz,) by the formulae (Fig. 4.4) I

w,, (x) = <

118


In this version of the finite element method the difference scheme has the form (36),(38) with a1 = a2 = 1 and

More valuable projective-difference grids can be found in an ample number of books on the finite element method. 4.2.8 DIFFERENCE SCHEMES WITH AN INCREASED ORDER OF

APPROXIMATION If both the coefficients and boundary conditions of an original differential problem have increased smoothness, we can construct difference schemes with an increased order of approximation. In the above we considered two examples of second-order equations with the second-order approximation. Difference schemes with higher orders of approximation can be constructed in several ways. Let us illustrate possible methods by the example of constructing difference schemes with an increased order of approximation for the equation

The first way is to approximate differential operators by difference ones on extended stencils. For instance, we can use a five-point stencil instead of the usual three-point stencil. As a result we get

We can employ the Taylor expansion to show that the operator A approximates the operator L with the fourth order, provided u ( x ) E C(". We thus relate to (65) the difference equation

where, in the simplest case, ~ ( x=) f ( x ) . We can approximate (65)with a higher order using compact approximations rather than formally extending the stencil. We take into account the integral definition of the derivative (21) with k = 2, apply the Simpson formula and derive difference equation (66j with

Y =-

'P(x)= & ( f ( x - hj

+ l O f ( x )+ f ( x + h ) ) .

(67)


119

Schemes of higher orders for one-dimensional problems can be constructed using exact difference schemes. The grid solution for the latter is identical to the exact solution of the differential problem at the nodes. The exact scheme for (57) is constructed by applying the averaging operator T (see (49)) to the original differential equation. Since (see Problem 1)

the exact difference scheme is written as

We can obtain from (68) schemes of desired accuracy using appropriate quadrature formulae. In particular, the above scheme (67) belongs to this class. The order of approximation can be efficiently increased on the solutions of the approximated equations. We have already encountered this phenomenon when constructing approximation (32) of the third-kind boundary condition. Let us consider approximation on the solutions for equation (65). We use the usual three-point scheme and obtain an increased order of approximation by correcting the right-hand side. Thus, the operator A in scheme (66) is defined by (67) and

Let us consider the approximation error ij, = p - Au. We have

Consequently,

h2 +O(h4) 12 because the approximation on the solutions is L2u = Lf. In order to get the fourth-order approximation, it is sufficient, for example, to put $=f+r-Lu+-Lf

I

in (69). The second version is completely identical to scheme (67). Thus, the order of approximation can be increased in different ways. Similar approaches can also be used in multidimensional problems. As a typical example, let us consider a scheme with an increased order of approximation

120


on the solutions of the Poisson equation (33),(35) (k(x) = 1) on a uniform rectangular grid. Let us consider the operator A defined by (38)with a, = 1,a = 1,2.Similar to the one-dimensional case, the error of approximation is expressed as

The solution of (33)yields

Substitution of this expression into (71) yields

We replace LlL2 by the difference expression

and get

+

h: h$ AIY + h y - 7 A I ~= Yd x ) ,

(72)

where, e.g.

h:

d x )= f ( ~+ ) 12f z W 1 +

hi

fz211-

(73)

Difference equation (72), (73) approximates the Poisson equation on the solutions with the fourth order, provided u(x) E C(6)(R)and f(x) E d 4 ) ( R ) . 4.2.9 PROBLEMS 1. S h o w t h a t t h e relations

du dx

S+ -(x)

= u,(x),

du S- -(x) = us(x), dx

d2u dx

T ( x ) = ( x ) (74)

hold for averaging o p e r a t o r s (48) a n d (49). Solution. According to the definition of the operator S+ (see (56)) we have

121


The operator S- is considered similarly. By definition, the operator T is the square of the operator S. After some algebra we find

We thus prove the third relation in (74). 2. C o n s t r u c t t h e exact three-point difference e q u a t i o n for secondorder o r d i n a r y differential e q u a t i o n (23). Solution.Let us integrate equation (23) from xi to x and divide the resultant by k(x):

Repetitive integration of (75) with respect to x from xi-1 to x, yields

Similarly, integration from x, to xitl results in

The last two equations yield difference equation (27), provided that the coefficient a is defined according to (52) and the right-hand side is =.+I

1 P' -h

(

I

z.

+

d

=.

d

)

.

(76)

=.-I

If we use quadrature formulae for computing the integrals in (52) and (76), we obtain a difference equation with the desired approximation error.

122


4.3 Uniform Convergence of Difference Schemes

4.3.1 CANONICAL FORM OF A DIFFERENCE EQUATION The maximum principle is used for studying stability and convergence of difference schemes in the uniform norm. It is employed for generic difference equations written' in a canonical form. Let g be the set of nodes (the grid) in a bounded domain II. A difference scheme is written on a stencil m ( x ) associated with the node x. We denote by 97.2' all points of the stencil 9Jl(x) forming a neighbonrhood of the node x except for the node x itself. We assume that the difference solution is defined as the solution of the difference equation written in the form

This is the canonical fonn of a dafference equation. As an example we consider the first-kind boundary value problem in a rectangle for the stationary heat equation in an isotropic medium

with smooth coefficients and solution. In a usual way (see Section 4.2), we consider the corresponding difference problem on a uniform rectangular grid L7

m,

Y(X)= xEa ~ , (5) where e.g. a l ( x ) = k(xl - 0.5hl,xz) and az(x) = k(x1,xz - 0.5hz). Let us rewrite difference equation (4) in the canonical form (1). The coefficients A(x) and B(x) and the right-hand side look like

F(x) = pi,, at the internal grid nodes.

x = x,j E w

123


For the boundary nodes we can consider m f ( x ) to be the empty set. Then boundary condition (5) is rewritten in the canonical form ( 1 ) as A ( X ) = 1, F ( X )= g ( x ) , E aw. (7) Thus, difference problem ( 4 ) , (5) is written in the canonical form (1) with

A(%)> 0,

> 0,

x E (z), A(x) =

x

B(x,F), x E w. (8) CE~C.) These properties of the coefficients turn out to be of most importance when formulating the maximum principle for difference equations. B(x,O

4.3.2 THE MAXIMUM PRINCIPLE The maximum principle holds for difference equations as for elliptic equations (see Section 4 . 1 ) . This is because a difference equation involves properties of continuous problems. We formulate the maximum principle for difference equations written in canonical form ( 1 ) . Let W (W E z ) denote a set of nodes involved by a difference scheme written in canonical form ( 1 ) and let w = u !JJr(x). 1EW

For example, W is the set of the internal nodes in the case of Dirichlet problem ( 4 ) , (5). The set of nodes is called a connected grid if for any x' E W and s" E W a sequence of nodes X I , x z , . . . , xb can be specified so that !JJr'(z1), Z, E m l ( x l ) , .. . ,Zk E m ' ( ~ k - ~ ) 2'' , Emf(xk), i.e. each sequential node belongs to a neighbourhood of the preceding one. Let us introduce the grid operator Zl E

B ( x ,O

A Y ~=) A b ) y ( x ) -

Y(~)

(9)

Cm'(.)

and

are satisfied for the coefficients of the grid operator A . Then difference equation ( 1 ) on the subset of nodes W can be rewritten as Ay(x) = F(x), x E W. (12) The maximum principle holds for grid equation (12) (cf. Theorem 1 in Section 4 . 1 ) .

124


rn

Theorem 1. Let y(x) $ const on a connected grid and let conditions (10) and (11) be satisfied. Then if Ly(x) _< 0, x E W (Ly(x) 0, x E W ) , then y(x) cannot achieve a positive maximal (negative minimal) value on W .

>

This is proved by contradiction. Let the condition

be satisfied. Assume that the maximal positive value of the grid functions y ( x ) is attained at a point x' E W , i.e.

Taking into account (9) and ( l o ) ,we find

>

0 under Given the conditions ( l l ) , we obtain from (15) that L y ( x l ) assumption (14). This does not contradict (13) if Ly(xl) = 0. The latter takes place if both terms in the right-hand side of (15) are equal to zero, i.e.

Since y ( x l ) > 0 and B(x', () > 0, we obtain

Since y ( x ) f const, there exists a point x" E W such that y(xl') < y ( x l ) . Since the grid W is connected, there exists a sequence of points x l , x z , . . . ,xk such that each subsequent term belongs to a neighhourhood of the preceding one. We take into account (16)to conclude that y(x1) = y(xl).We repeat the above reasoning first for the node X I , then for the node xz and so on. As a result, we get y(xl) = y(x1) = y(x2) = . . . = y ( x k ) Therefore

This contradicts condition (13),consequently assumption (13) is not valid.


125

The case of a negative minimum is considered similarly replacing y(x) by -y(s). This completes the proof. We emphasize once more that the maximum principle is established for any connected subset of nodes W of the difference grid E.

4.3.3 UNIQUE SOLVABILITY OF DIFFERENCE PROBLEMS The maximum principle can be used to prove uniqueness of a solution to boundary value problems for elliptic and parabolic equations. In the case of a difference problem the situation is better because existence of a solution can also be shown. A difference problem written in the canonical form (1) is actually a system of linear algebraic equations in which the number of unknowns is equal to the number of equations. Therefore, in order for this problem to be uniquely solvable for any right-hand side, it is sufficient that the corresponding homogeneous equation has only a trivial solution. Two cases appear when considering difference equation (1) on the whole grid 3 for each node x. In the first case, the neighbourhood of the node Dtl(x) is an empty set, and this node is called a boundary node (see e.g. (7)). In the second possible case (of an internal node) 'IlI1(x) contains at least one node. If all nodes are internal, then

Let us formulate the conditions, depending on whether there are boundary nodes, of unique solvability of the difference problem written in the canonical form (1) that immediately follow from the maximum principle (W = w ) . Corollary 1. Let condition (11) be satisfied for difference operator (9), ( I D ) on a connected grid E and let aw # 0 . Then diffe~enceproblem (1) is uniquely solvable. We should show that the homogeneous equation

has only a trivial solution. Obviously, y(x) 0, x E Z, is a solution of (17). Let us prove that there are no other solutions. For boundary nodes x E aw we have y(x) = 0, provided that (11) is satisfied. The maximum principle holds for internal nodes of w , which says that, on the one hand, y(x) 5 0 (a positive maximum cannot be attained) and, on the other hand, y(x) 2 0 ( a negative minimum cannot be attained). This is only possible for y(x) 3 0, x E E.

126


Let us now consider the case in which the grid contains no boundary nodes. Corollary 2. Let conditions (11) be satisfied for dzflerence operator (91, (10) on a connected grid 3 and let there exist at least one node I' of the grid w where x E w. (18) D(xl) > 0, Then difference problem (1) is uniquely solvable. We can apply the maximum principle to homogeneous equation (17) ( 3 = w). Therefore y(x) 5 0 (y(x) 0), i.e. the only possibility is y(x) = const, x E w. For such y(x) we have

>

at the point x'. Hence, y(x) = y(xf) = 0, x E w. Corollary 1 can be used to state unique solvability of difference Dirichlet problem (4), (5) with coefficients satisfying conditions (7) and (8). Corollary 2 is employed to prove unique solvability of problems with third-kind boundary conditions. As a typical example, we consider the problem of stationary heat conduction with convective heat exchange with the environment (see Section 4.1). The desired function u(x) satisfies equation (2) in the rectangle fl and the third-kind boundary conditions

The original equation (2) is approximated by difference equation (4) at the nodes of the uniform grid w inside the domain Q . Third-kind boundary conditions (19) are approximated with the second order on the solutions of problem (2), (19) as in the one-dimensional case (see Section 4.2). For instance, for x = (O,xzj), j = 1,2,. . . ,NZ 1, we have

-

127


This results in the second-order approximation that can be written as

Similar approximations are used at the other points on the boundary a n . When writing the difference scheme in canonical form (1)at nodes inside 0, we get expression (6). For the nodes on the boundary an we find D(x) 2 u(x). According to Corollary 2, the difference problem that corresponds to the thirdkind boundary value problem (2), (19) is uniquely solvable, provided that u(x) 0 and u(x) > 0 at least at one node on the boundary 80.

>

4.3.4 COMPARISON THEOREMS The maximum principle is used to prove comparison theorems for solutions of various problems. An example of such a result is provided by Theorem 2 in Section 4.1. Similar results also hold for difference problems (1).Let us separately consider the cases with (aw # 0)and without (aw = 0)boundary nodes. T h e o r e m 2. Let conditions (11) be satisfied for dqtference operator (g), (10) on a connected grid Z and aw # 0 . Let the inequalities

hold for grid functions y(x) and z(x). Then y(x) of the grid.

< z(x) at the internal nodes

To prove this, let us consider the function v(x) = y(x) - z ( x ) . According to (12) Av 0 and y(x) 0 , x E aw. By the maximum principle u(x) 0, x E w , which completes the proof. The following corollary of the comparison theorem is useful for estimating the difference solution.

<

<

<

Corollary 3. Let y(x) and Y(x) be the solutions of the problems

In this case, if

then Jy(x)l< Y(x) for all x

E

w.

The proof is based on Theorem 2 applied to the functions y(x) (-y(x)) and

Y(.).

128


It is logical to call the grid function Y ( x ) in this conclusion a majorant function for the solution of difference problem (22).It is defined as the solution of problem (23) under restrictions (24). If there are no boundary nodes, the comparison theorem is formulated even simpler.

Theorem 3. Let conditions ( 1 1 ) be satisfied for difference operator (91, (10) on a connected grid w = w and let there exist at least one node x' of the grid w at which condition ( 1 8 ) is satisfied. Then if Ay Az, x E w , then y ( x ) 5 z(x) at all the grid nodes.

<

This immediately follows from the maximum principle applied to the function v ( x ) = y ( x ) - z ( x ) . The majorant function is defined similarly.

Corollary 4. Let y ( x ) and Y ( x ) be the solutions of the problems

respectively. In this case, if

then \y(x)l

< Y ( x ) for

all x E w

The comparison theorems presented are used to construct simple a priori bounds for the solutions of difference problem ( 1 ) .

Corollary 5.

The estimate

holds for the solution of the problem

To prove this, we define a majorant function Y ( x ) as a solution of the problem

By Corollary 3, Jy(x)l5 Y ( x ) ,x E w. For the function u ( x ) = Ilg(x)llccaY)Y ( x ) we have Av > 0, x E w , and v ( x ) = 0, x E aw; therefore v ( x ) 2 0, x E w . Consequently Y ( x ) < Ilg(x)llccaul,and inequality (28) holds. Let us present the simplest a pr~oribound involving the right-hand side of difference equation ( 1 ) .


Corollary 6. If D ( x ) > 0 for x E

= w , then the estimate

holds for the solution of the problem

Let Y ( x ) be the solution of the equation

Then by Corollary 4 ly(x)l 5 Y ( x ) , x E w. The majorant function achieves its maximal value at a node x'. Let us write equation (30) at this node:

>

However, Y ( x l ) Y(E),therefore D ( x ' ) Y ( x l )5 IF(xf)l.Consequently

We thus derive the desired hound for the majorant function 4.3.5 STABILITY AND CONVERGENCE OF THE DIFFERENCE DIRICHLET PROBLEM The above results can be applied to deriving a priori hounds of solutions of the difference Dirichlet problem (4), ( 5 ) and to investigating the convergence rate. Difference problem (4), ( 5 ) is written in the form (22), where the grid operator A has relevant properties. We represent the solution of problem (22) as the sum y ( z ) = y ( l ) ( z )+ y ( 2 ) ( x ) , Z E z, (31) where y ( l ) ( z )is the solution of the homogeneous equation with nonhomogeneous boundary conditions

) the solution of the nonhomogeneous equation with homogeneous and y ( 2 ) ( x is boundary conditions

130


We thus derive a bound for the solution of problem (32) using Corollary 5, namely I l ~ ( " ( ~ ) l lS c (I~l d~ ~ ) l l c ( a ~ ) . (34) In order to derive bounds for the solution of problem (33) for the nonhomogeneous equation we need to construct the corresponding majorant function Z(x). Let us have (cf. the Dirichlet problem ior an elliptic equation, Problem 1 in Section 4.1) a grid function w(x) such that

According to Corollary 3 we can take the majorant function for (33) to be Z(x) = w ( ~ ) ~ l F ( z ) lTherefore \ ~ ( ~ ~ . we obtain the bound

where M = I l w ( ~ ) l I c ( ~ ) . In view of (31), (34) and (36) we find the a priori bound

for the solution of difference Dirichlet problem (22). The constant M in this bound is defined as the maximal solution of (35). In order to estimate a difference solution like (37) it is important to show that the constant M can be specified independently of the grid. Let us find a typical bound for M. Let R be a rectangle whose sides 11 and l2 satisfy the inequality 11 5 12. We assume that the function w(x) satisfying conditions (35) only depends on the variable xl. We can take into account the original difference scheme (4), (5) and determine w(x1) as the solution of the following one-dimensional problem:

Given k(x) 2 nl > 0 (the condition of ellipticity), the bound 0 5 w(x) 5 n;'lf is easily derived for the solution of one-dimensional problem (38), (39). Therefore, we can put

in the a priori bound in (37). Hence, the constant M in (37) does not depend on the grid steps. The bound in (37) reflects stability of the difference solution with respect to the right-hand side and boundary conditions.

131


Because of the stability of (37), (40) we can also derive appropriate estimates for the discrepancy of the difference solution. Indeed, let

4.) = ~ ( 2-) +), where u(x) is the solution of differential problem (2), (3). For this function we obtain the equation

where ,$(x) is the error of approximation on the solutions of problem (2), (3). We have (42) II@(x)IIc(~~ 5 MI (hf + h;) for smooth coefficients and solutions (see Section 4.2). Because of (42) and the bound in (37) for the solution of problem (41) we derive the bound for the error Thus, difference scheme (4), (5) uniformly converges with the second order. 4.3.6 THIRD-KIND DIFFERENCE PROBLEM The problem of stationary heat transfer (2) with third-kind conditions (19) is considered in a similar way. We shall not dwell on details and only touch some basic points of the investigation. Let w(') denote the set of the nodes of the grid g that lie inside the rectangle R and let w(') denote the set of nodes on its boundary (Z =a(') w(')). The difference solution is determined by the system of equations

+

The grid operator A in (44) is defined by (4) for x E w(') and corresponds to the approximation (20) for x E ~ ( ' 1 . Again, we represent the solution of the difference problem in form (31). The difference problem for y(')(x) becomes

A

=F

(

x E~('1,

)

Ay(') = 0,

x E J2).

(45)

The difference solution y(')(x) is defined by the conditions

hy(') = 0 , x

w

= F(')(%), x E w(').

(46)

Let us first derive a bound for the solution y(2)(z).We define a majorant function Y(2)(x)as the solution of the problem

132


According to the maximum principle (W = w ( ' ) ) ,the solution of problem (47) achieves its maximum on the set of nodes w @ ) .Similarly to Corollary 6 we have

Taking into account (see ( 2 0 ) )the condition D ( x ) 2 o ( x ) 2 oo > 0 we derive

from the latter inequality applied to problems (46) and ( 4 7 ) . Let us now consider problem ( 4 5 ) . The majorant for it is defined as the solution of the equation

For the case of a homogeneous medium ( k ( x ) = const in ( 2 ) ) we derive the desired bound for the solution of (49)

where the constant M does not depend on the grid parameters. Let us consider an auxiliary function w ( x ) that depends only on the one variable, namely x l . Let this function for constant k ( x ) be specified by the conditions

If x E w('), the operator A(') corresponds to approximation of third-kind boundary conditions with o ( x ) = oo on the portions xl = 0 and x2 = I1 and o ( x ) = 0 on the other portions (the Neumann conditions). Let us compare the solutions of problems (49) and (51). For their difference v ( x ) = Y ( ' )- w ( x ) we have

We thus find A - A(') = D ( ' ) ( x ) 2 0 and v ( x ) 5 0 for the solution of (52). Hence, w ( x ) is a majorant for problem (49). Difference scheme (51) is associated with determining the solution from the following one-dimensional problem:

STATIONARYPROBLEMS O F H E A T T R A N S F E R

For this problem we have the bound

i.e. the constant M does not depend on the grid steps. This leads to estimate (50). Combining (48) and (50) we find an a priori bound for the solution of problem (44), namely

This bound is in agreement with the corresponding bound for the solution of the differential problem (see Section 4.1). We can use (53) to determine the convergence rate of the corresponding difference scheme in the usual way. The problem for the error is posed as follows: Az = $(z), x E w ( ' ) , h z = u(x), x E w('), where $(x) is the error of approximation of the equation and u(x) is the error of approximation of the third-kind boundary conditions. An estimate like (43) holds for difference scheme (4), (20), i.e. this scheme converges with the second order. 4.3.7 PROBLEMS

1. S h o w t h a t t h e solution of t h e difference p r o b l e m

w i t h a(x) 2 n

> 0 can

b e e s t i m a t e d as

Solution. Let us represent the right-hand side of (54) in the form x, = 1 and obtain ayr x = c = const.

+

This yields the recursion relation

where hi = x,. We sum this relation with respect to i from 1 to k, take into

134


account the first condition in (55) and find

We now take into account the second condition in (55). For i = N we get the expression for the constant c:

Substitution into (57) yields

We take into account that Ak

< AN and conclude

i.e. the desired estimate (56) holds. 2. D e t e r m i n e u n d e r w h a t conditions t h e m a x i m u m principle holds for t h e difference Dirichlet problem of increased o r d e r of a p p r o x i m a t i o n (Section 4.2). Solution. The difference equation at the internal nodes of the rectangular grid Z is written as

where A-u = u ~ ~ a. =~1 , 2 . Let us rewrite (58) in the canonical form (1) and verify conditions ( l l ) , which provide that the maximum principle holds. After some algebra we find

135


We thus get D ( x ) = 0, x E w , and A(%) > 0 for any grid step. However, the conditions B ( x , E ) are only satisfied if

Condition ( 5 9 ) shows that we should use a rectangular grid whose steps are not very different so that the maximum principle holds for the difference Dirichlet problem with increased order of approximation.

4.4 Convergence of Difference Schemes in an Energetic

Space 4.4.1 EQUATIONS IN A FINITE-DIMENSIONAL HILBERT SPACE Difference schemes are studied in a finite-dimensional space, therefore we only consider properties of linear operators in finite-dimensional Hilbert spaces (see also the Appendix). First of all we note that every linear operator is bounded in a finite-dimensional space. We consider a Hilbert space H equipped with the scalar product ( y l ,y z ) and the norm IIY112 = ( y , y ) . A linear bounded operator acts in the space H . The norm of A is defined as

The operator A* that is adjoint of the operator A is defined as ( A y , , y z ) = ( y l , A * y z ) for all yl and yz. For a linear bounded operator A we have IlA'II = llAll. If A = A', the operator A is called self-adjoint, and if A = - A b , the operator A is called skew-symmetric. The operators AA* and A * A are both self-adjoint. We also note that (A*)' = A and (A')-' = ( A - I ) ' , provided that the inverse operator A-' exists. Any operator can be represented as a sum of a self-adjoint operator A . and a skew-symmetric operator A1, i.e. A = A0 A', where

+

A~ = ; ( A - A*). AO = ; ( A + A * ) , The identities ( A y , y ) = ( A o y , y ) and ( A l y , y ) = 0 hold. An operator A is said to be positive ( A > 0) if ( A y , y ) > 0 for all y E H except for y = 0. If ( A y , y ) 6 ( y , y ) ,6 > 0 , the operator A is referred to as a positive definite operator. The number ( A y , y ) is said to be the energy of the operator. The notation A 2 B implies that the inequality ( ( A - B ) y , y ) 2 0 holds for every y E H. If for operators A and B there exist constants yl yz > 0 such that

>

>

136


the operators A and B are called energetically equivalent operators. If

where E is the identity operator ( E y = y ) , then the numbers 6 and A are called the bounds of the operator A. Operators A and B are called commutative if ( A B ) y= ( B A ) yfor all y E H . For each self-adjoint nonnegative operator A there exists a square root B such that B2 = A, which is denoted All2. Given a positive self-adjoint operator A, we can introduce the energy space H A equipped with the scalar product ( y l , y z ) a= (Ay1,yz) and the norm llvlli = ( A Y , Y ) . The Cauchy-Schwarz inequality (or Bunyakowskii inequality)

in H A becomes 2

( A Y I , Y Z5) ( A Y I , Y I ) ( A Y ~ , Y ~ ) . If A is a positive definite operator, i.e. A 2 6E, 6 > 0, then there exists the inverse operator A-I and IIA-lll 5 6-'. In a finitedimensional space any positive operator is positive definite; therefore its inverse does exist. The 11 = IIAlIn for a positive definite operator A holds for any integer n. identity / [ A n We can introduce the negative n o m

for a self-adjoint positive operator A. We can use an alternative definition of this norm:

Let us consider in the first-kind operator equation

in a finite-dimensional Hilbert space H, where A is a linear operator and y is a sought-for element of H . Equation ( 1 ) is uniquely solvable if the homogeneous equation Ay = 0 has only the trivial solution y = 0. Let us derive some a priori bounds for the solution of (I), given different information about the operator A. Let A be a positive definite operator, i.e.


Then the solution of (1) can be estimated as

We take the scalar product of (1) by y and obtain

We use condition (2) to estimate the left-hand side of (4) and apply the Cauchy-Schwarz inequality (or Bunyakowskii inequality) I(ip, y)l 5 llipll . llyll to estimate the right-hand side of (4). We derive the desired bound (3) for the norm of the solution. Estimate (3) provides stability of the solution of (1) with respect to small perturbations of the right-hand side. If A is a self-adjoint positive operator ( A = A* > 0), we can derive an a priori bound of the solution of equation (1) in the energy space, namely we find the exact bound IIYIIA = Iliplla-1. (5) We take into account y = A-lip and obtain (5) directly from (4). If A is not a self-adjoint operator, we can derive bounds like (5) in the energy space generated by the self-adjoint positive part of the operator A . Let

then the a priori bound

I l ~ l l ~5o IIipllA,l holds for (1). In this case (4) becomes

According to the generalized Cauchy-Schwarz inequality applied to the righthand side we get

As a result we find bound (6). According to (6),

138

COMTUTATIONAL HEAT TRANSFER

provided that the operator Ao is positive definite ( A o 2 6E, 6 the following a priori bound holds:

Il~lla,i 6-11211vll.

> 0 ) . Thus, (7)

Unlike ( 6 ) ,in ( 7 ) the right-hand side of ( 1 )is estimated in an ordinary rather in a negative norm. Let us now assume that the non-self-adjoint operator A satisfies the condition 0 0 A 2 ?A, A = (A)' > 0. (8) Then we can derive a bound for the solution of ( 1 )in the energy norm. Namely, the following bound holds:

Equation (4) together with condition (8) implies

This relation yields the sought-for estimate ( 9 ) . If conditions (8) are satisfied, we can estimate the right-hand side of ( 6 ) in another negative norm by the inequality

This is because the inequalities A 2 yA and A-' self-adjoint positive operators.

2 A-' are equivalent for

4.4.2 SOME DIFFERENCE RELATIONS

Let us first recall some notation introduced before and introduce some new notation of the theory of difference schemes. For simplicity, we only consider uniform grids. We introduce the grid

on the interval [O, 11. We use the notation

w={xlz=ai=ih,

i = l , 2 ,...,N - I } ,

wt = { x / x= xi = ih, w-={xlz=x,=ih,

i = 1,2,, , , ,N } , i = O , l , ...,N - I }


139

for separate parts of the grid. We denote by &(xi) = &i the average grid step, namely

-

We define a Hilbert space H(G) on the grid G equipped with the scalar product (Y,~)YE Y ( x ) ~ x ) ~ ( x ) . zEu

The scalar product in H ( W ) is defined similarly:

where e.g. W = w , w+,w-. We also present grid analogues of the differentiation formula for a product of functions and the formula of integration by parts. Taking into account the above definitions of operators of the right- and left-hand side derivatives (Section 4.2) we can directly verify the formulae

Formulae (11) is a difference analogue of the differentiation formula

The grid identities

are grid analogues of the formula of integration by parts

We replace yi by a,=,, and derive the first difference Green fonnula

140


The second dzfference Green formula looks like

For grid functions y(x) and u(x) that vanish for x = 0 and x = 1, formulae (13) and (14) are simplified and become

Similar definitions are used for studying twodimensional (multidimensional, in the general case) grid functions. We introduce the uniform grid wl in the rectangle = {XI x = (xl,xz), 0 5 x, < l a ) , a = 1,2, with the step hl with respect to xl and the grid wz with the step hz along the other direction. Let w = wl x w:! and, hence,

Let w be the set of internal and aw the set of boundary nodes. Let now h(x) = h.1(x1)h2(x2). where the average steps hl(x1) and h(xz) along separate directions are introduced above. According to the notation introduced, for example,

Scalar products in other spaces of grid functions are defined similarly.

4.4.3 A PRIOR1 BOUNDS AND CONVERGENCE OF THE DIFFERENCE DIRICHLET PROBLEM Let us consider the construction of a priori bounds in the energy space. As an example we take a stationary problem of heat transfer in a nonhomogeneous medium with the simplest first-kind boundary conditions. We consider the elliptic equation

in the rectangle fl

141


We rewrite the corresponding boundary value problem (see Section 4.2) as

where

2

LY=

AY = x A , y ,

(22)

,=l

We take into account the homogeneous condition (21) and introduce the space H of grid functions that are defined on G and vanish on am. We define the scalar product in H by the relation

We denote

2

A ~ = A ~ = E A~ , Ev ,H

(23)

a=l

and rewrite difference problem (20)-(22) as a first-kind operator equation (1). We establish properties of the grid operator A by the following lemmas. Lemma 1. The operator A defined by (23) is self-adjoint and positive in H . We represent the operator A in the form

where

A,y = A m y ,

y E H,

a = 1,2

Let us show that each of the operators A,, a = 1,2, possesses the desired properties, i.e. A, = A: > 0, a = l,2. (25) For example, for the operator A2 we have

We apply Green's formula (16) to obtain

142


Thus, (26) becomes (Azy,v) = (y,Azv), i.e. the operator Az is self-adjoint. The operator A1 is considered similarly; therefore, because of representation (24), A = A ' . In order to prove that the operator Az is positive, we employ the first Green's formula (15). As a result we get

This implies that the operator A2 is positive. Properties of A1 are studied similarly. From (24) and (25) we conclude that the operator A is positive as well. If a, = 1, a = 1,2, then A is the grid Laplace operator henceforth denoted by A. Taking into account (27) we write

The operator A defined by (33) with 0 < K I 5" a,(%) 5 K I , " x E w , is energetically equivalent to the g r i d Laplace operator A and

Lemma 2.

This inequality is stated using (27) and assumptions of smoothness of the coefficients of the difference scheme. Let us derive a priori bounds for the solution of difference problem (20)(22) in the grid space that is an analogue of the Sobolev space W,'(n).Some a priori bounds for continuous problem (17)-(19) are given in Section 4.1. By analogy with functions of a continuous argument, let us define the space of grid functions that are defined on Z and vanish on aw.We equip this space with the norm Ilull? = llv~ll~ + lly1I2. (30) As in the continuous case, the a priori bound for the solution of difference problem (20)-(22) is based on the grid analogue of the Friedrichs equation.

Lemma 3.

The inequality

holds for a grid function y(x) defined on the grid ii vanishing on aw


143

Note that the bound in (31) is established for any rectangular grid in 0 with an improved constant M = ~(1;~+1;~).The inequalities with this constant M for a uniform grid can be based on the spectral problem for the grid Laplace problem (see Section 4.6). Let us consider a one-dimensional function v(xl) = y(xl, xz). We take into account the homogeneous boundary conditions and obtain expressions for u(x1):

To estimate the right-hand sides of (32) and (33) we apply the CauchySchwarz inequality in the form

We put ak = %,,kh:'2

and bk = h;l2 and derive from (32) and (33) that

We multiply (34) by l1 - xli, inequality (35) by xli, combine the resultants and find ~ 1 4 -XI;) 1 (~,)~hl. u2(x1;) 5 11

z

zl~w:

We now multiply (36) by hi, sum the resultant with respect to i (trapezoid formula for a segment of a parabola) and derive the following bound from (36):

We substitute the function y(x1,x2) for u(xl), multiply the resultant by hz and sum it with respect to x E wz and obtain

144


A similar relation holds for the bound of the solution of the norm of the difference derivative with respect to 2 2 . Thus, we take into account (28) and derive the Friedrichs inequality (31). By the Friedrichs inequality (31) we find from (30) that

We take into account (28) and (29) to derive the bound for the operator A

Let us relate the norm (30) with the corresponding negative norm defined by the formula

11 . 11-1

I(ip>Y)l

IIvII-1

= SUP -. y+o

IIyII1

Similarly to (6) and (9), let us derive an a priori bound for difference problem (20)-(22) written as the operator equation (I), (23).

Theorem 1.

The a priori bound

holds for dzfference problem (20)-(22) The proof is based on the identity (4) and estimate (37). We apply the inequality (9,y) 5 IIipII-lllyll-1 and find the bound in (38). Bound (38) implies, in particular, unique solvability of difference problem (20)-(22). Convergence of schemes with smooth coefficients is established in the usual way. For the discrepancy z of the difference solution of (17)-(19) we obtain the operator equation A z = 11, (39) where 11is the approximation error. It can be estimated as (see Section 4.2)

According to (30), we get

We take into account bounds (37) and (40) for the solution of (39) and find lialll M31hI2, M3 = n;lMlMz. Thus, the difference scheme (20)-(22) converges with the second order in the energy space.

<

145


4.4.4 NONUNIFORM GRIDS AND DISCONTINUOUS COEFFICIENTS We briefly mention some peculiarities of investigation of the accuracy of difference schemes in a more general case of nonuniform rectangular grids. As we showed in Section 4.2, the second-order approximation of the second derivative on the usual three-point stencil of a nonuniform grid is attained only in a special negative grid, while in C ( w ) and L z ( w ) the order of the approximation is only first. In the energy space considered, the second-order approximation is also preserved for a nonuniform grid. It is typical of nonuniform grids that the approximation error is represented as S b ) = SO(z+ ) **(x), (42) where @ ( x ) has a special divergent form. If nonuniform grids are applied for solving two-dimensional difference problem (20)-(22),the error is represented in the form (42) with

We estimate the error from equation ( 3 9 ) when (42) and (43) define the structure of the error. We thus have

The second term is estimated by ( 4 1 ) . We take into account ( 4 4 ) and find for the first term:

x 2

So)=

(2,

( 2 , %=.).

(45)

u=1

According to the grid identities (12) we obtain

for functions that vanish on aw. A similar relation holds for a = 1. We substitute this into ( 4 5 ) , apply the Cauchy-Schwarz inequality, take into M41h1211zII~. account ( 2 8 ) and (44) and derive ( z , @ ) 5 M41h1211Vzll We thus again come to the bound llzlll 5 MslhI2, which is obtained by representing the error in the form (43), (44). Let us recall that when nonuniform grids are used to find the smooth solution, the error has the same structure.

<

146


The error is also represented in the desired form (42)-(44) when solving problems of stationary heat transfer (17)-(19) with a discontinuous heat conductivity. This is obtained using the integreinterpolation method. According to Section 4.2, we find for the difference scheme (20)-(22) for problem (17)-(19) that

where the averaging operators are defined by the expressions

The approximation error for difference problem (20)-(22), ( 4 6 ) becomes

= 4 S z L u - Au = ~

( s ~ s- A,u). ~ L ,

(47)

o=1

Consider, for instance, the first term in the right-hand side of ( 4 7 ) . We apply the properties of the averaging operators to find

SlSzL1u - A I U = -

-

*)

(s2~ix) 8x1

;-1,2,j

)+

(aluz,

L,= n ~, ,

where

(48) We can similarly define


147

Thus, the error in difference scheme (20)-(22), (46) is represented in the divergent form

The scheme obtained by the integro-interpolation method has this desired form. As a typical example we consider the simplest situation in which the heat conductivity is discontinuous on the line x l = const, which passes through the nodes of the uniform rectangular grid J (al = xlk = khl E q ) .The homogeneous junction conditions (see Section 2.2)

are satisfied on this interface. In each separate subdomain, X I > xlk and x1 < xlk, qa(x) = O(lhlZ) because of (48) and (49), provided that the coefficients and the solution are smooth. Therefore the homogeneous difference scheme (20)-(22), (46) converges with the second order. A more complex situation arises when the discontinuity line xl = const does not pass through the grid nodes and, moreover, when the heat conductivity breaks down on an arbitrary curve. Similar reasoning is valid in the situation with a surface heat source. In this case, we consider nonhomogeneous junction conditions

instead of (51) (see Section 2.2). To simulate conditions ( 5 2 ) , we consider a boundary value problem for the equation

where XI) is the delta function. We use the integro-interpolation method and come to a difference scheme which only differs from (20)-(22), (46) at the nodes lying on the discontinuity line where

Conditions of a concentrated source and more general conditions are approximated similarly.

148


4.4.5 THIRD-KIND BOUNDARY CONDITIONS Let us dwell on the most important topics in investigation of elliptic problems with third-kind boundary conditions. As a model problem, we consider equations (17), (19) equipped with the boundary conditions

We consider the corresponding difference problem on the set H of grid functions defined at all the nodes of the grid G. We define the scalar product in H by the relation ( Y , "1 = ( Y >")Z3

To rewrite the difference problem in the operator form, we use the notation of (23) and (24). We take into account the second-order approximation of boundary conditions (53) on the solutions of the problem (see (20) in Section 4.3) and put

Under such a choice of the operators A,, a = 1,2, the operator A possesses the desired properties, namely the following statement is true. Lemma 4. The operator A defined b y (241,(54) and (55) is self-adjoint and positive in H , given u ( x ) > 0 . Let us prove that (25) holds. In our case,

( A z Y , ~=)

x x fLl

zlEu,

22EY2

AZY(X)U(X)~LZ.

(56)


According to the Green formula (14) we have

This implies that the operator Az is self-adjoint. It can be similarly checked that A1 is also a self-adjoint operator. The operators A,, a = 1 , 2 can be proved to be positive using the first Green's formula (13). Computations similar to the above yield

Consequently, if u(x) is positive, the operator Az is positive. The same is true concerning the operator A l . This completes the proof of the lemma. We take into account that y(x) # 0 on aw and define

and the norm according to (30). We define a grid analogue of the norm Lz(8w) by the formula

IIYIIL =

x

s,EY,

( Y ~ ( X ~ +y2(x1,12)) ,O) h.1

+

x

(y2(0,x2)

+ y2(11,x2)) h.2.

22EY2

The following Friedrichs inequality holds for the grid functions that do not vanish on aw.

150


Lemma 5. The inequality

where the positive constants m and M are independent of the grid, holds for any grid function y ( x ) defined on the grid Z. We do not prove this and only note that bounds like (58) can be derived in a way similar to the proof of Lemma 3. Instead of (32) and (33) it is sufficient to use the relations

and the &-inequality lab1 5 €a2 + b 2 / ( 4 € ) ,where E is an arbitrary positive number. The a priori bound like that in (38) can be derived for a difference problem with third-kind boundary conditions.

Theorem 2. The a priori bound

where M I = Ml(tcl, uo, m, M ) , hold for the problem ( I ) , (241, (54) and (55), provided that u ( x ) 2 uo > 0 . We take into consideration the Fkiedrichs inequality (58) and find

According to (57) and the restrictions on the coefficients of the differential problem we derive the bound

>

We combine (60) and (61)and derive the bound from below, namely ( A y ,y) M;'llyllt, which, as usual, yields the desired a priori bound (59). Convergence of a difference problem with third-kind boundary conditions is studied like that of the Dirichlet problem, hence we do not dwell on it.

4.4.6 PROBLEMS 1. Show energetic equivalence of the difference operator of the problem of stationary heat transfer in anisotropic media with firstkind boundary conditions.

151


Solution. We consider boundary value problem ( I ? ) , ( I S ) , where

with k,p(x) = kpa(x).The corresponding difference scheme (see Section 4.2) is written in the form (20), (21) with

to define the operator A = 0.5(A-+ A + ) on the set of grid functions vanishing on aw. We take into account difference relations (12) and verify that the operators A and A+ are self-adjoint, given k,p = kp,. For example,

where we take into account that y ( x ) = u ( x ) = 0 , x E aw. Because of the inequality (2) of Section 4.1, which expresses uniform ellipticity, we obtain

In particular, nl 5 k,, 5 n z , a = 1,2. Because of this inequality we derive from (62) 0

~ ( A Y , Y5 )( A V , Y )5 ~ A Y , Y ) . The operator A+ has the same properties. Thus, the difference operator of the Dirichlet problem for an elliptic equation with mixed derivatives is energetically equivalent to the difference Laplace operator with constants of equivalence nl and n2.

152


2. Derive t h e b o u n d

for t h e difference problem

Solution. Like in Theorem 1, we have the estimate

where a(x) 2 nl > 0 for scheme (64), (65) The inequality

holds for one-dimensional grid functions that vanish for x = 0, I . This inequality is proved similarly to Lemma 3. Inequality (36) in our notation becomes

This yields the bound in (67). We substitute

into inequality (66) and get the bound in (63). The bound in (67) is the simplest example of grid analogues of embedding theorems (in our cases, for the spaces C(0,l) and W,'(0,1)).

4.5 Direct Methods for Solving Grid Equations

4.5.1 METHODS FOR SOLVING SYSTEMS OF LINEAR EQUATIONS

The original differential problem is approximated by a grid problem. The corresponding difference (grid) equations comprise a system of linear algebraic equations for unknown values of a grid function. Methods of linear algebra that


153

maximally take into account special characteristics of the problem are involved to solve it. It is typical of grid problems that the corresponding matrix of the system of algebraic equations is sparse, i.e. contains many zero elements and has a band structure. When solving multidimensional problems, the matrix is of a large order, which is equal to the total number of the grid nodes. For example, we consider a two-dimensional problem on a grid with Nl nodes in one direction and Nz nodes in the other. Then the matrix has the order Nl N 2 . Therefore, grid problems require a lot of memory and an improved fast response of computers if standard mathematical software is used. We rewrite a system of linear algebraic equations in the form

where A is a square m x m-matrix with the elements a,,. For solving problems of linear algebra, direct (exact) and iterative methods are used. Direct methods are those giving the exact solution of the problem a t a finite number of arithmetic operations, given no rounding-off errors. The Gauss method is a classical direct method of linear algebra. It has been modified in several ways to more completely take into account special features of the problems being solved. The modifications include the Gauss method of choosing the principal element, methods for solving sparse systems, etc. The Gauss method is based on the representation of a nondegenerate matrix A as a product of a lower- and upper-triangular matrices (LU-decomposition):

A = LU,

(2)

where

The solution of problem (1) using the expansion in (2)-(4) is reduced to the solution of two simple problems

154


In order to determine the coefficients of the matrices L and U , recursion relations are used (the compact Gauss scheme), namely

Computational cost for LU-decomposition of the matrix A is Q = 0 ( m 3 ) arithmetic operations. When solving the equation by (2)-(5) the main cost is due to computation of the elements of the matrices L and U . Given L and U, the solution of a problem (5) with tridiagonal matrices requires only q = 0 ( m 2 ) arithmetic operations. For systems of equations (1) with a symmetric matrix A (A = A*) the method of a square mot (Cholesky decomposition) is used. It allows us to reduce computational work approximately by half. The method is based on the representation (7) A = LDL*, where D is a diagonal matrix with the entries +1 and -1. The elements of decomposition (7) are defined by the formulae

It is noneffective to use the above direct methods for solving multidimensional grid problems. The Gauss method or the Cholesky decomposition applied fur solviug a two-dimensional problem with N1N2 nodes would require Q = O(N:NZ) arithmetic operations. In several cases this cost can be reduced by taking into account a sparse structure of the matrix. 4 . 5 . 2 THOMAS ALGORITHM

When approximately solving one-dimensional problems of heat transfer, we encounter grid problems with tridiagonal matrices. For these problems, special

155


versions of the Gauss method are used that take into account special features of the problem. We rewrite the difference problem in the form

C I Y I- Biyz = Fi, - A .& .- I C,yi Biyi+i = F,, -A,U,-I + C,Y, = F,.

+

-

i = 2 , 3 , .. . , m - 1 ,

The solution of difference problem ( 9 )is sought in the form

Y , =a;+~yi+l +iRi+l,

=2 . - 1 ,

y,

=iR,+l.

The coefficients of this recursive representation are defined as follows:

a2 = C c 1 B 1 ,

PI

= C;'F1,

i = I , & . . ., m - 1, = (C, - Aiai)-' (F, +A$,), i = 1,2,. , , , m .

aitl = (C, - ~ , a , ) - ' B;, @,+I

Recursive relations ( l o ) , (11) for determining the solution of problem (1)with tridiagonal matrix (9) comprise the well-known Thomas algorithm

(forward-backward algorithm). The formulae for coefficients (11) are used in the forward algorithm and (10)are used in the backward algorithm. It is easily seen that only O ( m ) arithmetic operations are required for the solution of the problem. This essential and principal reduction in the computational work is because of the special sparse structure of the matrix. This version of the Gauss method completely takes into account this special structure. Therefore, different versions of the Gauss method for sparse methods are successively applied for solving grid equations that arise when using difference methods or finite-element method for approximate solution of problems of mathematical physics. In computational practice, different versions of the Thomas algorithm are used. In particular, we mention cyclic Thomas algorithms for solving problems with periodic boundary conditions, the flux version of the Thomas algorithm, the nonmonotonic Thomas algorithm, versions for problems with band matrices, etc. To employ computational formulae (11) of the Thomas algorithm it is necessary that the denominators c, - A,a, are other than zero. Let us now discuss the concept of the computational stability of the Thomas algorithm. The solution of the problem is determined by recursive relations (14) in which rounding-off errors can accumulate. For instance, assume that y , was = y, E,. The error in the computed with the error E,, i.e. we found other values, according to (14), is determined by the relations E, = a i + ~ c , + ~ , i = 1,2,. . . , m - 1. Thus, the rounding-off error can be accumulated and become large if the moduli of a , are greater than unity. For stability it is sufficient to demand la,[ 5 1 for all i.

c,,

+

156


Lemma. Let all real coeficients of the system of equations (9) satisfy the conditions

and at least one strict inequality hold in (13). Then in (10) and (11)

The above sufficient conditions (12) and (13) of computational stability of the Thomas algorithm are actually the conditions of the diagonal predominance in the matrix A.

4.5.3 TWO-DIMENSIONAL PROBLEM We consider methods for solving grid equations that correspond t o multidimensional stationary heat transfer problems as applied to the simplest twodimensional problem in a rectangle R with smooth coefficients and first-kind boundary conditions (problem (17)-(19) in Section 4.4). The corresponding difference problem ((20)-(22) in Section 4.4) on a rectangular grid that is uniform in both directions is written a s follows. We use the simplest approximations for the coefficients and right-hand side of the difference scheme that are written at internal nodes in the following way:

For the boundary nodes we have

Difference scheme (14), (15) is a system of linear algebraic equations, in which the unknowns are the values of the approximate solution at the internal Nz - 1. The total number of grid nodes, i.e. yij, 1 i 5 Nl - 1, 1 5 j the unknowns is ( N l - l ) ( N z 1 ) . If we directly apply the Gauss method

<

-

<

STATIONARYPROBLEMSOFHEATTRANSFER

157

not taking into account the sparse structure of the matrix, the computational cost would be Q = O(N:NZ) arithmetic operations. The m a t k Thomas algorithm, which is a generalization of the above scalar Thomas algorithm, allows us to substantially decrease the amount of comput&tional work. Let us introduce the vectors

i.e. the vector y; corresponds to the unknowns at the nodes of the ith column of the grid G. Then difference scheme (14), (15) can be rewritten as a three-point vector equation (9), where A;, Bi and C, are square matrices of the order Nz-l and m = Nl - 1. In our case, A, = (a?) and B, are diagonal matrices, while C; is a tridiagonal matrix. Computational formulae of the matrix Thomas algorithm have the form (lo), ( l l ) , where a; are square matrices and 0, are (N2 - 1)-dimensional vectors. The matrix Thomas algorithm requires a large amount of computer memory. In the process of computations the computer must store all the matrices at, hence the desired amount of memory is P = 0(N2Nl). Computational work for inverting of the matrices C, - A,=, (see (11)) is Q = O(NzNl). The method can be used for problems with Nz << Nl (the number of nodes in the xl-direction is substantially larger than that in the xz-direction). Besides, the matrix Thomas algorithm is successfully applied for solving of systems of differential equations of a low dimension. The reduction method is one of the most widespread methods for solving elliptic problems. It is used for solving boundary value problems with separating variables. In the case considered of model problem (14), (15) we assume k = k(xz). We rewrite the grid problem as the following three-point vector equation:

where rn = Nl. The general (see e.g. (11)) three-point vector equation is essentially simplified because A, and Bi are the identity matrices (A, = B; = E) and the matrices C; = C are constant. For simplicity, we only consider first-kind equations (see (18)). Currently, the reduction method is improved t o include the cases of grid problems with second- and third-kind conditions and periodic conditions, assuming that the variables are separated. The reduction method is a special version of the Gauss method for system of equations (30), (31) with m = 2'. It is based on subsequent elimination of the unknowns yi from equations (30). Firstly, the unknowns with odd numbers are eliminated, then the unknowns with even numbers are eliminated from

158


the rest of the equations, then the unknowns whose numbers are multiples of four, etc. Under such elimination, the number of unknowns is reduced after each step. As a result, a unique equation remains for determining y,,,. The inverse procedure is involved to determine all the unknowns. The reduction method requires Q = 0(N2Nl log, NI) arithmetic operations. This estimate is better than that for the matrix Thomas algorithm. The computational cost is O(log, NI) arithmetic operations per node which is close to the optimal asymptotic estimate O(1). Therefore the reduction method is said to be a fast method for solving grid equations. A marching algorithm is optimal ( Q = 0 ( N l N 2 ) ) for grid elliptic problems with constant coefficients. Nowadays, linear algebra software packages are widespread. They are oriented on grid problems that arise when solving problems in mathematical physics by difference and projective-grid methods. This applied software includes well-developed and tested routines for solving grid problems by the above methods.

4.5.4 THE METHOD O F SEPARATION O F VARIABLES Separation of variables is a classical approach to solving simple linear problems of mathematical physics (see Section 3.2). It is logical to expect that a similar idea can be applied to solving grid equations. Let us consider model problem (14), (15), where the coefficient k depends on a single variable (for definiteness, we assume k = k(xz)). Under these assumptions, we rewrite problem (14), (15) as

where

and the right-hand side is ~ ( x =) f(z)/k(xz), x E w. To apply the Fourier method for solving two-dimensional problem (19)-(22), we consider the eigenvalue problem with respect to 22:

We denote the corresponding eigenvalues and eigenfunctions by XI, and vk(xl), respectively; k = 1 , 2 , . . . , NI - 1. Since A is a self-adjoint operator, the

160


arithmetic operations. The fast Fourier transform is applied to the grid operator A l in the cases of first-kind boundary conditions (48), second-kind conditions, or periodic conditions with N1 = 2'. In these special cases, the eigenvalue problem has an exact solution (the eigenfunctions are sines and cosines). For problem (23), (24) we have (see Problem 2)

Algorithms of the fast Fourier transform are based on separation of the identical factors vx(xl), X I E wl in the sums like (25) and (26) and on regrouping of the terms. Various versions of the fast Fourier transform are described in an ample number of manuals. We considered an approach with the fast Fourier transform with respect to one variable and the Thomas algorithm with respect to the other. The Fourier transform can be applied to problems with constant coefficients with respect to both variables (expansion in eigenfunction of two-dimensional grid operator A ) . Among other possibilities, we mention FACR algorithms that combine the cyclic reduction (CR) method and Fourier analysis (FA) applicable t o the problems with constant coefficients.

4.5.5 PROBLEMS 1. Derive c o m p u t a t i o n a l formulae of t h e cyclic T h o m a s a l g o r i t h m for solving t h e s y s t e m of equations

w i t h periodic coefficients w i t h a period m:

Solution. Since the solution is periodic (y, = y,+,), we seek it for i = 0,1,2,. . . , m - 1. Therefore we rewrite problem (29), (30) in the form

161


Since the first equation in (31) includes a term with A. and the last equation in (31) includes a term with Bm-l, we cannot use the usual formulae of the T h o m a algorithm. In order to solve (31) we use the cyclic Thomas algorithm, which is based on the well-known method of linear algebra called bordering. We seek the solution of problem (31) in the form

where wi is the solution of the homogeneous problem wo = 0, -Aiwi-i

+ C,W,- Biw,+l = F,,

i = 1 2 .m - 1

(33)

For the second grid function we pose the three-point problem

with nonhomogeneous boundary conditions. Let us find the conditions under which (32)-(34) give the solution of (31). Equations (31) are satisfied for i = 1 , 2 , 3 , .. . , m - 1, and the first equations can be used to determine yo. We substitute (32) into the first equation in (31) and get Fo Aow,-I +Bowl YO = Co - Aov,_l - Bowl ' Each of problems (33) and (34) is solved by the standard Thomas algorithm. We should only note that these problems have a common coefficient, namely the solutions are represented as

+

The cyclic Thomas algorithm is applied to problems with periodic boundary conditions (heat transfer problems in cylindrical and spherical coordinates). 2. F i n d t h e values of t h e p a r a m e t e r X (eigenvalues) for which t h e r e exist nontrivial solutions p(xj) (eigenfunctions) of t h e difference problem

Solution. We directly verify that the eigenvalues of problem (35), (36) are XE =

4 .,rk sin -

h2

2m'

k = l , 2 , ..., m - l

162


and the eigenfunctions are

x

=

( )' I 2

rkzj sin 1

because xj = jh. The eigenfunctions p k ( x j ) are orthonormalized in H ( w ) , i.e.

where

is Kronecker's delta.

4.6 Iterative Methods of Linear Algebra

4.6.1 BASIC CONCEPTS

We discuss methods for solving systems of linear equations AY = f

(1)

by iterative methods. Here A is considered to be a linear operator acting in a finite-dimensional Hilbert space H , f is a given element, and y has to be found. An iterative method starts from an initial approximation yo E H and sequentially defines approximate solutions of equation ( I ) , namely yl, yz,. . . , yi;, . . ., where k is the iteration number. The values ykil are determined from the previously found yk, yk-1,. . . If only the values from the preceding iteration yk are used to compute yk, the iterative method is said to be one-step (two-level). Respectively, if yk and yk+l are involved, the iterative method is called two-step (three-level). Any one-step iterative method can be written in the form

where Bk and Ck are linear operators and rk~1are numerical parameters. The solution of ( 1 ) must satisfy (2), i.e. Bky = C k y + r k + ~f . We take into . we express Ck in terms of Ak account (1) and put Bk - Ck = r k + ~ AThen and Bk and rewrite ( 2 ) in the form


163

This is the canonical form of a two-level iterative method. Given yo, all subsequent approximations are found by (3). To estimate the accuracy of the approximate solution it is logical to introduce the discrepancy t k = yk - y. An iterative method converges in the energy space generated by a self-adjoint positive definite operator D in H if /(zk(lo-+ 0 as k -+ m. The relative error E is usually taken as a measure of convergence of iterations, i.e.

Since the exact solution y is unknown, the accuracy of the approximate solution can be estimated by the residual rk = Ayk - f = Ayk -Ay, which can be computed directly. For example, the iterative process can continue until

The convergence criterion (5) corresponds to the choice D = A*A in (4). Let no(€) denote the minimal number of iterations that provide the accuracy E ((4) or (5) is satisfied). When constructing an iterative method, we should try to minimize computational work needed to determine the approximate solution of problem (1) with a given accuracy. Let Qk be the number of arithmetic operations required to find the approximation yk and let n no(^) iterations be performed. Then the total cost is evaluated by the quantity Q(E) = Qk. AS applied to two-level iterative method (3), Q(E) is minimized by choosing the operators B k and iterative parameters T ~ + I .The operators B k are usually defined by some reasoning, while the iterative parameters are chosen to optimize iterative method (3). Two approaches are accepted in the theory of iterative methods. The first involves a priori information about operators of an iterative scheme (namely, Bk and A in (3)). In the second approach (variational iterative methods), iterative parameters are determined on each iteration by minimizing several functionals, while the a priori information about the operators is not used. We first describe in general iterative methods not specifying the structure of the grid operators Bk. Specific results for grid elliptic operators are presented in the next section. We consider the generic problem (I), in which the operator A is self-adjoint and positive definite (A = A* > 0) in a finite-dimensional Hilbert space H. We consider the iterative process

>

i.e. the operator Bk is considered to be constant (unchanged during the iterations), unlike the general case in (3).

164


4.6.2 METHOD OF A SIMPLE ITERATION

rk

The method of a simple iteration corresponds to the constant parameter T in (6), i.e. we consider the iterative process

=

assuming that A=A'>O,

B=B*>O.

(8)

The iterative method in (7) is said to be steady-state. Let us have the a priori information about the operators A and B in the form of the operator inequality

<

7 l B A 5 %B, 71 > 0, i.e. the operators A and B are energetically equivalent

(9)

Theorem 1. Iterative method (7)-(9) converges in Ho, D = A, B , for 0 < T < 2/71. The optimal value of the iterative parameter is T = 70. The number of iterations n necessary to obtain the accuracy E can be estimated as

where

Note that no(&) in (16) is, generally speaking, noninteger, and n is the minimal integer such that n no(&)is satisfied. Theorem 1 suggests how to optimize convergence of iterative process (7), (8) by choosing the operator B according to (9),i.e. the operator B should be close to A with respect to the energy. In this case, the most favourable situation corresponds to the choice B = A and TO = 1, n = 1 (a direct method).

>

4.6.3 THE CHEBYSHEV S E T O F ITERATIVE PARAMETERS

An optimal set of the iterative parameters in (6) is related to the roots of the Chebyshev polynomials, therefore this iterative method is referred t o as the Chebyshev iterative method (the Richardson method). Let us define the set of M , as follows:

The formula

is used to compute the iterative parameters

~k


165

Theorem 2. The Chebyshev iterative method (6), (a), (9), ( l l ) , (12) converges in H D , D = A , B. The necessary number of iterations n to obtain the accuracy E can be estimated as

where

Note that in the Chebyshev method (see (11) and (12)) the iterative parameters are computed using the given total number of iterations. Obviously, n = 1 corresponds to the above method of a simple iteration. The practical realization of the Chebyshev method is associated with the problem of computational stability. The thing is that the norm of the transition operator at several iterations is greater than unity, and consequently the local discrepancy can grow up and lead to an abend. The problem of computational stability is solved by a special ordering of iterative parameters (by choosing pk from the set Mk). Several algorithms were suggested to , the total determine optimal sequences of the iterative parameters ~ k given number of iterations rk.

4.6.4 METHOD OF ALTERNATING DIRECTIONS Among special iterative methods widely employed in computational practice, we mention the method of alternating directions. This method is applied, first of all, when approximately solving two-dimensional grid elliptic problems. We present the operator formulation of the method of alternating directions. Let us solve equation (1) with the operator A that is represented as a sum of two self-adjoint operators, i.e.

We pass from the kth iteration to the next, ( k + l ) t h iteration in two stages. At the first stage we determine the intermediate value yktl12 from the equation

At the second stage we solve the problem

166


The parameter w in (15) and (16) should be determined. In a more general case, different iterative parameters are used in (15) and (16). If the operators A,, a = 1 , 2 commute ( A 1 A 2 = A 2 A 1 ) ,the method of alternating directions is used with variable iterative parameters. Let us rewrite the system of equations (15), (16) as

Thus, passage to a new iterative approximation is associated with inversion of the operator E + wA1 and E + w A z . It is reasonable to use the method of alternating directions when it is much simpler to invert the operators E+wA,, a = 1,2, rather than the original operator A . Iterative process (15), (16) is written in the canonical form as the steadystate process (7) with the factorized operator

and

T

= 2w,

T h e o r e m 3. The optimal value of the iterative parameter in the method = 6~ = 6 and A1 = Az = A is of alternating directions (14)-(16) with w = (&A)-'.The estimate

where p is defined by

holds for the discrepancy.

4.6.5 TWO-LEVEL VARIATIONAL METHODS In the above we considered iterative methods for solving problem ( 1 ) under the conditions when a priori information about the operators B and A is given as the constants yl and yz (see (9)) of energetic equivalence. Optimal values of the iterative parameters are defined in terms of these constants. It may happen to be a complicated problem to determine these constants, hence it is reasonable t o try to construct iterative methods by calculating iterative parameters without this a priori information. This class of methods includes variational iterative methods. Let us start by considering two-level iterative method (6) under assumptions (8).


167

We denote the residual rk = Ayk - f and the correction wk = B - l r k and represent the computational formula for iterative parameters as

Iterative process (6) is then rewritten as

An iterative method is rendered concrete by choosing D = D* > 0. This operator can be chosen so that the iterative parameters are computable. Since formula (17) contains the value zk that cannot be solved, it is impossible t o choose D = B (see Theorem 1). The above possibility D = A leads to the method of steepest descent, in which

Among the other possibilities, we mention D = AB-'A, i.e. the method of minimal corrections. A two-level variational iterative method converges not slower than the method of a simple iteration. Let us formulate the corresponding result as applied to the method of steepest descent.

Theorem 4. Iterative method (7)-(9), (18) converges i n Ha, and the estimate (10) holds for the number of iterations n necessary to obtain the accuracy E . Three-level variational schemes are most common in computational practice. Their convergence rates are not greater than those of the iterative method with the Chebyshev set of parameters. 4.6.6 METHOD OF CONJUGATE GRADIENTS

In a three-level (two-step) iterative method a new approximation is found by the two preceding ones. Two initial approximations yo and y1 are necessary to start the process. Generally, yo is taken arbitrarily, and yl is determined by a two-level algorithm. A three-level method can be written in the following canonical f o n of a three-level iterative method:

where

and

n+lare

iterative parameters

168


Computations by (19) are based on the representation

where wk = B-Irk. Realization of a three-layer method is often associated with the representation

We compare (20) and (21) and associate the parameters X k and pk with the parameters a k and r k :

In the notation introduced, the computational formulae of the iterative method of conjugate directions become

The main property of iterative approximation in this method is

This implies, for example, that the method of conjugate gradients converges in a finite number of iterations, which is not larger than the dimension of the finite-dimensional space H. In the case D = A , according to (22), the iterative parameters are computed by the formulae

The iterative parameters of (19) computed by (23) define the method of conjugate gradients, which is most commonly used in practice.

169


T h e o r e m 5. The method of conjugate gradients (19), (23) converges in H A , given (8) and (9). The number of iterations necessary to obtain the accuracy E is evaluated as n 2 no(c) = ln(2~-~)/ln(p;'),

where

4.6.7 PROBLEMS 1. Show t h a t t h e n o r m of t h e o p e r a t o r

+

~ ( w= ) (E WA)-'(E

- WA)

(24)

with A = A*,

6E 5 A

6>0

5 AE,

(25)

is m i n i m a l for w = wo = (6A)-' a n d is equal t o

Solution. Let us consider the operator S as a transition operator, i.e. = Szk. The operator in (24) corresponds to the canonical form

zk+l

We take into account (64) and find (A-' +w)A 5 B 5 (6-' +w)A. Therefore, the constants of energetic equivalence are written as

71 = 6(l

+ w6)-',

72 = A ( l

The optimal value (Theorem 1) T = r0 = 2/(y1

+ w6).

+ yz) yields

This results in the desired expression (25) for w. The minimal norm is defined by the formula

Substitution of (25) and (27) in the latter equation yields (26).

(27)

170


2. O b t a i n c o m p u t a t i o n a l formulae (17) for a two-level iterative method. Solution. We study the error in H D by putting zk = D - % ~ . We obtain the equation for uk:

where C = D ~ I ~ B - ~ A D - ~ and I ~ S is the transition operator from one iteration to another. It is logical to choose the parameter T ~ + Ifrom the condition of minimal error zk+l in H D (UC+I in H ) . We get

This implies that the norm is minimal for the choice

We take into account the definition of C, denote the residual T & = Ayk - f and the correction wk = B - ~ T and ~ derive from (28) a computational formula for iterative parameters in the desired form (17).

4.7 Iterative Methods f o r S o l v i n g Grid Equations

4.7.1 DIFFERENCE PROBLEM OF STATIONARY HEAT TRANSFER

We consider a model problem of stationary heat transfer with first-kind boundary conditions in a rectangle. The difference problem on the uniform rectangular grid (see Section 4.2) becomes

171


Convergence of iterative processes is studied in the space H of grid functions defined on ?Jand vanishing on aw. For this purpose, the boundary nodes in difference problem (1)-(3)are eliminated according to (2). The scalar product = (y,v),. in H is defined in the usual way, namely (y,u)= (y,v)z We rewrite difference problem (1)-(3)as the first-kind operator equation

where we denoted

7

In this equation y(x) = 0 for x E aw, i.e. at the boundary nodes this function is not identical to the difference solution. We take a nonhomogeneous boundary condition by an additional term in difference equation (1). The right-hand side of (4)is different from that of difference equation (1) only at the nodes neighbouring the boundary. It is easy to check that f = i p i p ~ h ; ~i p ~ h ; ~ , where

+

+

A similar procedure is applied to difference problems with other types of nonhomogeneous boundary conditions. Properties of difference operator A are studied in Section 4.4.In particular, it is shown that A is a self-adjoint positive operator. In addition, if 0 < nl 5 a,(z) _< xz, x E w , the operator A is energetically equivalent to the grid operator Laplace operator, i.e. the operator inequality

holds

4.7.2 TWO-LAYER ITERATIVE METHOD We use the iterative method

172


for approximate solution of grid problem (1)-(3) written as the equation (4), assuming that an initial approximation yo is given. The choice of iterative parameters was discussed in the above. We now consider the choice of operator B in iterative method (7). Given A = A* > 0 and B = B ' > 0, the convergence rate is defined by the constants of energetic equivalence yl and 72, namely

or, more exactly, by the ratio = yl/yz. Therefore, the operator B chosen, on one hand, should be easily invertible and, on the other, should maximize the value E . Different classes of easily invertible operators are discussed in the following. Among these, we can separate the simplest class of operators of multiplication by a given grid function, which have diagonal matrices. Triangular grid operators that correspond to lower or upper triangular matrices are easily invertible as well. The same is evidently true for products of these operators (factorized operators B). The method of variable directions is also based on using a factorized operator. A regularization principle for iterative rnethodsis important for construction of iterative methods. When choosing B, we start with an operator R = R* > 0 (a regularizer) that is energetically equivalent to the operator A, i.e.

Let us now construct the operator B using the operator R, i.e. according to the inequality 0 7 l B 5 R YzB, YI > 0. (10)

<

Conditions (9) and (10) yield the desired inequality (8) with yl = cl+l and 0 y1 = c ~ Ywhere ~ ,

E

0

= (cl/cz)E,

0

0

E =7 1 1 ~ ~ .

(11)

The regularizer R is chosen so that the ratio cl/cz is independent of the grid steps and has a simple structure. In the case of problem (1)-(3), it is logical to take the Laplace operator Y as the regularizer. Because of (6) we have c, = K,, a = 1,2, i.e. the constants are chosen according to the given thermal conductivity. Consequently, the ratio c ~ / c zdoes not depend on parameters of the grid. Two possibilities arise in this choice of the regularizer R. The first possibility is when the ratio nl/nz in a problem of stationary heat transfer is not very small. The second is associated with too small a ratio nl/nz when the number of iterations depends on this ratio inappropriately. In the latter case, the

173


convergence rate of iterative processes constructed directly by the operator A without the regularizer R should additionally be investigated. 4.7.3 DIAGONAL OPERATOR B The simplest class of iterative methods is associated with the choice of a diagonal operator B. In this case,

B = b(x)E,

(12)

and a new approximation is computed by explicit formulae An example of such methods is given by the Jacobi iteration method, in which the operator B is taken to be the diagonal part of the operator A and r k = T = 1. The problem of constructing the optimal operator B in the class of operators (12) is solved. The ratio = yl/yz in operator inequality (8) for A = A* > 0 is maximal if B is taken to be the diagonal part of the operator A. In this sense, the Jacobi method is optimal. Let us investigate the convergence rate of iterative method (7), (12) using a regularizer R. We set R to be the grid Laplace operator 7. We take into account that

<

iy= -Yi+l.~ - 2y.j + Y , - I , - Yi.j+~- 2yo + Y i - 1 83-

'4

ht and put in (12)

+ h;')

b(x) = 2 (h;'

.

(13)

In order to derive the bound in (10) (for R = A) under the choice (12), (13), we can use the least (6) and greatest (A) eigenvalues of the Laplace operator, namely 6 E 5 A 5 AE. (14) 0

The eigenvalues of the problem Ay = Xy are composed of the eigenvalues of the corresponding operators with respect to the variables xl and x2 and are represented in the form (see Problem 2 in Section 4.5)

X I , = A(,:)

+

2

4 . k ah, -Cg s1n2 *'

Ira -

,=I

whence we find the least and greatest eigenvalues 2

2

4 . ah, 6=c6,=c-sm2->cT, 21a=1 a=1 h2,

2

8 a=1 1,

174


From (14) and (15), given (12) and (13), we derive the bound (10) with

where lhI2 = h: + hi, According to (16) and (11) we derive the corresponding bounds for the number of iterations of the method (7), (12), (13). According to Section 4.6, the number of iterations of the method of a simple iteration with the optimal value T = TO = 2(y1 +yz) and that of the method of steepest descent with the operator B chosen by (12) and (13) can be evaluated

Thus, the number of iterations is proportional to the total number of nodes (unknowns) and the ratio of the maximal and minimal values of the heat conductivity. For the Chebyshev iterative method and method of conjugate gradients instead of (17) we derive (see Theorems 2 and 5 in Section 4.6) the following estimate:

The method with the Chebyshev set of iterative parameters converges much faster than that of a simple iteration. In the case of a large ratio x = nzlnz (in problems of stationary heat transfer in compound bodies with different-scale thermal parameters) the number of iterations (18) can inappropriately depend on X. Therefore, it is reasonable to consider the iterative method with the diagonal operator B constructed directly using A (without a regulaxizer). Let us consider iterative method (7) for problem (4) with the operator B specified by (12), where according to (1)-(3)

In order to estimate the convergence rate of method (7), (12), (19), it is necessary to determine the constants y,, u = 1,2, in inequality (8). Let us first show that in our case

Let us consider a new grid function vij = u(xl,,zzj) = (-l)"'u(z~i,xz~) from the set H of grid functions defined on iii and vanishing on aw. For this function (Bv, v) = (Bu, u) (21)


175

because of (12). We take into account (1)-(4) for A v and derive

at the separate grid node z = ( z l ; , z 2 j ) . Consequently, ( A v , v ) = 2 ( B u , u ) - ( A u ,u ) The constants yl and y, in inequality ( 8 ) are defined by the relations

We take into account ( 2 1 ) and ( 2 2 ) to obtain

We thus come to relation (20). Relation ( 2 0 ) implies that TO = 2 / ( y 1 + y z ) is the optimal value of the iterative parameter in the method of a simple iteration, assuming that B is chosen according to ( 1 2 ) and (19). The bound for yl is derived using the following generalization of the Friedrichs inequality (Lemma 3 in Section 4.4). Let us consider a one-dimensional function y ( z ) on the uniform grid E. The following statement holds (Problem 1 ) . Lemma 1. Let p ( z ) Then the inequality

> 0 and a ( z ) > n > 0 for z 6 w , y ( 0 ) = 0 and y ( l ) = 0 . 2

( a y r , I),+ 2 ( P Y , Y), holds, where M-'

(23)

= m a x v ( x ) and v ( z ) is the solution of the problem =Em

We now use the bound in ( 2 3 ) to derive the left-hand inequality in ( 8 ) for ( 1 2 ) , ( 1 9 ) . According to (23) we deduce

176


The grid functions M;'(x2)

and M;'(xl),

where M i 1 = maxua(x,), rEu,

u = 1,2, are defined by the three-point problems

in view of (24). According to (25) we have

We compare this with (26) and set yl to be 2

71 = min sew

M i 1 = Emin ~ E WM;'(x2)

+

min M;'(X~).

*IEW

(27)

a d

The investigation of the influence of thermal conductivity (the ratio x = r;l/n2), because of (20), is reduced to studying the quantity 71, which is specified by (26) and (27). We can employ asymptotic analysis (as lhl --, 0) to make some conclusions. Instead of boundary value problems (26), we consider the boundary value problems

taking into account (19). The analysis of boundary value problems (27) shows, for example, that for media with piecewise-constant thermal conductivity (compound media) it is possible that yl substantially depends on the coefficient ratio x = ~ z / n l(yl X-'). Thus, in several cases the number of iterations does not depend on x = n2/nl, no matter whether a regularizer is used. If the operator B is constructed by the diagonal part of the operator A, this dependence can only be weaker. It should be mentioned that this reasoning also relates, to an extent, to other techniques of choosing the operator B, which are discussed below, in particular t o the method of variable directions and triangular iterative methods. Advantages of variational iterative methods become more evident in problems with compound bodies with large ratios of heat conductivities. In the Chebyshev iterative method the choice of iterative parameters is only limited by the constants of energetic equivalence yl and y, of the operators A and B. The iterative parameters in variational methods are chosen taking into account the behaviour of the error in the approximate solution and are determined not only by the bounds of the spectrum (71 and y2) of the corresponding

-

177


generalized eigenvalue problem but also by the distribution of the eigenvalues. Thus, the number of iterations in variational iterative methods applied for solving heat transfer problems for compound bodies depends only slightly on x =mlnl. 4.7.4 TRIANGULAR ITERATIVE METHODS We can take the operator B not to be a diagonal operator (cf. (12)) but an operator corresponding to an upper (lower) triangular matrix or a product of such operators. Iterative methods of this class are extremely popular in applied computations. Firstly, we consider the case in which we choose B as an operator corresponding to one triangular matrix (asymmetric case). Further we will consider iterative methods with factorized triangular operators. Let us consider problem (4) for the positive self-adjoint operator A that is represented in the form

Let the operator D ' correspond to the diagonal part of A and let C correspond to the lower triangular matrix. Then since A = C 'D C*, we find

+ +

for the operators A,, a = 1,2, in view of representation (29). In order t o solve (4) by iterations, we use the following version of the method of a simple iteration

which is called the treangular iterative method. The method in (31) represented in the canonical form corresponds to the choice

Triangular iterative methods are often written as

We take into account (30) to get r = 2w/(2 - w). Among triangular iterative methods, we mention the Seidel method, for which r = 2 (w = 1).We also mention the method of over-relaxation, which corresponds to T = 2(6(2 - 6))lI2, where 6 is the constant from the inequality A 2 6'D, 6 > 0.

178


A priori information in triangular iterative methods can be given as the following operator inequalities:

T h e o r e m 1. Iterative method (291, (31)-(93) with the optimal T = 70 = 2/(6A)'I2 and with 2) = D* > 0 converges in HA,and the e n o r can be estimated as llznlla pnllzolla, where

<

E

Because of (34), the number of iterations necessary to obtain the accuracy can be evaluated in the usual way as

Note that in the above reasoning, D is not necessarily the diagonal part of the operator A. The over-relaxation method thus corresponds to the iterative parameter T close to the optimal value. The bound in (34) for the number of iterations also holds for the over-relaxation method. Let us illustrate the advantages of triangular iterative methods (31) by the 0

grid Dirichlet problem for the Poisson equation (equation (4) with A = A). We thus consider problem (1)-(3) with a,(x) = 1, a = 1,2. We take D to be 0

the diagonal part of the operator A, namely

The operators A,, a = 1 , 2 , in representation (29) can be represented as

on the set of grid functions that vanish on aw. Let us find the constants 6 and A in inequalities (33). According to (35) and the specified minimal eigenvalue (see (15)) of the Laplace operator, we have 2h; ?rhl 2ht 2 ?rhz 6= - sin + sin h: + h i 21, h? + h; 212

,


Using (35), (36) we get

The right-hand side is rewritten as

We take into account

We thus come to the inequality ( A I D - ' A 2 y ,y) 5 0.5(Ay,y ) . We compare this with the second inequality in (33) and conclude that A = 2. For the optimal value of the iterative parameter T = T O , we evaluate the number of iterations using (34), namely

This bound is close to that for the Chebyshev iterative method (see (18)) with a diagonal D . The Seidel method is c h a r ~ t e r i z e dby T = 2. The number of iterations is estimated as

no(.) = o ( + - l n t ) . Hence, the Seidel method converges much more slowly than the overrelaxation method. The number of iterations in the twedimensional problems considered, as in the method of a simple iteration, is proportional to the total number of nodes.

4.7.5 ALTERNATING TRIANGULAR METHODS The operator B in triangular iterative methods (31) is constructed according to ( 3 2 ) . Of course, the operator B can also be constructed using the operator A2. Alternating use of the operators A1 and A2 is typical,


181

Since then operator V is positive, we derive (By, y) 2 2w(Ay, y), i.e. A 5 yzB, where 1 yz = -. (43) 2w Now we can choose the parameter w in (40) from the condition of maximal ( = ((w) = yl/yz. According to (42) and (43) we obtain 71 ((w) = = h

2w6

1

+ w6 + w26A/4'

The maximum of ( ( w ) is attained for

and is equal to

In view of the bounds derived, we can formulate the corresponding result on convergence of the alternating triangular method, where the parameter w is taken to be optimal.

Theorem 2. The alternating triangular iterative method (7), (29), (33), (do), (44) with the Chebysheu set of iterative parameters converges in HA and H e . The number of iterations is estimated as n 2 no(&)= ln(2~-')/ln(p;'), where 1 - <'I2 p=1+ ('/2 and ( is defined by (45). We take into account that q is small, and derive a simpler expression for the number of iterations:

It is essential that the a priori information for the alternating triangular method is given in the form of operator inequalities (33). The alternating triangular method can be realized in a version of the conjugate gradient method. In this case, the number of iterations is characterized by (46). The class of alternating triangular methods is characterized by the choice of the operator B in the form (40) and is rendered concrete by specifying the operator V. Let us mention some possibilities.

182


We again consider problem (4) with A = A. The standard version of the alternating triangular method is characterized by

The constant 6 in the first inequality in (33) is the least eigenvalue of the operator A, namely

,

4 . ahl 4 . ah2 6 = - sin - + - sm ht 211 hi 212

-

The operators A,, a = 1,2, are defined by (36). According to the above estimate (37) we conclude

We take into account (48) and (49) to find the optimal value of the iterative parameter w for which the number of iterations in the Chebyshev iterative method is estimated as

Thus, the number of iterations is proportional to the square root of the number of nodes in one direction (to the fourth root of the total number of nodes in our two-dimensional problem). The bound in (50) is preferable as compared with the above bounds for other iterative methods. When solving problem (1)-(3) with variable coefficients, the alternating triangle method can be constructed using a regularizer R, e.g. the grid Laplace operator. In this case, instead of the bound in (50) we find

The alternating triangle method for such problems is constructed by choosing a diagonal operator D. As in the above iterative methods with the diagonal operator B, in several cases the dependence of the number of iterations on the ratio K I / K ~can be weakened. We expand the operator A corresponding to problem (1)-(3) and obtain, similarly to (36), that

183


+

We use the notation a:'(x) = al(x1 h1,xz) and a Z 1 ( x ) = a z ( x l , x 2 The operator 'D is given as 'Dy = d ( x ) E , where

+ hz).

The grid functions b, and c,, a = 1,2, are determined by solving three-point boundary value problems. In order to find b, = max v m ( x )we solve the problem z,E%

Similarly, c, = max w e ( x ) , where ="cum

Under this choice of the operator D ' the following inequalities hold:

We can analyse how the number of iterations depends on the ratio n l / n 2 by solving the above one-dimensional grid problems. The parameter w in alternating triangular method (7),(40) can be included in the operator V.We can thus use (7) with B = ('D+ A1)D-'('D A 2 ) . The iterative process can be optimized only by changing the operator 'D. It is worth noting the alternating triangular iterative method in the form

+

B = ('D + L)D-'('D + C).

(53)

If we take 'D to be OD,where'6 is a constant, the choice of ( 5 3 ) ,as in the case of triangular iterative methods, is equivalent to the choice above. Of course, if V # OD,these two versions of the alternating triangular methods are not equivalent. The version of the alternating triangular method in (7), (53) is referred to as the approximate fnctorjzntaon method. Let us suggest how t o choose the operator D ' in (53). Let a diagonal operator 'D he such that the row sums are equal to each other, i.e. Be = Ae, (54)

184


where e is a 'unit' vector (e = ( 1 , 1 , .. . , I ) ) . We can use (54) to construct D not using any a priori information about the bounds of the operator A. Let us consider the construction of the operator B under the conditions (53), (54) as applied to model problem (1)-(3).In particular, we should show that D > 0 and, hence, B = B' > 0. We rewrite problem (1)-(3) in the form

The appropriate coefficients in (55) vanish if y ( x ) , x E aw. In the opposite case they are defined as follows: a,, = h;'al(x) and b i j = h;'a2(x). For the diagonal part, ( 1 ) - ( 3 ) yield

The condition of diagonal predominance

is satisfied for problem ( 5 5 ) .The strict inequality holds for nodes ueighbonring theboundary ( i = l , N l - 1 , j = l , N ~ - 1 ) . Let us derive a formula for computation of the elements gij of the grid operator 2)taking into account that the operators A and B are 'close' because of (54). We obtain

therefore

Since the row sums are equal to each other according to ( 5 4 ) ,we derive


185

' can be found by the This implies that the elements of the grid operator D following recursion formula:

where the computations start with 911. Let us show that if the condition of diagonal predominance (56) are satisfied then q,, determined by (57) are positive. Let

.,

- b,jpij-l. This formula is We can then rewrite (57) a s gij = d - a ,,p,-1, . . transformed as follows:

Relation (59) together with (56) can be used to prove by induction that the conditions gij > 0, pij 5 1 are satisfied, i.e. B = B* > 0. To investigate the convergence rate of the alternating triangle method in (7), (53), (54), let us estimate the constants of energetic equivalence in inequality (8). According to (57), we have the representation

where C is the grid operator defined by the expression

After some algebra we find

Consequently, C 2 0 and yl = 1 in (8). It is more difficult to evaluate yz. Among all possibilities, we single out the case in which p 1 (>O, XEU. (62) Relation (61) yields

186


+

+

We apply the €-inequality ( a b)' 5 ( 1 € ) a Z+ (1 + E-')b2 with ai+l,,/bi,j+l to the right-hand side of (63) and get

E

=

Since for diagonal predominance

we come to the bound ( C y ,y) 5 ( 1-C)(Ay, y ) , provided that (62)is satisfied. Thus, yz = C-' in inequality (8). Hence, if (62) is satisfied, iterative method ( 7 ) , (53), (54) with the Chebyshev set of iterative parameters converges, and the number of iterations can be estimated as

where 5 is a constant in (62). The estimate C = O ( N l+ N z ) holds for a wide class of grid elliptic problems, therefore the alternating triangle method in the version of the approximate factorization method converges in the number of iterations proportional to the square root of the number of grid nodes in one direction. 4.7.6 PROBLEMS 1. Let p(x) > 0 , p(x) $ 0 and a ( x ) > n > 0 on a uniform grid w . Show that the estimate (a& 2 M(m,vLI (64)

where M-' = max,~, u ( x ) and u ( x ) is the solution of the problem

holds for any grid function y ( x ) such that y(0) = 0 and y(1) = 0. Solution. Let us define a grid operator A = A* > 0 on the functions that vanish at the nodes of the closed interval [O,I] by the relation Ay = -(a%)., x E w. Let us consider the eigenvalue problem

187


for all y(x), where ();. = (., .),. Equality holds on an eigenfunction corresponding t o the minimal eigenvalue Amin. We denote this eigenfunction by w(x). Since (Ay, y) = (ay$, I ) + , because of (67) the desired hound in (64) will be established if we show that M 5 X,i,. First of all, let us explain that w(x) > 0, x E w (more exactly, the function w(x) has a constant sign). Let us assume the contrary, i.e. w(x) changes its sign on the grid w. We consider the function lw(x)l and obtain

This inequality contradicts the assumption that w(x) minimizes the ratio (my$, l ) , + l b y 2 , 1). Let us now consider the relation Aw = X,i,p(x)w. This, because of W(X)> 0. x E w, yields

We take into account (66) to evaluate the denominator (pw, 1 ) = (Av, w) = (u,Aw) 5 maxu(x)(Aw, l ) , =Em

because Aw 2 0, x E w. Thus, (68) yields Xmi, proof. 2. Let in the iterative method (7)

> M, which completes the

and let the operator B he represented in a factorized form:

Determine the optimal value of the parameter w. Solution. The convergence rate is defined by the constants yl and y2 in (8). Thus, let us first choose w in (70) so that the ratio -(l/yz be maximal. This investigation is similar to that for the alternating triangle method. According to (70),

+

+

B = E - w ( A i + Az) W ~ A I A Z Zw(A1 + A2) = ( E - wAt)(E - wA2) + 2wA. Therefore

= 1/(2w). We take into account (69) and derive

188


Let 6 = min6, and A = max A,, then a

Consequently,

0

x

We minimize ( = ((w) = 711%to determine the optimal value w = wo = (6A)-'I2. Factorized operator (70) corresponds to the method of alternating directions with an autonomous definition of iterative parameters r k + , in iterative method (7) (independently of the parameter w).

4.8 Numerical Solution of Problems in Irregular Domains

4.8.1 CURVILINEAR ORTHOGONAL COORDINATES

Difficulties arise when numerically solving boundary value problems of heat tracsfer in complex computational domains. We have thus far considered problems in a rectangular domain (a regular computational domain). The simplest approach to the solution of problems in nonlinear coordinates is to use curvilinear coordinates in which the computational domain becomes regular. We first consider orthogonal curvilinear coordinates. In the physics of heat, much attention is paid to cylindrical coordinates when modelling thermal fields in cylindrical domains. As the second example, we mention the spherical coordinate system (see Section 2.1). The stationary thermal field in an isotropic medium in a cylindrical coordinate system is described by the equation

Another example of a twwdimensional problem of heat transfer is provided by an axisymmetric problem, in which coefficients, right-hand side and solution do not depend on the angle ip. In this important special case equation (1) becomes i a Equation (1) can also be simplified if thermal characteristics and thermal field do not depend on the coordinate z (a plane problem of heat transfer).


189

Equation (1) in this case corresponds to the use of polar coordinate system (T,91, namely

Equations (2) and (3) can be rewritten in the following divergent form:

Except for the notation, equations (4) and (5) belong to the above considered class of heat equations written as second-order self-adjoint elliptic equations with variable coefficients, namely

In (4) we used the notation

and replaced xlf (2)by f (x).Equation (5)is written in the form (6)with the anisotropic 'heat conductivity'

Equation (6) is solved in a rectangle R, which corresponds in the original cylindrical variables for (4) to the computation of the temperature in a cylinder (hollow cylinder if X I 2 1 > 0) in a cross-section ip = const. Similarly, the rectangle R in (6) corresponds to the solution of (5) in an infinitely long cylinder whose cross-section is a disk, a circular sector, a ring or a sector of a ring. Since the equations are reformulated, topics in the construction of difference schemes, investigation of convergence of a difference solution to an exact one, and solution of grid equations are considered as in the above. It only remains for us to mention the main typical features of boundary value problems (6), (7) and (61,(8). Equation (6)with (7) and (8) degenerates for x1 = 0, therefore one should be careful when solving problems in which this point lies on the boundary of Cl (a solid cylinder in (4), disk or circular sector cross-section in (5)). It is

190

COMPUTATlONAL HEAT TRANSFER

natural to require the solution of equation (6), (7) (or (6), (8)) to be bounded for zl = 0, which is equivalent to the requirement

au

lim zlk(z) - + 0. axl

=I-0

Condition (9) corresponds to the absence of a thermal flux for zl = 0. Let us discuss special features of the construction of difference schemes satisfying condition (9). For this purpose, we separate the operator L1 with respect to the variable 21:

We have to solve the boundary value problem for equation (6),(7) (or (6), (8)) in the rectangle R = {XI x = (XI,ZZ),1; < z, < 1;) for 1; = 0. The equation is equipped with boundary condition (9). It is convenient to use the following quasi-uniform grid with respect to the variable z l grid

i.e. the uniform grid shifted by half a step (a flux grid). Further, difference approximation of (6), (7) (or (6), (8)) is constructed by the integrointerpolation method (Section 4.2). We denote ql(x) = -xlk(x)au/ax and integrate equation (6) with respect t o xl from 0 to hl (over a neighhourhood of the node XI = hl). We thus get

The flux ql(O,xz) = 0 because of condition (9), and the succeeding transformations are made in the usual way. We thus come to a difference scheme for equation (6), (7) (or (6), (8)) with condition (9), for which the difference solution for zl = 0 cannot be found. The second possibility of the construction of a difference scheme for (6), (7) (or (6), (8)) is associated with the use of an ordinary grid. It is typical of difference problem (6), (8) that all the difference solutions are equal to each other for xl = 0. We should separately discuss realization of the latter condition and can recommend the flux grid with respect to the radial variable. When solving problems like (6), (8), we should specifically mention periodic boundary conditions with respect to the variable zz (1; = 0, 1: = 2a). We have already discussed an appropriate algorithm for solving one-dimensional grid problems with such conditions (Section 4.5, Problem 1).


191

4.8.2 IRREGULAR GRIDS Irregular computational grids are traditionally widely used for the approximate solution of problems of stationary heat transfer. We consider the model Dirichlet problem for the Poisson equation (stationary heat transfer in a homogeneous medium)

to illustrate possible approaches. In several cases, a consistent grid can be constructed for irregular domains that is formed by the nodes of an ordinary nonuniform rectangular grid together with nodes lying on the boundary. Figure 4.5 shows an example of

Fig. 4.5.

Fig. 4.6.

192


Fig. 4.7.

a consistent grid. In order for the boundary to be formed by nodes, strongly nonuniform grids must be used. Difference grids for problem (lo), (11) on a given grid are constructed as usual. A consistent grid can be constructed only for a narrow class of domains. Therefore the problem of irregular computational domains is solved within other approaches. The simplest method is to use an ordinary rectangular grid inside the computational domain and approximate boundary condition (11) at the grid node closest to the boundary a R (Fig. 4.6). We actually replace problem (lo), (11) by another problem in another domain whose boundary is consistent with the grid (an approximation of the boundary). It is most natural and universal t o use a grid that consists of the nodes of a regular (uniform) grid (the internal nodes) and additional irregular boundary nodes x E aw lying on the boundary of the domain fl (Fig. 4.7). These nodes are the intersection points of lines passing through the nodes of the regular grid with the boundary of the computational domain. Let us discuss the construction of &fference schemes on these grids in more detail on the example of model problem (lo), (11). It is reasonable to approximate the equation at a node 0 near the boundary (Fig. 4.7) using the irregular five-point stencil

Difference scheme for problem ( l o ) , (11) constructed using approximation (12) is studied in the usual fashion. In particular, the maximum principle (see Section 4.3) holds for these schemes. It can be applied to estimate the accuracy of difference schemes like that in (12) in the uniform norm. It is more difficult to study properties of a difference operator in a grid Hilbert space H. For the Dirichlet problem, it is logical (see Section 4.4) to


193

define H as the set of grid functions vanishing on aw equipped with the scalar product (Y,w) = x y ( x ) w ( x ) h ~ h z , (13) rEu

where w is, as usual, the set of internal nodes. It is easy t o check that the operator A corresponding to an approximation like that in (12) is not self-adjoint in the space H introduced. This important property is lost after passage from the differential problem to the difference one, which significantly complicates construction of effective schemes for solving the grid problem. When we studied iterative methods (see Sections 4.6 and 4.7), we only considered problems with self-adjoint grid operators A. This difficulty can be overcome in two different ways. Let us introduce a new space H, in which the scalar product is defined in a more complicated fashion. In our case, the grid step depends on the node (it is no longer essential for us that the grid is quasiuniform). We determine the average grid steps in different directions for each node. For example, for (12)

Let h(x) = hl(xl)h2(xz) and let the scalar product in H be defined as

According to this definition of H, the difference operator A corresponding to approximation (12) is self-adjoint, as desired. Alternatively, we can preserve the Hilbert space with the scalar product in (13), hut slightly change the grid operator itself. Instead of difference equation 1121 we use

It can be shown that this modification does not make the monotonicity properties of the difference scheme worse, its accuracy is preserved, while the operator A is self-adjoint in a simple Hilbert space H with scalar product (13). We can thus choose (12), (14) or (13), (15). We suggested several possibilities for solving problems of stationary heat transfer on irregular orthogonal (rectangular) grids. It is natural that nonorthogonal grids provide a wider range of possibilities. For instance, one can use nonorthogonal quadrilateral grids. This technology has its own peculiarities both in construction of difference schemes and in their investigation. These problems are beyond the subject of this monograph.

194


4.8.3 METHOD OF FICTITIOUS DOMAINS The method of fictitious domains is useful for solving boundary value problems in irregular domains because it is easy to code. It is based on supplementing a computational domain to a regular domain Ro ($2 C n o ) , e.g. to a rectangle in two-dimensional problems (Fig. 4.8). Then the problem is solved in Ro by the usual methods. We should extend the solution of the original problem into the fictitious domain R1 = R \ Ro so that the difference solution of the problem in the extended domain gives an approximate solution in the original domain R. The problem in the extended domain contains small (large) coefficients in the differential equation. Thus, accuracy and computational realization of iterative methods should be specially investigated as applied to these problems. Let us illustrate some advantages of the method of fictitious domains in the example of problem (lo), (11) with homogeneous boundary conditions. We only consider the convergence of the approximate solution to the exact one in terms of differential calculus. We denote the approximate solution of (lo), (11) with g(x) = 0 by u,(x), where E is a parameter of extension. We define this solution as the solution of the following Dirichlet problem:

Among the basic versions of the method of fictitious domains, we single out the method of extension with respect to the principal coefficients. For problem (lo), (11) with g(x) = 0, we specify the coefficients of the problem in the extended domain as follows:

Fig. 4.8.


195

where E is small. The version of the method of fictitious domains (16)-(18) relates to the problem of stationary heat transfer in a compound body RQ = 01n R , given a large thermal conductivity in a portion of the computational domain (in 0 1 ) . It is logical to expect that under these conditions the temperature in R I level and the conditions on the boundary (17) yield u E ( x )+ 0 as E 0. The corresponding bound for the discrepancy between the approximate and exact solutions is 11u,(x) - U ( X ) I [ ~ ; ( ~ ) 5 ME'. Another common version of the method of fictitious domains for approximate solution of the Dirichlet problem for elliptic equations is the version with extension with respect to lowest terms. In this case, the second derivatives are continuously extended into the fictitious domain. In order to approximately solve problem ( l o ) , (11) with homogeneous boundary conditions, we involve the boundary value problem (16), (17) with the coefficients

-

The corresponding bound for accuracy of the version (16), (17), (19) has the form (20) IIuc(x) - 4 x ) l l w ; ( n ) 5 M E . This version of the method of fictitious domains allows us to consider problems like ( l o ) , (11) with nonhomogeneous Dirichlet conditions. For this purpose, we extend the function g ( x ) into the fictitious domain Ol and define the righthand side of equation (16) to be f,(x) = ~ - ' g ( x ) for x E 0 1 . Boundary conditions on 80, which lies inside the extended domain 00,are most naturally specified in the version of the method of fictitious domains with a surface delta function. We extend f ( x ) into the fictitious domain 01 and instead of ( l o ) , (11) consider the problem for the equation

with boundary conditions (17). Let us estimate the accuracy of the version of the method of fictitious domains in which the surface delta function is used.

Theorem 1. The bound in (20) holds for the difference between the approximate solution u, defined by (21), (17) and the exact solution u ( x ) of problem (101, (11). Let us define the solution of problem ( l o ) ,(11) in the fictitious domain as the solution of the problem

196


IL(X)= g(x), U(Z)= 0,

x E 80, x E an,.

We multiply equations (10) and (22) by the discrepancy w,(x) = u,(x) u(x), integrate the resultants over R and 0 , respectively and find

Here (.I stands for the jump of a function when passing through the boundary 8 0 . Similarly, for problem (21), (17) we have

We subtract (25) from (26), take into account the boundary conditions ( l l ) , (23) and come to the relation

The Cauchy-Schwarz inequality applied to the right-hand side of (27) yields

We discard the first term in (27) and derive from (27) and (28) that

If (29) is satisfied, according to the Friedrichs inequality (see Section 4.1, the homogeneous boundary conditions in (17), (24)), the bound 11uC(x)u ( ~ ) I / q ( ~5, M ) Efollows from (28). Moreover, this bound holds for a portion of the domain R c Ro, which yields the desired inequality (20). Nowadays, versions of the method of fictitious domains for more complex problems than those in (lo), (11) are developed. For instance, boundary conditions of the second or third rather than first kind can be given.

197


Large difference in coefficients of an elliptic operator is typical of boundary value problems like (16), (17) solved by the method of fictitious domains. In the version with extension witb respect to the principal coefficients (see e.g. (18)) a large (small) parameter multiplies the leading derivatives, and we thus encounter a singularly perturbed problem. The version with extension with respect to the lowest coefficients is not associated with such essential transformation of the problem. Hence, the latter is preferable for applied mathematical modelling. When constructing difference schemes of the method of fictitious domains, it is useful to apply the integreinterpolation method, which was developed just for problems with discontinuous coefficients. Grid problems are solved by various iterative methods. As for the convergence rate of iterative processes, the versions with extension witb respect to lowest coefficients are preferable as compared with those employing extension with respect to principal terms.

4.8.4 DECOMPOSITION METHODS WITHOUT OVERLAPPING A decomposition method, in which the computational domain is separated into simple subdomains, gives another approach to solving boundary value problems in irregular conlputational domains. This approach is actively discussed as applied to developing methods for solving boundary value processes on parallel computers. In each subdomain, its own problem is solved. The domains are connected via boundary conditions. In each subdomain, its own grid is introduced, which can be inconsistent with grids in other subdomains. Therefore, difference schemes of decomposition methods can be treated as difference methods on composite grids. There are two important classes of decomposition methods. The first is associated with partitioning the domain into nonoverlapping subdomains. In the second class of methods, separate subdomains may intersect one another.

Fig. 4.9a.

Fig. 4.9b.

198


In order to explain the essence of the decomposition methods, we consider a model problem in an L-shaped (Fig. 4.9) domain. It is convenient to divide this irregular domain R into two regular subdomains (rectangles) nl and 0 2 without overlapping. Figure 4.9 shows two simple versions of such a partitioning of the L-shaped domain. In order to solve difference problems in the subdomains, in several cases we can employ fast direct methods for solving grid equations considered in Section 4.5. Therefore, particular attention in the following is paid to methods for determining the approximate solution on the common boundary y of the subdomains R1 and Rz (y = awl n 802). Assume that boundary value problem (lo), (11) is solved in the domain S1. As for the method of fictitious domains, we only consider the main points of the decomposition method from the viewpoint of differential calculus. If the boundary condition on y were known, the solution of the problem in R would be reduced to solving two separate problems in subdomains 01 and Rz. Let us denote the exact solution of problem (lo), (11) in the subdomain Rl by ul(x) and that in R2 by u ~ ( x ) The . junction condition on the boundary y in this notation becomes

Here n,, a = 1,2, is the external normal relative t o the domain S1,, u = 1,2. Let us construct the simplest iterative process to explicitly refine the first-kind boundary condition on y taking into account (30) by the formula

where u(s) = ul(z) = u2(x). The boundary condition is refined according to the flux unbalance. In each separate subdomain R,, u = 1,2, we solve the boundary value problem

u

= s(x),

2:

E an,\y,

=u x ,

X

E y.

(32)

An alternative to (31) is iterative refinement of the second-kind boundary condition (using the temperature unbalance), namely


where now

is the general second-kind boundary condition on 7 . In a more general operator form, the implicit iterative refinement of a boundary condition like (31) is written as follows:

where

Iterative methods like (33), (34) are known as iterative methods of capacity matrix. Let us mention the basic topics of investigation of convergence of iterative decomposition methods. It is convenient to consider the iterative process for the discrepancy wk(x) = uk(x) - u(z). In this case (33) yields

where the operator Aw defined by (34) is given for x E 7, where (see (lo), (32))

Let us show that the operator A defined on the set of functions satisfying (36) is self-adjoint in 71 = L2(y). In this case, because of Green's formula

i.e. A = A * . Similarly, the energy of the operator A is written as

200


We take into account the Friedrichs inequality and find

with some positive constants c,, a = 1,2. Further improvements are associated with estimates of traces of the functions w,(x) from Wi(Cl,) on y. Since the conditions are nonhomogeneous only on y, the inequalities

hold. Thus, (37) becomes

We take into account the embedding ~ : / ~ ( yinto ) L2(y) and obtain

where m = cs > 0, i.e. the operator A is positive definite in X. The derived inequality (38) points to a preferable choice of the operator B in iterative process (35). Let B = n' and

then the operator B is energetically equivalent to the operator A. In particular, the operator B can be constructed using energetic norms in ~ i ~ ~ ( y ) . We can deduce similarly in terms of difference schemes as applied to problem (lo), (11) considered in the L-shaped domain. In particular, we can examine the convergence rate of method (31), (32) with explicitly specified boundary condition. This consideration yields the following bounds for the corresponding difference operator:

6E < A

< AE,

6 = O(1) > 0,

A = O(h-I),

(39)

where h is the grid step with respect to the variable xz (Fig. 4.9a) or XI (Fig. 4.9b). The bounds in (39) are used to estimate the convergence rate of specific iterative methods. When realizing the iterative decomposition methods, it should be taken into account that on each iteration the conditions are refined only on a part of the boundary, and it is necessary to find the solution of the grid problem in a subdomain in a portion of nodes. This problem in several cases can be solved

201


by direct methods faster than the problem of determining the solution at all nodes of the grid. Other versions of the decomposition method became widespread in computational practice. Among these, we can mention approaches with differenttype conditions on common boundaries of the subdomains (the DirichletNeumann condition). When decomposition methods are solved by parallel computers, increased attention is paid to the dependence of convergence rate of the corresponding iterative processes on the number of subdomains. 4.8.5 DECOMPOSITION METHODS WITH OVERLAPPING SUBD0MAI.W It is more prospective for approximate solution of boundary value problems in complex irregular domains to use decomposition methods with overlapping subdomains. When we decompose an irregular domain, it is logical to try t o divide it into regular subdomains. In this case, the problem in a regular subdomain can be solved most effectively (generation of grid, construction and investigation of difference schemes and solution of grid equations are simplified). This approach can be realized more successfully if decomposition methods with overlapping subdomains are used (because there is much freedom in choosing regular subdomains). Figure 4.10 shows examples of decomposition of the L-shaped domain into overlapping subdomains. We consider the iterative method for solving model problem (lo), (11) using the solution of the problem in separate subdomains. Iterative decomposition methods for boundary value problems for secondorder elliptic equations are usually constructed by the classical alternating Schwarz method and its modifications. Let us introduce the necessary notation. Let r, = annn,, y, = 80, \r,, cu = 1 , 2 , therefore aR = rl u rz and 80, = r, U y,, a = 1 , 2 . In iterative decomposition methods, boundary value problems are successively solved in the subdomains 01 and Rz, consequently we should specify boundary

Fig. 4.10a.

Fig. 4.10b.

I I

202


conditions on y,, a = 1 , 2 on each iteration. In the alternating Schwarz method, Dirichlet conditions are formulated on these parts of the boundary. We denote the approximate solution of problem ( l o ) , ( 1 1 ) on the kth iteration hy u t ( x ) in the domain Rl and by u,k(x) in the domain R2. Given u : ( x ) , z E y2, we can find u,k on yl solving a boundary value problem in the subdomain R2, namely

where

Given u i t l ( x ) , we refine v l ( x ) on the boundary y l . For instance,

The classical alternating Schwarz method corresponds to the choice T = 1 in ( 4 3 ) . In order to specify u:+'(x) in the whole subdomain R 1 , we solve the boundary value problem

The convergence rate of iterative process (40)-(45) in C ( R ) is established by the maximum principle for second-order elliptic operators (see Section 4 . 1 ) . We introduce the notation '

Let us define v , ( x ) as the solution of the boundary value problem Lv, = 0 , v,(x)=O,

x E R,, xEr,,

v , ( x ) = 1,

x E y,.

203


Theorem 2. Iterative process (40)-(45) with 0 < T < 2/(1+9192) converges at the rate of a geometric progression to the solution of problem (lo), (11). If the iterative parameter is set to be optimal, i.e. T = TO = 1, the following bound holds: Iuk(x)- 4 x ) I 5 M ( q l q d k , x E 0. (49) In order to prove that u k ( x )converges to u ( x )in R , because of the maximum principle, it is sufficient t o prove that u y x ) converges to u ( z ) only for x E yl. Let w,(x) = u,(x) - u ( x ) ,(Y = 1,2. We define the operator A by the relation

Let G,(x, a ' ) denote Green's function of the Dirichlet problem for equation (46) in the domain a,, a = 1,2. According to (43)-(45) we have the representation

Similarly, we define the operator Sz by (40)-(42), namely

We take into account (51),(52) and (see ( 4 2 ) )

w ~ + ' ( x= ) w:(z),

x E 7.2

for the operator A and find

Awl = w i ( x ) - . ? ~ S ~ w l ( x ) , x E 71 We take into account that a G l ( x , x ' ) / a n l is positive and obtain

It is easily seen that

204


where vl(x) is the solution of problem (46)-(48). We thus derive from (55) the bound l l ~ i l l c (5~~i11~i11c(~,)~ ~) IISill 41. (56)

<

Similarly, from (52) we get

We take into account (53), (56) and (57) to write

According to (54) and (58) we get the inequality

We substitute (54) into (43) and derive

The transition operator S = S(T) (W:+~(X)= Swk(x)) can be represented as S = (1 - TIE TSZSI.We take into account the bounds for the norms of the operators S,, cr = 1,2, to write

+

The sufficient condition for convergence llS(~)ll5 1 yields restrictions on the parameter T (0 < T < 2/(1 + q1q2)) that appear in the formulation of the theorem. Expression (61) immediately yields that minl[S(~)IIis attained for 7 T = T~ = 1. For this value of the iterative parameter we deduce from (60)

This immediately yields the bound in (49) to be proved. Currently, there are several different generalizations of the alternating Schwarz method. For instance, an asynchronous version is worthy of note. The classical (synchronous) version of the Schwarz method considered in the above corresponds to the solution of the problem firstly in one subdomain ( n l in our case), refinement of the boundary condition (on yl) and solution of the problem in the other subdomain (in nz). This strict (synchronous) sequence of computations can prevent effective use of the alternating Schwarz


205

method by parallel computers. The asynchronous method does not have this disadvantage and is based on the iterative procedure

rather than on (43). That is, the boundary condition on yl is refined by the solution in the domain Rz on the kth iteration. As compared with the classical Schwarz method (Problem 2), the convergence rate of the asynchronous version is much slower (twice as slow for Q1 = qz). The maximum principle for grid problems (see Section 4.3) can be applied to investigation of grid analogues of the alternating Schwarz method, as was shown in the above. 4.8.6 PROBLEMS

1. Consider t h e a p p r o x i m a t i o n of t h e third-kind b o u n d a r y condition au E an (63) a n U(X)U= g(x), for e q u a t i o n (10) o n a consistent grid. Solution. Let us consider a node 0 on the boundary (Fig. 4.5). We rewrite (63) in the form

+

where cos(n,x,), a = 1,2, are the direction cosines of the external normal. The simplest approximation of (64) yields

We rewrite the error of approximation as

Thus, (65) approximates boundary condition (64) with the first order. In several cases, the order of approximation on the solutions of equation (10) can be increased by specially choosing the grid steps h*, a = 1,2, of the consistent grid. We rewrite (66) in the form

206


If the grid steps are related as follows:

we take into account (10) for the approximation

and write the error in the form $(a) = 0((hi)2 + (h;)2). Of course, the consistent grid and, what is more, the grid satisfying (67) can only be constructed in several special cases. 2. Study the convergence rate of the asynchronous version of the alternating Schwarz method for the case in which the boundary condition is refined according to (62). Solution. Let us study convergence of the alternating Schwarz method (40)(42), (44), (45), (62) on the direct sum of the spaces 'H = 'Hi 'Hz, where H ' , = C(y,), a = 1,2. We define the norm of an element U = (u1,uz) E 'H by ! . iterative process (40)-(42), the expression [IUll = IIulllcct,) l l ~ l l ~ ( + ~ The (44), (45), (62) for the error W = (wl,w2) is written as

+

According to the above notation, we express the operator A

Because of (69) we rewrite (68) in the form

We take into account the above bounds for S,, a = 1,2 to derive from (69), (70) that

Hence, iterative process (40)-(42), (44), (45), (62) converges for 0 < T < 2/(1 q), where q = maxq,, at the rate of geometric progression to the m

+

k solution of problem (lo), (11). The bound llu:llc(,,)+ lluzllC(72) 5 Mqk holds for the optimal value of the iterative parameter T = TO = 1. We compare this bound with that in (49) and conclude that the asynchronous version of the Schwarz method converges much slower than the synchronous version. The number of iterations for the former is at least twice that for the latter.

4.9 Nonlinear Problems of Stationary Heat Transfer

4.9.1 BOUNDARY VALUE PROBLEM FOR A QUASI-LINEAR

EQUATION Let us discuss some topics related to numerical solution of nonlinear problems of heat transfer. As a model problem, we consider the problem of stationary heat transfer in a rectangular domain R when the thermal characteristics depend on the temperature. The temperature is determined as the solution of the following Dirichlet problem for the second-order quasilinear elliptic equation:

As usual, we assume that the conditions

are satisfied. Let us first establish sufficient conditions of uniqueness of the solution of ( I ) , (2). Uniqueness of solutions to nonlinear problems is studied using the corresponding linear problems. Let us illustrate this general statement by the example of problem ( I ) , (2). We assume that two solutions exist, namely u4(x), 0 = 1,2, i.e.

Let ut = tuZ(x)+ (1 - t)ul(x) and w(x) = uz(x) -%I(%), then

208


According to the notation introduced for the difference of the solutions, we find from (4) and (5):

Thus, the problem of uniqueness of the solution to the nonlinear problem is reduced to studying the uniqueness of the solution of a homogeneous Dirichlet boundary value problem for linear equation ( 6 ) .The simplest situation occurs when the thermal conductivity does not depend on the temperature (k = k(x)).Then if

the classical maximum principle (Section 4.1) holds for equation (5). We thus conclude that boundary value problem ( 6 ) , ( 7 ) has only trivial solution w ( x ) = 0, i.e. the solution of problem ( I ) , (2) is unique, provided that the thermal conductivity is linear and ( 8 ) is satisfied. The situation with k = k ( x , u ) is more interesting. It turns out that the maximum principle also holds for linear equations like ( 6 ) (see Problem 1). This is the case if the derivative of the thermal conductivity with respect to temperature is bounded, i.e.

in addition to ( 6 ) . Condition ( 8 ) is also assumed to be satisfied. In view of this we also establish uniqueness of solution for problems more general than ( I ) , (2). In this connection, we only mention the case in which the thermal conductivity depends both on derivatives and temperature and problems with nonlinear boundary conditions. Some important examples of such problems are presented in the following. 4.9.2 DIFFERENCE SCHEMES

Let us present some difference schemes for the quasi-linear problem of heat transfer. We assume that the uniform rectangular grid Z = w U dw with the steps hl and hz with respect to the variables xj and x2, respectively, is


is introduced in a rectangle difference scheme

n. We

209

relate to problem ( I ) , (2) the nonlinear

The right-hand side of (10) for smooth functions f (z, u) can be rewritten as 'P = f h y ) ,

" E w.

(12)

The difference operator in (10) is represented as

The coefficients of the nonlinear difference operator, in the simplest (see Section 4.2) case, are taken in the form

Other approximations can also be used, e.g

I t is convenient to rewrite difference scheme (10)-(12) with approximation (15) as

For nonlinear difference schemes (10)-(14) ((10)-(13), (15) or (10)-(12), (16)), the error is studied similar to linear problems (Section 4.2). Each of these schemes approximates the original boundary value problem with smooth coefficients and solution with the second order. It is much more complex t o study the accuracy of nonlinear difference schemes. In particular, only for linear schemes do stability and approximation yield convergence of the difference scheme (see Section 4.2). A general theory

210


of nonlinear difference schemes is not yet developed. Therefore each specific nonlinear scheme should be studied separately. Besides, such research involves more advanced mathematics and is usually cumbersome. Therefore, in this book we only consider the simplest case of problem (I), (2), in which the thermal conductivity is linear and nonlinearity is concentrated on the righthand side. Our consideration is based only on the maximum principle for difference schemes formulated in Section 4.3. 4.9.3 CONVERGENCE OF THE SIMPLEST DIFFERENCE SCHEME We consider difference scheme (10)-(12) with the linear difference operator

The first problem we encounter is associated with the unique solvability of the difference problem. Unlike linear problems, nonlinear grid equation are not necessarily solvable if the corresponding homogeneous problem has only a trivial solution. Solvability should be additionally studied. Difference problem (10)-(12), (17) is uniquely solvable if condition (8) is satisfied. We omit the proof because it is not so simple. Let us examine the unique solvability of difference problem (10)-(12), (17). As for differential problem (1)-(3), we assume that problem (10)-(12), (17) has two solutions yp, P = 1,2. We obtain a problem for the difference v(x) = yz(x) - yl(x), namely

where

Because of the maximum principle (Section 4.3) for grid problem (la), (19) we obtain v(x) = 0, x E a,provided that (8) is satisfied. That is, if (8) is satisfied, we have a solution of nonlinear difference problem (10)-(12), (17). Let us now formulate the problem for the error z(x) = y(x) - u(x), x E ii. We have

211


where g is defined similarly to (20) and the error of approximation is $ ( x ) = Au - f ( x , u ) .For smooth coefficients and solution of the differential problem , J h J 2= hy hz. The estimate (see Section 4.2) we find $ ( x ) = 0 ( l h J 2 )where in the uniform norm

+

Ild~)llcc~) C Mll$(~)llc(w)

(23)

holds for the solution of difference problem (21), (22), provided that ( 8 ) is ( ~ w) ( x ) is defined by the conditions satisfied. Here M = I l w ( ~ ) l l ~and AW

2 1,

E w,

W(X)

2 0,

E

aw.

Thus, if (8) is satisfied, nonlinear difference scheme (10)-(12), (17) converges with the second order. Note that the maximum principle yields that the solution of the difference problem is bounded. We thus have

where

We rewrite difference equation (10)-(12) taking into account (24) and get

If ( 8 ) is satisfied, in view of (23) the solution of problem ( l l ) ,(25) can be ( ~ ) , the constant M is defined estimated as JJy(x)JJc(,) MJl~ ( X , O ) J J ~ where in the above.

<

4.9.4 ITERATIVE SOLUTION OF NONLINEAR GRID SCHEME Nonlinear difference equations can be solved by iterative methods. We do not consider in detail topics in the general theory of iterative solution of nonlinear difference problem (10)-(12), (17) and again only investigate convergence of the simplest scheme in (10)-(12), (17). We consider nonlinear difference problem (10)-(13). The simplest iterative process involves computation of the coefficients of the grid operator A by the preceding iteration. If the right-hand side is calculated in the same way, the new approximation yk+l(x) is determined by solving the linear problem

212


Nonlinear problems are usually solved by methods related to the classical Newton method (quasi-linearization methods). Linearization of the right-hand side

results in the iterative process that is based on the solution of the difference equation

with boundary conditions (27). It seems to be more systematic to apply quasilineadzation to the whole difference equation (10)-(13), which corresponds to the use of the Newton method. In this case, the new approximation is determined from the difference equation A1(yk)yk+l = F(x,yk), Ew (29) and boundary conditions (27). The grid operator A1(yk) in (29) has the form

For the right-hand side we have

Solution of nonlinear difference problem (10)-(13) by the Newton method (27), (29)-(31) requires inversion of the elliptic grid operator A1(yk) on each iteration. This operator, unlike the operator A(yk), is not self-adjoint, which essentially complicates the problem. Iterative method (27), (28) with partial linearization (only with respect to the right-hand side rather than the thermal conductivity) is preferable in this sense. However, it should be mentioned that the Newton method has a faster (quadratic) convergence rate. Various generalizations of classical iterative methods for solving linear systems of equations are commonly used for solving nonlinear systems of


213

equations. For example, we mention nonlinear relaxation methods. We write the simplest nonlinear difference scheme (10)-(12), (17) in the form

where the notation of Section 4.7 is used. The iterative relaxation method with iterative parameter w = 1 corresponds to specifying the new approximation by solving the problem

In iterative process (32), a one-dimensional nonlinear equation is solved at each internal grid node. Possibilities of optimization of iterative methods for nonlinear difference problems are narrower than those for linear grid equations. In particular, this is the case for the choice of iterative parameters. When using iterative processes like (27), (28) and (27), (29), an elliptic grid problem is solved on each iteration. This linear problem in turn can be solved by iterative methods. We thus come to two-step iterative methods, which are characterized by external and internal iterations. The two-step iterative processes can be optimized by limiting the number of internal iterations (there is no need to compute the solution of the linear problem until it converges, i.e. to obtain the accuracy of the entire problem). 4.9.5 CONVERGENCE OF ITERATIVE METHODS

We use the simplest nonlinear difference scheme (10)-(12), (17) as an example t o discuss convergence of iterative processes like (26), (27) or (27), (29). Let us first consider the method of a simple iteration

Iterative process (33) corresponds to the use of (26) with r = 1 (successive approximations of nonlinearity, the Picard method) for approximate solution of problem (10)-(12), (17). The ~roblernfor the error uk = yk - y of iterative process (27), (33) becomes

214


On the set of grid functions y(x) vanishing on the boundary we define the grid operators in the usual way: Bu = Av,

af A ~= v AV- -(x, ay

-

yk)u,

x E W.

(35)

Iterative process (34) is rewritten as

Convergence of iterative process (36) (see Section 4.6) is determined by the constants y,, a = 1,2, in the inequality

The definition in (35) with -M 5 aflay 5 0 implies that yl = 1 and = 1 6-'M. Here 6 > 0 is the constant in the bound B 2 6E. The optimal value of the iterative parameter T in (36) is written as

+

The following bound holds for the error: IIuk+111 5 pollvkII, where PO = (1 -()/(I + E), E = yl/yz. Thus, convergence of the simplest iterative process (27), (33) is in general like that for convergence of the method of a simple iteration (Section 4.6) for linear problems. Let us now consider the Newton method for approximate solution of difference problem (10)-(12), (17). Because of (28), the new approximation is determined from the difference equation

completed by boundary conditions (27). We have the problem for the error

The right-hand side of (39) is represented by


According to the expansion

Equation (39) thus becomes

On the basis of the investigation of linear problem (40), (41) we can draw a conclusion about the convergence rate of the Newton method. According to the maximum principle (see (23)) we derive the bound

Let q = M M l , then we obtain from (42) that

This implies that the Newton method converges if the initial approximation 1, the iterations converge is close to the solution. Namely, if qllyo - yllc(,) at a quadratic convergence rate. When considering iterative processes (including those for nonlinear problems), considerable attention is paid t o monotonicity of the approximate solution. For example, a n~onotonicprocess can give an approximation from below, i.e. yo 5 yl 5 . . . 5 yk yt+t 5 . . . 5 y. An example of a monotonic process for nonlinear problem (10)-(12), (17) is considered in Problem 2. The Newton method gives an approximation from above or below depending on whether the function f(x, y) is convex or concave with respect to the second argument. For instance, let f ( x , y) be a concave function, i.e.

<

<

Then (41) (the maximum principle) implies that uk+l = yk+l - y 5 0, i.e. the Newton method gives an approximation from below (yk+l 5 y).

216


4.9.6 PROBLEMS

1. Show t h a t t h e Dirichlet problem for t h e elliptic equation

>

<

h a s a unique solution if c(x) 0 a n d lb.(x)I M. Solution. We should show that the homogeneous problem

<

has only a trivial solution. For this purpose, let us show that u(x) 0, x E 0, for f ( x ) 5 0, x E R, and u(x) 0, x E a0 (the maximum principle). It can similarly be shown that u(x) 2 0, x E R, for f (x) 2 0, x E 0, and u(x) 0, x E a n . Under these conditions problem (44), (45) has only a trivial solution. 0, x E a, Let us prove this by contradiction. We assume that for f ( x ) and U(X) 0, x E 8 0 , u(x) > 0, x E R+ (46)

<

>

<

<

n+

in a subdomain ( 1 + c 0).We integrate equation (43) over the domain n,. Since u(x) = 0, x E an+,we have -

/

an+

I(,),

au dx +

/

an+

C(Z)Udx =

/

f (x) dx.

an+

Under assumptions (46), the left-hand side of this inequality is positive while the right-hand side is nonnegative. We thus come to a contradiction, which shows that u(x) 5 0 in the entire domain R. 2. Use t h e i t e r a t i v e m e t h o d of a s i m p l e iteration (33) for a p p r o x i m a t e solution of p r o b l e m (10)-(12), (17) t o c o n s t r u c t a m o n o t o n i c iterative process, assuming (8) t o b e satisfied. Solution. Instead of (33) we consider the following iterative process:

We choose c(x) and iterative parameter 7 in (47) to obtain a monotonic . . . Y. approximation from below, i.e. yo y~ 5 ... yk yk+l Let the initial approximation be chosen so that

<

< <

< <

217


and let the boundary conditions be satisfied exactly. For the error we have

According to the maximum principle zo 5 0 , i.e. the initial approximation is given from below: yo 5 y. Let wk+l = Y ~ + I- yk. For W I we obtain from (47)

+

( A c(z)E)

+ Ayo - f ( z ,yo) = 0 :

therefore wl >_ 0 because of assumption (48). From (47) we derive

for an arbitrary k. We rewrite this equation in the form

where

+

+

Fk = ( 1 - T ) ( A c(z)E)wr: r

(50)

As for the parameters of the iterative process, we assume

Under these conditions FI 2 0 in view of (50),provided that (48) is satisfied. Besides, (49) yields ( A c ( z ) E ) w z 2 0 because of (49). Let us show that similar relations hold on all the other iterations. 0 and We will prove this by induction. For k = 1 we have Fl ( A c ( z ) E ) w z 2 0 . Let us show that a similar relation holds for k - 1, i.e. Fk-1 2 0 and ( A c(z)E)wk 2 0. Let us now consider Fk. Under the above assumptions, this function is positive because of (50), given that conditions (51)are satisfied. The second necessary inequality ( A + c ( ~ ) E ) w k is + obtained ~ from (49). Because of the maximum principle, the inequality ( A + ~ ( z ) E ) w 2 ~ +0 ~ yields yk+l - yk 0. We take into account yo 5 y and find that if conditions (50) are satisfied and the initial approximation, according t o (48), gives a monotonic approximation of the solution to problem ( 8 ) ,(10)-(12), (17) from below. If the initial approximation satisfies the condition

+

+

>

+

>

AYO- YO) 2 0 , the iterative process gives a monotonic approximation from above.

218


4.10 Bibliography and Comments

4.10.1 GENERAL NOTES 4.1 Boundary value problems for elliptic problems are the basic problems of mathematical physics. Much space is given to them in all textbooks on partial differential equations (13, 30, 321. The maximum principle is a classical tool for studying the unique solvability of the first-kind boundary value problem for second-order elliptic equations [3, 8, 17). According to comparison theorems, the corresponding bounds for solutions are derived in the uniform norm. Boundary value problems for elliptic equations with nonsmooth solutions are considered in Sobolev spaces (8, 13, 14, 321. Special attention is paid to derivation of a priori bounds in the Hilbert spaces W,*(O),k = 1,2. We presented only the simplest bounds, which are inherited by the corresponding difference elliptic problem. 4.2 Difference methods (2, 15, 22-28] and the finite element method [5, 16, 18, 291 are widely used for approximate solution of elliptic problems. Construction of difference schemes by direct approximation is traditionally covered in primary manuals on numerical solution of problems in mathematical physics [20, 21, 341. Tikhonov and Samarskii 1221 were the first who systematically presented the general integrw interpolation principle for construction of difference schemes. This m e thod is also discussed in this monograph and is further improved in 1221. Other approaches to the construction of difference schemes for elliptic boundary value problems (e.g. methods of approximation of a quadratic functional and methods of integral identities) are mentioned in the above work. 4.3 Accuracy of difference schemes for problems with smooth coefficients is conventionally investigated by the maximum principle for difference equations. A difference scheme is written in a canonical form, which allows us to study main boundary value problems for elliptic equations from a general viewpoint. In our presentation, we followed [24, 251. 4.4 Methods of Hilbert space are used to obtain basic results concerning the accuracy of difference schemes. In particular, difference problems on nonuniform grids, problems with discontinuities, etc. can be studied. In our presentation, as in (24, 251, we imposed strict requirements of smoothness on the exact solution. More advanced results for elliptic boundary value problems with generalized solutions are given in [27]. 4.5 Direct methods for solving linear algebraic equations are presented in all traditional courses of linear algebra (see e.g. [6, 341). Difference and finite element methods result in sparse matrices. Because of this structure, the amount of computational work can be substantially


4.6

4.7

4.8

4.9

219

diminished [7]. The most widespread method for band matrices (in particular, tridiagonal) is the Thomas algorithm and its versions. Detailed presentation of these algorithms and the grounds for them are given in (281. Special direct methods are used for grid elliptic problems with separating variables, e.g. the reduction method and fast Fourier transform. These methods are discussed in detail from various viewpoints in 110, 281. We present the general theory of difference schemes following [28] but more concisely. In particular, we do not cover topics related t o computational stability of Chebyshev iterative methods. Earlier, the method of variable directions was very popular. However, an optimal set of iterative parameters can only be constructed for grid elliptic problems with separating variables, for which the direct methods are faster. Therefore we only consider the general case of the method of variable directions with noncommuting operators. Other advantages of the methods of variable directions are thoroughly studied in [24, 281. In order to make the presentation of variational (both two- and three-layer) systematic, we choose the iterative parameters that minimize the residual on the next iteration, which seems to be natural. In other books on iterative methods (see e.g. [9]), more attention is paid t o a more complex and less clear interpretation of the method of conjugate gradients. In this chapter, we do not discuss the solution of grid problems with operators which are not self-adjoint. We estimate the effectiveness of iterative methods and choose iterative parameters of iterative methods following 128). The dependence of the convergence rate of classical iterative methods on discontinuous coefficients is presented in [31]. The alternating triangle method was suggested by A. A. Samarskii in 1964. Its more detailed presentation is given in [28]. The method of approximate factorization, which in the symmetrical version was treated as an alternating triangle method, is described in [4]. The general theory of iterative methods in subdomains is studied in [12]. Curvilinear orthogonal coordinates and locally irregular grids are traditionally widely used in computational practice [24, 27, 281. We do not touch upon problems of generation of nonorthogonal grids and solution of boundary value problems on such grids. The method of fictitious domains has been developed since the 1960s. This approach is thoroughly considered in [31]. Decomposition methods are almost absent in monographs and textbooks. The only exception is (151, where one chapter is devoted to this method. Decomposition methods without overlapping are discussed in the review [I]. Methods for approximate solution of nonlinear boundary value problems for elliptic equations are considered from different points of view in

220

COMPUTATIONAL HEAT TRANSFER t h e literature o n difference a n d finite element methods. T h e difference schemes a r e studied in [ll]a n d iterative methods for a class of nonlinear difference schemes a r e discussed in (281. We also mention (191 as a reference book o n t h e systems of nonlinear equations.

4.10.2 LITERATURE 1. Agoshkav V. I. (1991) Methods of decomposition of a domain in problems of mathematical physics. In: Cornputationol Pmcesses and Systems, issue 8 [in Russian]. Nauka, Moscow, pp. 3-51. 2. Bakhvalov N. S., Zhidkov N. P. & Kobel'kov G. M. (1987) Numerical Methods [in Russian]. Nauka, Moscow. 3. Bers L., John F . & Schechter M. (1964) Partial Differentiol Equations, Interscience, New York. 4. Bulaev N. I. (1989) A Spatial Model of lbrbulent Exchange [in Russian]. Nauka, Moscow. 5. Ciarlet Ph. (1977) The Finite Element Method for Elliptic Problems. NorthHolland, Amsterdam. 6. Fadeev D. K. & Fadeem V. N. (1963) Computational Methods of Linear Algebra [in Russian]. Fizmatgiz, Moscow. 7. George A. & Liu J . (1981) Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall, Englewood Cliffs (USA). 8. Gillbarg D. & Trudinger N. (1983) Elliptic Partial Differentiol Equations of Second-order. Springer-Verlag, Berlin. 9. Hageman L. & Young D. (1981) Applied Iterative Methods. Academic Press, New York. 10. Hockney R. & Eastwood J. (1987) Computer Simulation Using Porticles. McGraw-Hill, New York. 11. Karchevskii M. M. & Lyashko A. D. (1976) Difference Schemes for Nonlinear Problems of Mathemotieal Physics [in Russian]. Kazan' Lobachevskii State University, Kazan'. 12. Kuznetsov Yu. A. (1985) Computational methods in subspaces. In: Computotional Processes and Systems, issue 2 [in Russian]. Nauka, Moscow, pp. 265-350. 13. Ladyzhenskaya 0. A. (1973) Boundary Value Problems of Mathematical Physics [in Russian]. Nauka, Moscow (1973). 14. Ladyzhenskaya 0. A. & Ural'tseva N. N. (1973) Linear and Quasi-linear Equations of Elliptic Type [in Russian]. Nauka, Moscow. 15. Marchuk G. I. (1975) Methods of Numencol Mathematics. Springer-Verlag, New York. 16. Marchuk G. I. & Aproshkov V. I. 11981) ~. Introduction to Proiective-orid Methods [in Russian]. Nauka, Moscow. 17. Miranda C. (1955) Equozioni alle derivote parziali di tipo ellitico. SpringerVerlag, Berlin. 18. Oganesyan L. A. & Rukhovets L. A. (1979) Variational-difference Methods for Solving Elliptic Equations [in Russian]. Armenian SSR Acad. Sci., Erevan. 19. Ortega J. & Rheinboldt W. (1970) Iterative Solution of Nonlinear Equations i n Several Variables. McGraw-Hill, New York. 20. Richtmyer R. (1957) Difference Methods for Initial Value Problems. Interscience, New York.

-


221

Methods for Initial-value Problems. 21. Richtmver R. & Morton K. (19671 ~ , Difference Interscience, New York. 22. Samarskii A. A. (1971) Introduction to the Theory of Difference Schemes [in Russian]. Nauka, Moscow. 23. Samarskii A. A. (1987) Introduction to Numerical Methods [in Russian]. Nauka, Mmrow. 24. Samarskii A. A. (1983) Theory of Difference Schemes [in Russian]. Nauka, Moscow. 25. Samarskii A. A. & Andrew V, B. (1976) Difference Methods for Elliptic Equations [in Russian]. Nauka, Moscow. 26. Samarskii A. A. & Gulin A. V. (1989) Numerical Methods [in Russian]. Nauka, Moscow. 27. Samarskii A. A,, Lazarov R. D. & Makarov V. L. (1987) Difference Schemes f o r Differential Equations with Generalized Solutions [in Russian]. Vysshaya Shkola, Moscow. 28. Samarskii A. A. & Nikolaev E. S. (1989) Numerical Methods for Grid Equations. Birkhauser-Verlag, Basel. 29. Strang G . &Fix G. (1973) An Analysis of the Finite Element Method. PrenticeHall, Englewood Cliffs (USA). 30. Tikhonov A. A & Samarskii A. A. (1972) Equations of Mathematical Physics [in Russian]. Nauka, Moscow. 31. Vabishchevich P. N. (1991) Melhod of Fictitious Domains i n Problems of Mathematical Physics [in Russian]. Moscow Lomanosov State University, Moscow. 32. Vladimirov V. S. (1976) Equations of Mathematical Physics [in Russian]. Nauka, Moscow. 33. Voevodin V. V. (1977) Computational Methods of Linear Algebra [in Russian]. Nauka, Moscow. 34. Wasow W . & Forsythe G. (1960) Finite-difference Methods for Portiol Differential Equations. John Wiley & Sons, New York.

Nonst ationary Problems of Heat Transfer

The main problem of calculation physics of heat is to study nonstationary heat fields described by the heat equation, which is classified as a secondorder parabolic equation. This part of the book is devoted to construction and investigation of difference schemes for nonstationary boundary value problems of heat conduction. Specific features in constructing economical difference schemes fat approximate solution of multidimensional problems are considered separately (see Chapter 6). We begin with the reference material concerning the properties of boundary value problems for second-order differential equations of parabolic type. The maximum principle, in particular, is formulated for the first boundary value problem. The simplest a priori estimates of solution to the differential problem in the Hilbert space are presented. The estimates characterize the stability of solution with respect to small perturbations of initial data and the right-hand side. The construction of difference schemes for nonstationary problems is discussed. Substantial attention is paid to general issues of stability of difference schemes. The difference solution considered a s an approximate solution of a well-posed boundary value problem should be stable with respect to small perturbations of initial conditions. In estimating the error of the difference solution the main attention is paid to investigation of stability of the solution with respect to the right-hand side. The accuracy of the difference solution in the uniform norm can be studied on the basis of the maximum principle. Purely implicit difference schemes for the heat equation are studied; these schemes are classified as unconditionally stable (stable without any conditions on the spatial and time steps of the grid).

224


Constraints are imposed on the time step for other schemes (conditionally stable difference schemes). In studying the stability in Hilbert spaces, we use the general theory of stability for difference schemes. The latter is based on writing rnanylevel difference schemes in the canonical form and formulating the stability conditions in one or other norms as operator inequalities. The theory is accurate in the sense that the necessary and sufficient conditions coincide. The general theory of stability for difference schemes is applied in studying two- and three-level schemes for the heat equation. Thus, in particular, conventional schemes with weights are studied. The corresponding results on accuracy of difference schemes are formulated on the basis of stability investigation. The regular mode of heat conduction is investigated separately. The notion of asymptotical stability is introduced to describe the developed stage correctly. The asymptotical stability of difference schemes is investigated for the heat equation. In particular, we show that the conventional symmetrical difference scheme is unconditionally stable in the common sense and is asymptotically stable. Difference schemes are also considered for the hyperbolical heat equation. Using the simplest boundary value problems of heat conduction as examples, we illustrate the specific features in using difference schemes for approximate solution of nonlinear problems.

5.1 Boundary Value Problems for Second-order Parabolic Equations

5.1.1 LINEAR NONSTATIONARY HEAT EQUATION

We describe the heat state of a solid body, which has volume 0 , beginning with the initial time t = 0 up to a final time t = T , T > 0. Let Q = {(x, t ) I x E R,O < t 5 T}, and let r = {(x, t) / x E aR,O < t 5 T } be the lateral surface of Q. The propagation of heat in an anisotropic medium (see Section 2.1) is described the heat equation

where

NONSTATIONARYPROBLEMS OFHEATTRANSFER

225

under the conventional constraints

Equation (I), (2) is the classical linear parabolic second-order equation, which is classified as a basic equation of t,he mathematical physics. In the general case of a movable medium equation (1) is written as

Further we consider problems for heat equation (1) in an isotropic medium with

as basic problems. Let us write the heat equation separately for a homogeneous medium where both the heat conductivity k(x) and the specific heat capacity c(x) are constant. Reducing the problem (see Section 3.1) to a dimensionless form, we arrive at the simplest second-order parabolic equation with constant

Equation (1) (or (3) and (6)) is complemented by the necessary boundary conditions. The most attention is paid to the Dirichlet problem, where

A more general situation (the convective exchange with the environment) corresponds to the boundary conditions of the third kind, namely

>

where u(x, t ) 0. In (8) we use the notation from Chapter 4. For nonstationary equations we can point out the problem with u(x, t ) = 0 as a separate problem with boundary conditions of the second kind. The well-posed problem for equation (1) is that with the known initial temperature x c R. (9) u(x,O) = uo(x),

226


The temperature field u(x, t) at any point of the calculation domain R a t T is determined from the heat equation (1) and (2) (or each time 0 < t (1) and (3), etc.), boundary conditions (7) (or, for example, (8)) and initial condition (9).

<

5.1.2 THE MAXIMUM PRINCIPLE

The maximum principle holds for the nonstationary heat equation. The principle states that the maximum temperature is attained either on the boundary or at the initial time, provided that no volume sources are present. The maximum principle for parabolic equations is formulated as follows. Let us determine the operator L by the relation

and consider equations (I), (3), and (4)

<

T h e o r e m 1. Let Lu 0 (Cu 2 0) be the bounded domain Q. Then the maximum (minimum) of the function u(x, t ) is attained on the boundary of l and (or) at t = 0, i.e. domain X maxu(x,t) = max{maxu(x, t), =EQ ~ c r

yEyu(x,~)},

m a x u ( s , t ) = max{maxu(x,t),

maxu(x,o)}

r€T

(10)

rER

Based on the maximum principle we can easily prove the uniqueness of solution to the first boundary value problem ( I ) , (3), (4), (7), and (9). As in stationary problems, the comparison theorems, which follow from the maximum principle, are important. The following statement can be used as an example. T h e o r e m 2. Assume that the inequalities

hold for the functions u(x, t) and ~ ( t ) ;then u(x, t) domain Q.

v(x, t) over the whole

Let us present a priori estimates in the uniform norm for parabolic boundary value problems. The estimates are based on the maximum principle.

NONSTATIONARY PROBLEMS OF HEAT TRANSFER

227

T h e o r e m 3. The following estimate holds for the solution of the first boundary value problem (I), (3), (4), (7), and (9):

where constant M depends on the diameter of domain R and the coeficients of equations ( I ) , (3), and (4). It follows from estimate (12)that the solution of the first boundary problem depends continuously on the initial data, right-hand side and boundary conditions. The constant M is determined similarly to that in stationary problems (see Section 4.1). Statements similar to Theorems 2 and 3 can also be formulated for problems with boundary condition of the third kind (8) under logical constraints a ( x ,t ) 2 a, > 0.

5.1.3 OPERATOR FORMULATION OF NONSTATIONARY HEAT TRANSFER PROBLEM We consider problems of nonstationary heat conduction under uniform boundary conditions ( g ( x , t ) = 0, ( x , t ) E r, in (7) and (8)) in the Hilbert space Fl = L z ( 0 ) Assume that A is the operator of heat conduction which corresponds to L determined by (2) under boundary conditions of the first and third kind. The operator A was introduced in Section 4.2. We define additionally the operator B = c ( x ) E .As a result, we write heat equation ( 1 ) as a first-order evolutionary equation in 'H, namely

The latter is complemented by the initial condition

" ( 0 ) = 110.

(14)

For the boundary value problems in hand the operator A is self-conjugated and positive definite, i.e.

A=A'>6E,

6>0.

(15)

Assuming that c(x) 2 co > 0, by virtue of definition of operator B we have

In the most important specific case of homogeneous medium (see ( G ) ) , we have B = E. Besides, the operators A and B are constant, i.e. do not depend on t. The estimates of stability for nonstationary problems are based on using the Gronwall lemma. We formulate this in the simplest form.

228


Lemma. Assume that w ( t ) ,s ( t )

> 0 , m > 0 , and the inequality

t < T , T > 0. Then is satisfied for all 0 i

Making estimate (18) more approximate, we obtain the inequality

Based on the Gronwall lemma, let us obtain the simplest estimates of stability for solutions to problem (13) and (14) with the operators subject to constraints (15) and (16). We will tend towards these estimates in obtaining the corresponding estimates for the difference solution. Multiplying equation (13) by v ( t ) in a scalar way, we arrive at the equality

We connect the Hilhert space 'Ka with the operator B so that llulli = ( B u , ~ ) . Taking into account that the operator d is positive, and using the inequality

( f , ~ 5) IlfIla-1 I I ~ B , from (19) we obtain d

< Il4la-1. dt By virtue of the lemma, from this we obtain the estimate of stability with respect to the initial data and right-hand side for problem (13)-(16) - Il~ll5

Il4t)llo 5 Il40)llo + tllf Ilo-1.

(20)

Let us present some other a priori estimates for problem (13)-(16) which are not optimal for differential problems, hut are useful in obtaining the corresponding estimates for difference solutions. Taking into account (15),from (19) we obtain the inequality

For the right-hand side we use the estimate (f?.)

< ~11. 1 12

1

+ ,llf1I2

This results in the inequality

and the estimate

follows from this. Unlike (20), here we estimate the squared norm of the solution, and use the norm of 'R in estimating the right-hand side. A similar estimate for problem (13)-(16) can also be obtained (see Problem 2) in 'Ha. Let us also show how to obtain estimates of stability ' = 'D* > 0. in some general spaces ED generated by a constant operator D Assume that the operators B and A are permutahle. By virtue of conditions (16), there exist the inverse operator B-', and we can switch from equation (13) t o the equation

where A' = B-'A. By virtue of (15) and (16), for the permutable operators A and B we obtain A' = (A')' > 0. The estimate (20) for equation (22) is now like Il.(t)ll Il.(O)ll + t m p IID-lf II. (23)

<

The operators A and B are permutable for the considered problems of heat ' is permutable with conduction if c ( x ) ? coust. If, additionally, the operator D A', then from (22) we obtain the estimate

which generalizes (23)

5.1.4 PROBLEMS 1. O b t a i n t h e a priori estimate in t h e uniform n o r m for t h e first b o u n d a r y value problem for parabolic equation

230


Solution. To obtain the a priori estimate of solution to the first houndary value problem for equation (25) in the uniform norm, we use the transformation u(x, t ) = exp(pt)u(x, t). For v(x, t ) we obtain the equation

>

If the condition d(x, t ) - pc(x) is satisfied, then the a priori estimate like (12) holds for this equation. For the solution to the first boundary value problem (25), (7), and (9) at d(x, t) - pc(x) 2 0, p < 0, we have the estimate

Estimate (26) implies that growth of solution (temperature) can be bounded due to a heat source proportional to the temperature (the term in (25)). for p r o b l e m (15)-(18). 2. O b t a i n a priori e s t i m a t e i n Solution. Let us show that the inequality

holds for the problem in hand. Multiplying equation (15) in a scalar way by duldt, we obtain

Taking into account (la), we have

and use the estimate

for the right-hand side. Substituting these in (28), we obtain the inequality

and the sought estimate (27) follows from this

231

NONSTATIONARY PROBLEMS O F HEAT TRANSFER

5.2 Difference Schemes for Nonstationary Problems

5.2.1 M A N Y - L E V E L DIFFERENCE SCHEMES Let us mention some general features in difference solutions of nonstationary problems. We assume that we consider an approximate solution to the problem in the domain R on the segment [O,T].In R we introduce the simplest spatial grid wh and relate some finite-dimension space Hh with it. For simplicity, we introduce a uniform time grid w, = { t I t = t , = nr,n = 0 , 1 , . . . , N, N T = T } with the step T > 0. The approximate solution is considered as a function yh(t,) of the discrete argument t , E w, with the values from space Hh (yh(t,) E Hh). Further we denote y, = yh(t,). Assume that B,, a = 0, I , . . . ,p, are some linear operators acting in Hh, and rp, is a grid function. We call the (p + 1)-level operator difference scheme the difference equation

which relates the difference solution on ( p + 1 ) time levels. To determine uniquely the solution from ( I ) , we should prescribe p initial values y,, a = O , l , ...,p - 1 . In our further consideration most attention is paid to two- and three-level schemes. At p = I we have the two-level difference scheme

at given yo E Hh. Similarly for p = 2 we determine the three-level difference scheme and write it as

B>yn+2+ &y,+i

+ Boy,

= vn,

n = 0,1,. . . ,

(3)

assuming that yo and y, are given

5.2.2 T H E CANONICAL FORM OF T W O - A N D THREE-LEVEL DIFFERENCE SCHEMES It is convenient to study difference schemes for nonstationary problems by reducing these to a single canonical form. Any two-level difference scheme can be written as

232


where T ~ + I= tn+l - tn is the time step (generally speaking, nonuniform). In order t o switch from ( 2 ) t o the canonical fown of the two-leuel difference scheme, it is sufficient t o take into account that y,+, = y, + ~,+~(y,+, Y,,)/T,+I and put B = T,,+~B~ and A = B1 Bo. We have mentioned that it sufficient t o use the uniform time grid t o consider the specific features of the difference schemes. Therefore we consider the two-level difference scheme

+

Yn+1 - Yn +Ay,=ip,,

n = 0 , 1 , ....

T

(4)

Theory of difference schemes widely uses the subscriptless notation Y=Yn,

Yt=

(P='Pn,

Yn+l

- Yn

T

Therefore the twc-level difference scheme (4) is written in a shorter form

Byt

+ Ay = ip.

(5)

The scheme (3) can be written in the canonical fonn of the three-level difference scheme as

Yn+1 - Yn-1 27

+

pR Y"+l - 2yn + Yn-1 T2

+ AY, = (P,,

(6)

n = 1 , 2 , ..., assuming that the time grid is uniform. In order to switch from (3) t o ( 6 ) , we Put

B=r(Bz-Bo),

R=.$(Bz+Bo),

A=Bo+Bl+Bz.

Using the subscriptless notation Y; =

+ Y:)

$ ( ~ t

=

Yntl

- Yn-1

2T

>

%t =

Y n t l - 2yn

+ Yn-I

T2

we write the three-level scheme ( 6 ) as

By.

t

+ .r2RZt+ Ay = ip.

(7)

In order to determine the solution a t the new time level in case (4),we should solve the equation


233

The two-level difference scheme is called the ezplicit scheme if B = s ( x ) E , s ( x ) # 0. Otherwise ( B # s ( x ) E ) we deal with an implicit scheme. For the three-level difference scheme (6) we have

and therefore it is classified as an explicit scheme at B + 2 r R = s ( x ) E . For B + 2 r R # s ( x ) E we have an implicit three-level difference scheme. 5.2.3 THE STABILITY OF TWO-LEVEL DIFFERENCE SCHEMES The stability of difference schemes is most important in approximate solutions of nonstationary problen~s.Omitting general definitions, we clarify the problem using two-level difference scheme ( 4 ) as an example. The scheme ( 4 ) is well-posed if the conditions of existence and uniqueness of solution as well as that of continuous dependence of the solution on initial data are satisfied. The unique solvability of difference scheme (4) (see (8)) can be proved by assuming that B-I exists. The initial data for the two-level difference scheme (4) are the functions yo and 9, from some grid spaccs. We assume that two norms are prescribed, namely 11. lllh for the solution and I/.llzh for the right-hand side. Let us present the simplest definition of stability for two-level difference scheme ( 4 ) . A two-level difference scheme is said to be stable if there exist constant M , > 0, a = 1,2, independent of h , T , and n such that the inequality

is satisfied for any input data yo, 'pa,n = 0,1,. . . , N . We can consider separately the stability with respect to the initial data for the uniform difference scheme

Difference scheme (10) is stable with respect to the initial data if the inequality

is satisfied. It is logical to introduce the notion of stability with respect to the right-hand side for a nonuniform difference scheme with zero initial data. The difference scheme Yn+l - Yn + A y , = i p n , n = 0 , 1 , ..., y o = O , (12) r

234


is said to be stable with respect to the right-hand side if the estimate

is satisfied. The introduced notions of stability with respect to the initial data and right-hand side reflect more precisely the sense of the two terms in the general condition of stability (9). In some cases this allows us to restrict ourselves by investigating only the stability with respect to the initial data. The stability of many-level schemes can be investigated by reducing to an equivalent two-level scheme. Further we discuss the ways to do this in considering three-level difference schemes. This allows us to restrict ourselves to considering only the stability of twdevel difference schemes.

5.2.4 THE CONNECTION BETWEEN THE STABILITY WITH

RESPECT TO THE RIGHT-HAND SIDE AND THAT WITH RESPECT TO THE INITIAL DATA Let us show that for consistent norms the stability with respect to the righthand side follows from that with respect to the initial data. We say that a difference scheme is uniformly stable with respect to the initial data if there exists constant p > 0 and constant MI independent of h, 7 , and n such that the estimate II~nfllllh ~ I I ~ n l l l h (14)

<

<

holds for all y, E Hh for the uniform difference equation (lo), with pn MI. If p # 1, we say that the uniform pstability of difference scheme (10) with respect to the initial data takes place. The p itself may depend on T (only constant MI should not depend on 7). We can choose p so that

>

where c 0 does not depend on h, T , and n. Using (15), for the constant MI we obtain ( M I 2 p") M I = exp(cT). The uniform difference scheme (10) can be written a s

where

s = E - TB-'A

(17)

is the operator of transition from one time level into another. The operator S , generally speaking, can depend on n. By virtue of (16) and because y, E Hh

235


are arbitrary, the uniform stability of difference scheme (10) with respect t o the initial data is equivalent to that the norm of operator S is bounded by the constant p, namely IISll 5 P . (18) The estimates of stability like (9) for two-level difference scheme (4) can be obtained on the basis of the Gronwall diffewnce lemma (cf. Section 5.1). L e m m a . Let E, and 3" be nonnegative functions defined on the grid w,, and let p > 0. Then the inequality

implies the estimate

Estimate (20) is proved by induction. For n = 0 (20) is clearly satisfied (it coincides with (19)). Assume that (20) is satisfied for some n = rn - 1. From (19) and (20) we have

Thus (20) is also satisfied for n = m . The relation between the stability with respect to the initial data and that with respect to the right-hand side is given by the following statement. T h e o r e m 1. Assume that difference scheme (4) is uniformly stable with respect to the initial data in the norm 11 . l ) l h . Then difference scheme (4) is also stable with respect to the right-hand side and the estimate (9) holds for its solution at ll$okll2h = IIB*l$okl[lh and Mz = M l T . Taking into account that the operators of the difference scheme may depend on the time level, we rewrite equation (4) as yktl = Sk+lyk +rB-'$oh. Hence it follows immediately that IIYI.+Il l ~ h5 ll~k+~ll llylcll~h T IIB; f $oklllh. The condition of uniform stability of scheme (4) with respect t o the initial data (see (18)) allows us to pass to the inequality

+

236


Using (20), from (21) we obtain the estimate II~n+l\llh5 ~ ~ + ' l l y d l+h

.,

C T P ~ - ~ I I B(22)~ ~ ~ I

k=O

Due to our assumption on the uniform stability with respect to the initial data, we have Ml and pn-k 5 M l , and therefore from (22) we have

<

we arrive at estimate (9) of stability with respect to both the initial data and right-hand side at l)(ok)/2,, = JJB;lp*lJlh and M2 = M I T . Thus the theorem is proved. Definitions of stability can use the corresponding estimates for the squared norm of the difference solution (see estimates (21) and (27) for the differential problem in Section 5.1). Instead of (9),we can require that the inequality

is satisfied. In this case, however, we cannot obtain the stability with respect to the right-hand side from that with respect to the initial data. One can pass to estimates baaed on the squared norms by transforming the difference Gronwall lemma (Problem 1). 5.2.5 THE REPRESENTATION OF THE THREE-LEVEL SCHEME AS

A TWO-LEVEL ONE Assume that H = HA and introduce the space H Z = H @ H as the direct sum of two spaces H. The addition and multiplication by number for vectors Y = {yl,yz} from H Z (yl, yz E H ) are coordinate-wise. Let Y = {yl, yz), let V = {ul ,wz), and let a and b be numbers; then a Y + bV = {ayl bwl, ayz buz}. The scalar product in H z is determined by the formula (Y>V)= ( Y , , v ~+) (yz,uz). Let us represent the three-level difference scheme written in the canonical form (6) as the two-level difference scheme

+

+

237


To do this, we introduce the vector

The vector Y ncan also he chosen in other ways (a variant different from (25) is considered, for example, in Problem 2). Calculating directly, for the operators of difference scheme (24) we obtain the formulae

and for the right-hand side of (24) we have

The representation of three-level scheme (6) in the form of two-level scheme (24)-(28) allows us t o define the stability of three-level scheme with respect to the initial data and right-hand side. 5.2.6 THE CONVERGENCE O F DIFFERENCE SCHEMES FOR NONSTATIONARY PROBLEMS The concepts of approximation, convergence and accuracy of difference schemes are introduced for nonstationary problems similar to the corresponding concepts for stationary problems (see Section 4.2). Let u(x, t) be a precise solution of a differential problem. We assume, for example, that the approximate solution is searched for on the basis of difference scheme (4). For the error 2, = y,-uh(tn) of the difference solution we have the difference problem &+I

- Zn T

+ Az,

n = 0 , 1 , . . .,

= &,

For the approximation error

11,

zo = yo

- uh(0).

(29)

we have the formula

+

Difference scheme (4) has the approximation O(JhJm T'), m on the solutions u(x, t), if max ll$k112h OSkSN

< M3 ((him + 7')

,

> 0, 1 > 0, (30)

238


Assume that the initial condition is approximated within the same accuracy, i.e. the estimate IIyo - 4 J ) l l l h M4 (Ihlm T') (31) is satisfied. If difference scheme (4) is stable, i.e. estimate (9) is satisfied, then for the error we have

+

<

It follows from stability (condition (32)) and approximation (estimates (30) and (31)) that the difference scheme (4) has the accuracy O(lhJm TI), i.e. the convergence of the difference scheme follows from the stability and approximation.

+

5.2.7 PROBLEMS

1. Show t h a t if t h e e s t i m a t e

is satisfied o n t h e g r i d w, for nonnegative En a n d F,,,t h e n t h e inequality

holds w i t h a r b i t r a r y E. Solution. It follows from (33) that

for any E , where p = exp((2c + E)T).Using the difference Gronwall inequality, from this we have estimate (34). 2. W r i t e t h e three-level scheme (6) as t h e two-level s c h e m e Y"+' = SY"

+an,

w i t h t h e vector Y" = {Y,-~,y,}. Solution. We use the representation of the threelevel scheme as B2Yn+1+B1Yn+BoY,-~=9,,

n = l , 2 ,....

Then for elements of the matrix S = (S,@) we obtain

Sll = 0,

Sl2= E,

szl= - B ; ~ B ~ ,

Sz2= -B;~B~,

239


and for the right-hand side we have (6),we have

an = {-O,B;'~,}.

Comparing (36) with

Formulae (37) and (38) determine the operator of transition in the twcblevel difference scheme (35).

5.3 Uniform Convergence of Difference Schemes for the Heat Equation

5.3.1 DIFFERENCE SCHEMES FOR THE HEAT EQUATION

The basic problem of nonstationary heat conduction in our consideration is the two-dimension problem of heat propagation in a solid isotropic cylindrical bar having the rectangular cross-section 0, with the temperature mode being given on the boundary. We consider the heat equation

Equations (1) and (2) are complemented by the boundary and initial conditions

The specific features in constructing difference schemes for parabolic equations are revealed in choosing the time approximation. In Section 4.2 we discussed the spatial approximation, for example, on the basis of integrointerpolation method. As usual, we assume that a uniform rectangular grid w is introduced in the rectangle 51 and has the steps hl and hz with respect to the variables xl and x2, respectively. In order to obtain a conservative spatial approximation, we integrate equation (1) over neighbourhoods of each internal node R,, = {x I x = ( X ~ , ~ , X I , 5, -5 1~ 5 ~Z X ~ , , + I / Z , X Z , ~ - Ixz / Z5 X Z , , + I / ZDenoting ~.

<

240


by u(x, t ) , x E w, the approximate solution to equation (1) at the time t we arrive at the system of ordinary differential equations

Here the function b(x) corresponds to the approximation of specific heat capacity c(x), and the operator A is associated with the approximation of differential operator L defined by (2). For sufficiently smooth coefficients and solutions we can put (see Section 4.2) 2

Au=

IA,~, A,v= -(a,(x)wze)==,

a=1,2,

(6)

a=1

b(x)=c(x),

(7)

XEW.

The system of equations ( 5 ) is complemented by the conditions following from (3) and (4), Similar to (6) and (7) we use the simplest approximations U(X,t ) = g(x, t), u(x, 0) = u0(x, t),

E aw, x E W.

o < t 5 T,

(8)

(9)

The transition from the parabolic boundary value problem (1)-(4) to the system of ordinary differential equations (5)-(9) corresponds to the method of lines.

In order to obtain the difference scheme for problem (5)-(9), it is necessary to use one or other time approximations. We again apply the integrointerpolation method. In using two-level schemes, values at two time levels, namely at t = t, and t = tn+l, enter the difference equation. Let us integrate equation (5) over the segment t, 5 t 5 t,+l (time averaging). Integrating the first term in ( 5 ) , we obtain

t"

In integrating functions which depend on time, we use the formula

where u is a numerical parameter (the weight of quadrature formula). Taking this into account, in integrating (1) we obtain the difference scheme

241


under the conditions

Difference scheme (10)-(12) is a difference scheme with weights. Let us mention some important specific cases of the scheme (10)-(12). If a = 0, we deal with an explicit scheme. In this case, in order to determine the solution at the new time level, we use the formulae y,+l = y, - bF1(x)~(Ay,ip,), x E w , with conditions (11) and (12) taken into account. The most widespread among implicit schemes are the symmetrical scheme and the purely implicit one. The symmetrical difference scheme (the CrankNzcolson scheme) corresponds to CT = 0.5. In this case (10) is like

If a = 1 we deal with the purely implicit diffe~encescheme

The latter is sometimes called the difference scheme with advancin.q 5.3.2 APPROXIMATION ERROR OF THE SCHEME WITH WEIGHTS Let us consider the approximation error for the scheme (10)-(12). For the error we have 1~= -b(x)u(t) - A(& (1 - u)u) ip, (15)

+

+

where u = u(x, t,), and ?Z = u(x, t,+l) are the solutions of differential problem (1)-(4) at the corresponding moment in time. Let us denote 'ii = u(z, t,+l12) and u = a u / a t , and use the expansions

We assume (see Section 4.2) that the grid elliptic operator A approximates the differential operator L within the second-order accuracy, i.e.

242


Taking into account (16)-(IS),let us represent the approximation error (15) -

+ (v- f ) - (u- 4) rAZ + 0 ( r 2 ) If, in addition to (16)-(19), we assume that 9 = f + 0 ( r 2+ lhI2), then for =b(~)ri+ J - (A?i - Ln)

the error we have the formula The general representation (20)for the approximation error of the difference scheme with weights (10)-(12) for heat conductivity problem (1)-(4) allows us to obtain 0 ( r 2 lhI2), u = 0.5, O ( r (hI2), u # 0.5. Thus the symmetrical difference scheme (13) has the second order of spatial and time approximations, while the other schemes with weight have the first order of time approximations and the second order of spatial approximation. The error of spatial approximation can sometimes be increased for the scheme with weights on the solutions of the heat equation (see Problem 1).

+ +

5.3.3 T H E MAXIMUM PRINCIPLE

In Section 4.3 we formulated the maximum principle for difference equations. The canonical form used represents the difference scheme by expressing the difference solution at a node in terms of the difference solutions at neighbouring nodes. The maximum principle in such a general form can also he used for investigating difference schemes for nonstationary problems. Here, however, the pattern of difference scheme includes the values at different time levels. Let us write the difference scheme with weights (10)-(12) in the canonical form with respect to the node ( x ,t ) = (x,, , t,+l). We represent the difference operator A (see Section 4.3) in the form


243

The coefficients of scheme (10) can be determined, for example, a s

We use the same notation for the pattern with respect to spatial variables Then the difference scheme (10) is written as

For the latter we can formulate conditions sufficient for the maximum principle to be satisfied. The condition that all the coefficients in the right-hand side of (22) are nonnegative results in

Condition (23) is quite logical and does not restrict the time step. Condition (24) is satisfied for all T only if a = 1, i.e. the maximum principle is satisfied unconditionally only for the purely implicit scheme. From the other a from (24) we have

Taking into account the formulae for the coefficients of difference scheme ( l o ) , from (25) we obtain the sufficient conditions for the maximum principle to be satisfied for the scheme with weights, namely

1

minc(x) ( h c 2+ h ~ 2-I) 2(1 - a ) max k(x)

75-

Estimate (26) shows that the maximum principle is satisfied for schemes with weights o # 1 under severe conditions for the time step T = 0(lh12). 5.3.4 THE CONVERGENCE OF DIFFERENCE SCHEME Using the maximum principle we can find the corresponding estimates of stability and convergence for difference schemes in the uniform norm. For the difference scheme (10)-(12) we can also obtain the a priori estimate reflecting

244


the stability of the scheme with respect to the initial data and right-hand side. The estimate can be obtained in the norm llyll = max Ily,llccw). In O
considering nonstationary problems, we try to obtain the estimates in the norm of difference solution at one time level. This, in fact, corresponds to using the maximum principle at a separate time level (for the grid elliptic operator). In order to estimate the solution at the (n 1)th time level (n 2 0), we rewrite (22) as

+

where

Under the formulated conditions (23) and (24) for the parameters of the difference scheme, we consider the grid elliptic problem (27) and (28). It is most simple to obtain the estimate of solution to the difference equation (27) with uniform boundary conditions (g(z, t , + ~ )in ( l l ) ) , which is sufficient for investigation of convergence of the difference scheme. For equation (27) with uniform boundary conditions we have (see Corollary 6 in Section 4.3) the estimate

In our case

If (23) and (24) are satisfied, then for such D(x) from (28) we have

Thus for ~ , + ~ ( zwe) obtain the estimate

245

NONSTATIONARYPROBLEMSOFHEATTRANSFER

Using the difference Gronwall lemma, from (29) we obtain the required estimate for the difference scheme (27) and (28) with the uniform boundary conditions, namely

The latter reflects the stability of difference scheme (10)-(12) with weights with respect to the initial data and right-hand side in the uniform norm. Recall that the estimate is obtained assuming that the maximum principle is satisfied (i.e. the conditions (23) and (24) are satisfied for the parameters of the difference scheme). In studying the accuracy of difference scheme (10)-(12),we formulate the corresponding problem for the error 2, = y,-u(x, t,), x E w. From (10)-(12) we obtain q X )&+I - Z" A(u2,+1 ( 1 - a ) z n ) = , 7 (31) X E W , n = 0 , 1 , ...,

+

+

*,

with the uniform conditions

Estimate (30) for problem (31)-(33) is like

Taking into account estimate (21) for the approximation error of difference scheme (10)-(12), we obtain

where u = 2 for a = 0.5 and v = 1 otherwise. We again emphasize that both the stability and convergence of the difference scheme are proved if the conditions (23) and (24) are satisfied. We shall show further that for weaker norms the estimates (35) can he obtained under less restrictive conditions on the scheme parameters. 5.3.5 THREE-LEVEL SCHEMES FOR THE HEAT EQUATION

Three-level difference schemes with weights are widely used in calculation practice, although more rarely than two-level ones. In applying three-level

246


schemes, one should store the solution at the (n - 1)th level. Instead of (10) we use the difference scheme

where iL = u n _ ~under , the additional conditions (11) and (12). In order to begin the calculations by the three-level difference scheme, we should know yl along with yo. In order to find y l , one can use some two-level schemes. Difference scheme (36) is characterized by three weight parameters 8, a1, and 0 2 . Two classes of three-level schemes are usually used, each including only one parameter. First we mention the symmetrical difference schemes which are written as

The scheme (37) corresponds to the weights 6' = 0.5 and a1 = 02 = a.Taking into account (19), it is easy to check that the symmetrical three-level scheme provides the second order of spatial and time approximations. The other class of three-level difference schemes for the heat equation corresponds to the purely implicit approximations of the elliptic operator. In this case the difference equation is

i.e. a1 = 1 and u 2 = 0 in (36). For the approximation error of the scheme (38) we have 0(-r2 + lhI2), a = 1.5, $ n = { o(7 (hI2), 0 # 1.5.

+

Thus the class of three-level schemes (38) also includes that providing the second order of approximation. It can be shown (Problem 2) that one cannot formulate simple sufficient conditions of the maximum principle like (23) and (24) for symmetrical threelevel schemes (37) at a # 0.5. The asymmetrical threolevel difference scheme is more fruitful here. To investigate the difference scheme (38), let us write it in a form similar to (22), namely

Hence it can be seen that the maximum principle is satisfied if the conditions 26' - 1 0 and 1 - 9 0 are satisfied, i.e. at 0.5 5 6' 5 1. However, the

>

>


247

second-order scheme (0 = 1.5)does not satisfy these conditions. Note that the maximum principle is satisfied at 0.55 0 5 1 without any constraints on the grid parameters. 5.3.6 PROBLEMS 1. C o n s t r u c t t h e two-level difference s c h e m e of h i g h e r approxim a t i o n o r d e r ($ = 0 ( r 2 h4))for t h e one-dimension u o n s t a t i o n a r y p r o b l e m of h e a t conduction

+

Solution. In this case the accuracy of the scheme is increased due to approximation of solutions, as in some stationary problems (see Section 4.2). On the uniform grid w for approximate solution of problem (39)-(41)we use the scheme with weights (10)at Ay = -ye,, x E w . For the approximation error we have

Expanding in the Taylor series at the point (x,t,+llz) we obtain

Substituting (43) into (42)and taking into account that

for the error we obtain

ip = f , we arrive at the formula (20)for the approximation error. On the solutions of equation (39)we have

If we put

248


and therefore (44) is transformed to

In order to obtain the scheme with higher approximation order, we put a = 0.5 - hz/(12r), and define the right-hand side of the difference scheme as

2. Formulate conditions under which the maximum principle is satisfied for the symmetrical three-level difference scheme (37). Solution. We write the scheme (37) as

The coefficients in the right-hand side can be positive only at a = 0.5. If u = 0.5 then the constraints on the time step are caused by the condition such as (24), namely b(x) - 2raA(x) 0. From this we obtain the constraints for the time step min c(x) (h;Z h;Z)-l 75; min k(x)

>

+

in the three-level symmetrical difference scheme (37) at a = 0.5.

5.4 Theory of Stability of Difference Schemes

5.4.1 THE NECESSARY AND SUFFICIENT STABILITY CONDITIONS

First we separate the class of two-level difference schemes with self-adjoint operators. We consider the stability with respect to the initial data and righthand side for the scheme written in the canonical form (see Section 5.1)


249

with stable (independent of n) grid operators

B = B*,

A =A*.

(2)

The simplest estimates of stability for the two-level difference scheme follow from estimates of stability with respect to the initial data. We discuss this and other issues later. The stability of difference scheme with respect to the initial data is considered as the uniform stability in the sense of the definition in Section 5.2. We study the stability by using the method of energetical inequalities. Therefore first we obtain the simplest energetical identity for the difference scheme (1).Let us multiply the latter in a scalar way by 2ryt = 2(yn+1 - y,). This results in the inequality

Taking into account the formula (3), we transform the second term in the right-hand side of (3): Yn

1

+

= ? ( ~ " + l yn) -

T

5Y t .

Substituting (4) into (3) we obtain 27 ( ( B -

S)

yt,yt)

+ ( ~ ( y , + l + yn). yn+t - yn) = ~ T ( ( P , , Y ~ ) .

For the self-adjoint operator A we have (A(Y,+I + ~ n ) , ~ n +- yln ) = (AY,+I,Y,+I) + ( A Y ~ , Y ~ + I ) - (Ayn+i,yn) - (Ayn,yn) = (Ayn+lryn+i) - (Ayn,yn). This allows us to rewrite (5) as 27 ( ( B -

i)

yt,yt)

+ (Ayn+l,yn+l) = (Aynryn)+ 27(ipn,yt).

The difference scheme with ip, = 0 is stable (more precisely, uniformly stable) in H D , D = D* > 0, if the estimate

is satisfied. The main result of the stability theory for difference schemes is formulated by the following theorem.

250


Theorem 1. The necessary and suficient condition of stability of the difference scheme (1) at A = A* > 0 with respect to the initial data in Ha is that the inequality

~

~

2

2

3

:

~

(8)

is satisfied, where Bo = 0.5(B + B * ) Here we do not assume that the operator B is self-adjoint. To prove the sufficiency, we should show that estimate (7) follows from (8).For a uniform difference scheme (9, = 0 ) the energetical identity ( 6 ) is like

If ( 8 ) is satisfied, we obtain

i.e. the estimate ( 7 ) at D = A. The necessity is proved if we show that the operator inequality ( 8 ) follows from the stability condition (estimate ( 7 ) at D = A). Inequality ( 8 ) implies that T (Bu,v ) 2 5 W , v ) (10) is satisfied for any u E H. Let us write the energetical identity ( 9 ) at n = 0:

Taking into account that the scheme is stable with respect to the initial data, we have

For any v = yt(0) E H we find yo = -A-'Bu E H . The stability holds for any initial conditions yo and therefore estimate (10)is satisfied for any v , i.e. (8) is satisfied. Thus the necessity is proved. We emphasize once again that the presented result cannot be improved, and the stability conditions are precise, because the necessary conditions coincide with the sufficient one. Similar conditions can also be presented for the stability of difference scheme (1) and ( 2 ) with respect to the initial data in H g . Namely, condition ( 8 ) is


251

necessary and sufficient for the stability in Hg at B > 0. The sufficiency of this condition is proved on the basis of energetical inequalities in Problem 1. 5.4.2 p-STABILITY OF DIFFERENCE SCHEMES

Some important applied problems require that the pstability of difference schemes be investigated for p # 1, with the estimate ((&+I 5 p ( ( y , , ( ( ~ satisfied instead of (7). As an example we can mention the asymptotical stability ( p < 1) for the ordinary heat equation. A situation where the norm of solution grows in time, i.e. p > 1, often occurs in inverse problems of heat conduction. Therefore, generalizing Theorem 1 we formulate the corresponding result on pstability for arbitrary p > 0. Theorem 2.

The conditions

are necessary and suficient for the difference scheme (1) and (2) to be p-stable in HA for A > 0 (in Hg for B > 0 ) . We prove the theorem by obtaining the corresponding operator inequalities. In doing so the implicit scheme (1) is reduced to an explicit one, and we estimate the corresponding transition operator. We restrict our proof to the case of B > 0 in Hg. Since B > 0, for the uniform difference scheme (1)we have

By virtue of B = B'

where

2,

> 0, there exists B1f2;applying it

= B 1 f Z y ,and

S=E-rC.

is a self-adjoint operator. The inequality for the transition operator S, namely

is equivalent to the stability condition

to (12) we obtain

252


Taking into account that zn = B1lZy,, we rewrite (17) as the required estimate of stability in Ha, namely

Next we switch from inequality (16) to inequalities for the operators of difference scheme (1). For the self-adjoint operator S inequality (16) is equivalent to the two-side operator inequality

Taking into account (14), from (19) we obtain

-E
~ + P E . T

7

Substituting (15) into (20) and multiplying this from both sides by BIIZ (in doing so the inequality still holds), we just obtain the two-side inequality (11). The stability in HA is similarly proved by passing to the explicit scheme for s, = A1JZy,. 5.4.3 STABILITY WITH RESPECT TO THE RIGHT-HAND SIDE Let us present some estimates which characterize the stability of difference scheme (1) and (2) with respect to the initial data and right-hand side. First we formulate the statement following from the stability with respect t o the initial data and Theorem 1 of Section 5.2 on the relation between the stability with respect to the right-hand side and that with respect to the initial data. Here we restrict ourselves to the case p = 1. To pass to a more general case of arbitrary p > 0 is not difficult.

Theorem 3. estimate

If A

>

0 and inequality (8) is satisfied, then the a priori n

IIYn+lIIA

< I ~ Y o ~ ~ A+ CTIIB-~IP~IIA(21) k=O

holds for the difference scheme (1) and (2)). If B

> 0 then

Estimates (21) and (22) follow immediately from the proved general estimate (23) from Section 5.2 (with MI = 1, (1 . l l l h = 11 . ID, D = A,B). The estimates of stability of the difference solution with respect to the righthand side in other norms are useful. Some of these have been mentioned for solution of the differential problem in Section 5.1. Similar estimates can be obtained by making condition (8) more approximate.


Theorem 4. If A > 0 and the inequality B > E E + ~ A ,

2

where

E

> 0 is satisfied, then the a priori estimate

holds for the difference scheme ( I ) and (2) The right-hand side of (24) is estimated in the simplest norm. Then (24) is proved on the basis of energetical identity (6). We have

Substituting this into (6) and using condition (23), we arrive at the inequality

Taking into account the difference Gronwall lemma, from this inequality we obtain the required estimate (24).

Theorem 5. If A > 0 and the inequality I+€ B>TA, 2

where

E

> 0, is satisfied, then the a priori estimate n

llr/ntlll2a

I+€

IIl~oll2a+ -- C~ll~nlli-1 2E

lid

holds for the difference scheme (1) and (2)). We again consider the energetical identity (6). The inequalities

<

+

7

~ T ( P ~ , Y ~~ )T I I ~ ~ ~IIY~IIB I I B - 5L 2 ~ P l l ~ t l l Z e -Il~nlIi-1 2P

hold for any

0 > 0. Substituting (27) into (6), we obtain

27 (((1 - 0 ) B

- '2 A )

yt,yt) + ( A ~ n t ~ i ~ n t l ) T

= (AY,,YJ

+5ll~~11~-~.

254


+

Taking into account (25), we choose 0 so that 1 E = 1/(1 - 0 ) . Then (28) gives the following estimate of the solution on a level:

and the a priori estimate (26) follows from this.

5.4.4 STABILITY OF THREE-LEVEL DIFFERENCE SCHEMES Let us cite some stability conditions of three-level difference schemes written in the canonical form

under the condition that

are constant grid operators. We investigate the stability of three-level difference scheme (29) and (30) by reducing it to the corresponding twelevel scheme and use the formulated results on stability of two-level schemes. Let '~n-~n-lr then (Section 5.2) the three-level scheme (29) is written as

The operators A and B in (32) are determined a s

and the right-hand side of (32) is

mn = {%>01

(31)

255


Under the formulated assumptions (30), the operator A is self-adjoint, and the operator B is not so. We have already mentioned that the necessary and sufficient condition for two-level difference schemes to be stable with respect to the initial data at A = A* > 0 is the inequality (Theorem 1 also holds for B # B*): B > ~ A . 2 (36) The stability holds in the space H i , and, by virtue of (31) and (33), for V = {UI,212) we have

Theorem 6 . If A > 0 and

then the following estimate of stability with respect to the initial data holds for the difference scheme i n (29) and (30):

We should check the inequality (36) and condition A > 0, in order to obtain the estimate (40). If inequality (39) is satisfied, then, by virtue of (37), we have A > 0. From the definitions of operators A and B (see (33) and (34)), we have T B B--A= 2 -T ( R - + A ) 0

I

Therefore for any element V = {ul,vz} from Hz we have

Hence it follows that ((B-$A)V,V) = ( B V I , U I ) + T ( ( R - + A ) ~ ~ , W I )

- T ( ( R - : A ) U I , ~ ,= ) (BV,,VI)

>

Therefore inequality (36) is satisfied under the condition B 0 (inequality (38)). Thus the theorem is proved. Note that in proving the stability for both the two-level scheme (Theorem 1) and three-level scheme (29), in practice we do not use the condition that the operator B is self-adjoint. Note also that if (29) is satisfied, then condition

256


(38) is not only sufficient, but also necessary for the difference scheme (29) and (30) to be stable. The presented result can be obtained on the basis of corresponding energetical identity for three-level difference schemes, and by estimating the norm of transition operator in writing the difference scheme a s in Problem 2, Section 5.2. Anyway this entails cumbersome calculations, and therefore we restrict ourselves to the considerations presented.

5.4.5 + S T A B I L I T Y OF T H E THREE-LEVEL SCHEME In order to obtain the conditions of pstability of the three-level difference scheme (29) and ( 3 0 ) , we reduce it so that the stability conditions of the transformed scheme (with p = 1 ) result in the conditions of pstability of the initial scheme. A similar approach can be applied for two-level difference schemes which results (Problem 2 ) in the stability in less convenient norms. Assume that the transformation

is applied to difference scheme (29). The simplest estimate a new grid function vn+l like [lun+l[[5 [I D , [ [ corresponds Ily,+l(l 5 plly,ll, which is related to the pstability. A similar in using norms in HZ. By taking into account (41), the three-level difference transformed to

g ""+I - "n-1 +72E U"+1 - 221, ZT

+

r2

Un-1

+ Av,

for the norm of to the estimate situation occurs scheme (29) is

= &,

(42)

n = 1 , 2 , ... . The operators in (42) are determined in terms of the operators of initial difference scheme (29) by means of the relations

Let us see how the conditions of stability for difference scheme (42) are transformed into operator inequalities for the initial difference scheme (29). By taking into account (43), the condition > 0 is reduced to

A


Now let us check the condition

Taking into account (43), we obtain

Therefore inequality (45) is transformed to

The necessary and sufficient condition

5 2 0 is like

It only remains for us to formulate the corresponding estimate of stability. Under the conditions imposed on the operators, for the difference scheme (42) (see Theorem 6) we have the estimate

where ,vn

- n"-1

Taking into account (41), from (49) we obtain

Therefore we can define

In this case estimate (48) has the necessary form

IlY"+'II,- 5 PllY"II,-. Thus we can formulate the following statement.

(49)

258


T h e o r e m 7. If inequalities (441, (46) and (47) are satisfied, then the estimate (50) and (51) of p-stability with respect to the initial data holds for difference scheme (29) and (30). Note that the estimate of stability (51) is obtained in a more complex norm, in particular, depending on p. Estimates in simpler norms can be obtained under stricter constraints on the operators of difference schemes.

5.4.6 STABILITY OF THREE-LEVEL SCHEMES WITH RESPECT TO THE RIGHT-HAND SIDE Let us present some estimates of stability of the three-level difference scheme (29) and (30) with respect to the right-hand side. The simplest estimates can be obtained by writing (29) as the equivalent two-level difference scheme (Problem 2, Section 5.2)

at Y" = {y,-~,y,} and a" = {O,B;'~,), where Bz = R + (2.r)-'B. For the vector Y = {yl, y2} we determine the norm a s

T h e o r e m 8. I f A the estimate

> 0 and the inequalities (38) and (39) are satisfied, then " IIynt'll~5 llylll~ + IlB~l~nll~ (54)

C

k=1

holds for the dafference scheme (29) and (30). Under the conditions formulated for the operators of (29) and (30) the latter are stable with respect to the initial data. The corresponding estimate of stability (40) now implies the stability in the norm determined according to (53). From (52) we have

Due to the stability with respect to the initial data, we have

and it only remains for us to transform the second term in (55). We have


259

Substituting (56) and (57) in (55), we obtain the proved estimate (54). As for two-level difference schemes, we can extend the set of estimates of stability with respect to the right-hand side by using various norms for the right-hand side of (29) and (30). 5.4.7 PROBLEMS 1. O b t a i n t h e estimate of stability with respect t o t h e initial d a t a i n H s by t h e m e t h o d of energetical inequalities for t h e difference scheme (1) a n d (2) w i t h positive operators A a n d B assuming t h a t t h e condition (8) is satisfied. Solution. In order to obtain a new energetical identity, let us multiply the uniform difference scheme (1) in a scalar way by 27G, namely

Taking into account (4) and a similar formula

Substituting these in (58), we obtain

If inequality (8) is satisfied, then we have

and therefore estimate II$IIB 5 llylle follows from (59), i.e. the difference scheme (1) and (2) is stable in H s . 2. O b t a i n t h e conditions of pstability for t h e two-level difference scheme (1) a n d (2) o n t h e basis of transformation (41). Solution. The difference scheme (1) is written as

260


For the operators of scheme (60) we have

A difference scheme is stable in H z if

These conditions result in the two-side operator inequality (11) of pstability. In doing so, the estimate of pstability with respect to the initial data like IJ~,,+~llx5 pIIyn!la holds for the initial difference scheme. Thus we obtain the estimate of stab~lityin a norm more complex than earlier (see Theorem 2).

5.5 Stability and Convergence of Difference Schemes for the Heat Equation

5.5.1 STABILITY OF TWO-LEVEL SCHEMES WITH WEIGHTS

We begin by considering two-level difference schemes with weights, which were constructed in Section 5.3 for the first boundary value problem for the heat equation. For convenience, we, as usual, assume that the boundary conditions are uniform. We write differencescheme (10)-(12) from Section 5.3 as

Here the operator A is defined on the set of grid functions being zero on aw and

where, for example,

We write the scheme with weights (1) in the canonical form


with the operators

B = b(x)E + UTA,

A = A.

261

(6)

The difference scheme (5) and (6) belongs to the class of schemes with weights for which B=D+mA. (7) Assume that A = A* > 0 in the scheme (5) and (7); then the necessary and sufficient condition of stability in H A with respect to the initial data (Theorems 1 and 2 in Section 5.4), i.e.

is now

D + ( o - i)T A 2 0.

(9)

>

Let D 0; then condition (9) is satisfied for all u 2 0.5 independent of the time step (the time stability). If we know additionally a positive constant 6 such that A AD, (10)

<

then condition (9) is satisfied at

This inequality can be interpreted as a condition for the time step for u namely 1 T
< 0.5, (1'4

Thus, for example, for the explicit scheme the time step is like

We specify the obtained condition for the difference scheme (1) and (2) written in the form (5) and (6). In this case the mentioned properties of operators A and D clearly hold. Therefore the difference scheme (1) and (2) with weights is unconditionally stable with respect to the initial data at u 0.5 in H A . This, in particular, holds for the symmetrical scheme. Earlier (see Section 5.3) we proved the unconditional stability (in the uniform norm) for only the purely implicit scheme. If o < 0.5 the scheme with weights is conditionally stable. Let A be the constant in the inequality A Ab(x)E, (14)

>

<

262


then the stability condition is as in (11) and (12). Making (14) more approximate we can obtain explicit constraints for the time step depending on the spatial parameters of the grid. Let co = min c(x) and ~2 = max k(x). Taking into account the inequalities (15) b(z)E 2 COE, A 5 KZA5 KzA,&, where is the Laplace grid operator, and A,, is its maximum eigenvalue. (see Section 4.7) we can use the estimate For A,. A,, < 4(hy2 + h i 2 ) . (16) Taking into account estimates (15) and (16), we see that the inequality (14) is satisfied with K2 A = 4-(hc2 + hy2). (17)

A

CO

From (12) and (17) we obtain the following constraints for the time step:

The presented estimate (for u < 0.5) can be compared with the estimate of stability of difference schemes with weights (see (26) in Section 5.3) in uniform norm If u < 0.5, the limit time step in estimate (18) exceeds the time step in (19) (the latter can be increased by (1 - u ) / ( l - 20) times). Estimate (18) reflects (like (19)) the essential dependence of the maximum step on h (T = O(lhI2)). 5.5.2 ACCURACY OF TWO-LEVEL DIFFERENCE SCHEMES In order to investigate the accuracy of difference scheme with weights (1) and (2), we should consider the problem for error z,(x) = y,(x) - u(x, t,), x E w, namely

X E ~ , n = 0 , 1 , ..., zo(x) = 0, x E W . For the approximation error we have (see Section 5.3)

(21)

(22) $L(x) = O ( r Y+ llhl12). where Y = 2 for u = 0.5 and Y = 1 if 0 # 0.5. The corresponding estimates for the error can be obtained by using the results on stability of the difference scheme (20) and (21) with respect to the right-hand side (see Theorems 3-5 in Section 5.4). Here we use an analogue of Theorem 4 for the difference scheme (5) and (6).

263


Theorem 1. If A = A* > 0, D = D* > 0, and u

estimate

> 0.5, then the a priori

n

\\Yn+ll\2A5

/\YO\\;

+

7\I(Pk\\'D-1

(23)

k=O

holds for the dzfference scheme (5) and (6) Under the theorem condition, B >_ D identity ((6) in Section 5.4) results in

+ 0.5rA, and the main energetical

Using the estimate

we obtain the inequality

From this we obtain the proved estimate (23) of stability for the difference scheme (5) and (6). Now let us apply the theorem to the problem for error (2) and (21). Taking into account (6), we obtain

From this estimate we obtain a simpler one

Taking into account (22), the estimates (24) and (25) ensure that the difference scheme (1) and (2) converges with second order with respect to spatial variables and with order v with respect to time. 5.5.3 THREE-LEVEL SCHEMES WITH WEIGHTS

Let us consider the conditions under which the three-level difference scheme for the heat equation is stable. The common three-parameter scheme is (see scheme (36) in Section 5.3) like

with given yo(z) and yl(x), x E w , and uniform boundary condition.

264


First we write the scheme (26) in the canonical form

By checking directly we see that in this case we should take the coefficients

The operators B, R, and A are self-adjoint in (27) and (28), with A > 0. The stability of the scheme with respect to the initial data is ensured (Theorem 6 from Section 5.4) by the inequalities

We consider the difference scheme (27) and (28) under the additional constraint 8 0.5. The other case (8 < 0.5) should be considered separately (see Problem 2) and is not of interest for the problems on heat conduction in hand. Taking into account the operator inequality (14), from (28) and (29) we obtain

>

These conditions are satisfied at

The conditions of stability for the family of oneparametric three-level difference schemes introduced in Section 5.3 follow from the conditions (30) and (31) of stability for the general class of three-level difference schemes (26). For symmetrical three-level difference schemes we have

For the latter the conditions of stability (30) and (31) are reduced to the simplest inequality 1 u>z. (33) The class of purely implicit threelevel schemes corresponds to the parameters


265

in the scheme ( 2 6 ) . Condition ( 3 0 ) is clearly satisfied in the case of ( 3 4 ) , and ( 3 1 ) is satisfied at 02;. (35) Thus, in particular, the scheme providing the second order of time and space approximation with 8 = 1.5 is also stable (see Section 5 . 3 ) . In order to study the convergence of three-level difference schemes, we use the estimates of stability with respect to the right-hand side. The latter is formulated by Theorem 8 in Section 5.4. If the difference scheme ( 2 7 ) and ( 2 8 ) is stable with respect t o initial data, then for the error of difference solution Zn = {z,-l, t,} we have the estimate

where

1 & = R + - B27 ,

and the norm is determined by the formula

It follows from estimate ( 3 6 ) that the accuracy of solutions at the zero and first levels should be consistent with the approximation of the threelevel scheme. Let us obtain a simpler estimate for the norm of approximation error. Taking into account ( 3 7 ) and the positive definiteness of operators B and R , we have

Further considerations take into account the specific form of expressions ( 2 8 ) for grid operators. Thus, for example, for the symmetrical scheme ( 2 7 ) , ( 2 8 ) , and ( 3 2 ) we have B = b ( x ) E and R = uA, and therefore ~TIIB-'+,,~IR= ~ r r l [ b - ' + ~ l We l . can also obtain more acceptable norms for the approximation error, but do not dwell on this here. Based on the considerations above, we can draw a conclusion on the convergence of threelevel difference schemes in the corresponding norms. An additional, although obvious, point is that the accuracy of yo and yl should be consistent with the approximation error of three-level schemes. Thus, for example, if we use a scheme providing the second order of time approximation, then yl should be found within the same accuracy by using one or other difference scheme.

266


5.5.4 PROBLEMS

1. S t u d y t h e convergence of difference scheme (5) for A = A', if

Solution. In our case the necessary and sufficient conditions of stability in Ha result in

By virtue of this, the difference scheme (5) and (38) is unconditionally stable at a 16.

>

2. By regularizing t h e explicit Richardson difference scheme

c o n s t r u c t a n unconditionally s t a b l e explicit difference scheme.

Solution. The scheme (39) belongs to the class of symmetrical schemes with weights with e=;, o,=02=o=0. Here the stability conditions (30) and (31) are never satisfied, i.e the Richardson scheme is absolutely unstable. Let us regularize the difference scheme by perturbing the diagonal part of the grid operator which is denoted by D. Thus we arrive at the Dufort-Frankel difference scheme

The latter differs from (39)by replacing y, in the diagonal part of the operator A by a half-sum of solutions from the (n+l)th and ( n - 1 ) t h levels. It is written in the canonical form (27) with

B = b(z)E,

R = D,

A=A.

(41)

The scheme (27) and (41) is stable with respect to initial data, provided that the inequality R-+A=~(zD-A)>o (42)


267

is satisfied. For the grid elliptic operators A determined by (3) and (4) we have the estimate A < 2 0 , which has been proved in Section 4.7. This implies that the condition of stability (42) is satisfied. The disadvantages of difference scheme (40) are caused by the fact that it has the conditional approximation, namely $, = 0 (.r2 lhI2 ~ ~ I h l - i.e. ~ ) , = O(lhI2) at T = O(lhI2), as for the conventional explicit scheme.

+

+

5.6 Asymptotical Stability of Difference Schemes for the

Heat Equation

5.6.1 ASYMPTOTICAL STABILITY In order to describe the process of heat conduction, we separate the stage of regular mode (Section 3.4), which is characterized by the attenuation of the solution according to the asymptotics of the main (minimum) eigenvalue of the corresponding heat conduction operator. It is logical t o construct difference schemes which reflect such an asymptotical hehaviour of the solution at large times. Usually the condition of stability with respect to the initial data consists in the fact that the norm of solution does not increase in time, i.e. one investigates the stability of difference schemes with p = 1. To describe the regular mode of heat exchange correctly, it is necessary that the solution attenuates according t o a definite attenuation law. The difference schemes satisfying this condition are said to be asymptotically stable. We consider the simplest heat equation

au axu -=Cax; ' at

~ € 0 ,t > O ,

o=l

with the initial and boundary conditions U(X,t ) = O, u(x,O) = ug(x),

t > o, x E R.

E an,

(2) (3)

For the asymptotically developed stage of regular heat transfer the solution is specified by the formula

where XI is the first eigenvalue, and the eigenfunction wl(x) corresponds to it.

268


According to Section 5.2, we relate to the boundary value problem (1)-(3) the difference problem

in the finite-dimension space H of grid functions under the initial condition

The operator A in (5) is self-adjoint and positive definite, i.e

where 6 is the minimum eigenvalue of the Laplace difference operator A. The solution of problem (5) and (6) on the asymptotical stage can be represented as in (4). In order to formulate adequate conditions of stability, we present the corresponding estimate for the norm. Multiplying (5) in a scalar way by v(t) and taking into account (7), we

From this inequality we immediately obtain the following estimate of stability with respect to the initial data for problem (5) and (6):

It is quite reasonable to require that the norm of the difference solution to (5) and (6) decrease according to the estimate (8). To do this, we should specify the definition of stability for the corresponding difference solutiou. A difference scheme is asymptotically stable, if the estimate of stability with respect to the initial data

holds. The latter holds if the corresponding difference scheme is pstable with

In these frameworks the investigation of asymptotical stability is based on considering the pstability with p determined by (10).


5.6.2 T WO-LEVEL SCHEMES

For (5) and ( 6 ) let us write the conventional scheme

with the weight a. Recall (Section 5.4) that the two-level difference scheme

is pstable in Ha with A > 0 (or in Hg with B > 0 ) at A = A' and B = B* if and only if the conditions

~7- P B S lA+ pSg T

(13)

are satisfied. Let us check the conditions (13) of pstability for the scheme with weights ( 1 1 ) . The latter can he written in the canonical form (12) with the operators

First we consider the right-hand side of the two-side inequality ( 1 3 ) .For (14) we have A< l + p ~ + ~ ( l + ~ ) ~ . (15) T The latter is satisfied for all

T

if the weight is chosen according to

The usual condition of stability (for p = 1) results in the conventional condition of unconditional stability, i.e. a 112. By taking into account (14), the left-hand side of (13) is transformed to

>

The latter is satisfied at

270


Hence we obtain the upper estimate for the weight, namely

For problem (5)-(7) we consider pstable difference schemes with p determined by (10). Let us check the conditions (16) and (18) under such assumptions. For p = exp(-67) we have

since sinh(P) > p for fi > 0. Thus we cannot point out the weight u under which both the conditions (16) and (18) are satisfied, i.e. no two-level difference scheme with weights (11) is unconditionally asymptotically stable. We can, however, mention some classes of conditionally pstable schemes with weights. Along with (17), for the operator A we have the upper estimate

A 5 AE,

(19)

where A is the maximum eigenvalue of the Laplace difference operator A. First, let us find admissible time steps without fixing the weight of the difference scheme. If (19) is satisfied, then, instead of (16), we have

In this case

where 17 = 6/A. The admissible time steps are determined by the condition by the inequality

The latter is equivalent to the condition

01

- uo

2 0, i.e


271

where ~ ( p = ) p/sinh(p). Let us show that the inequality

holds for the function ~ ( p for ) p 2 0. Inequality (22) is equivalent to the function

being nonnegative. In investigating the nonnegativeness of functions, we use the statement proved in Problem 1 (see later). In our case we have

for p > 0. The inequality (24) follows from this. It follows from (21) and (22) that $ 5 6q and therefore even at the optimal (in sense of asymptotical stability) weight a = uO(rO)= ol(ro) for the time step we have 0 < T TO, 7 : % 6/(6A). (23)

<

By taking into account that 6 = O(1) and A = O(lhI2), the condition (23) of asymptotical stability results in TO = O(lh1) for the maximum step. It should be noted that, taking into account the error of spatial approximation, the condition of asymptotical stability can be replaced by the condition of p-stability with p = exp(-&T), where 6h = 6+O(lh12). However, the maximum time step does not essentially increase in this case. Now in inequality (20) we can put p = 6hr. Instead of (23), we obtain the estimate 0 < T 5 TO, T: = 66/(6:A), which is not essentially weaker than (23). Thus for any asymptotically stable difference schemes the time step should be constrained by the condition (23). The latter can only be replaced by more severe conditions for specific schemes (with a given weight a ) . Let us mention the most interesting difference scheme with weights. Let us obtain the corresponding conditions of pstahility for the symmetrical (u = 112) and explicit ( a = 0) schemes. The left-hand side of (13) (inequality (18)) is satisfied in both cases. This is led from inequality

273


condition for p, we can obtain stable schemes under weak constraints for the time step. Let us consider the purely implicit scheme as an example. In this case inequality (16) (and even more, inequality (20)) is satisfied. Let us put p = exp(-y&), where 0 < y < 1 (less restrictive constraints for the solution decrease). Then let us choose the maximum step TO so that the inequality (18) is also satisfied. This is the case for u = 1 if

The latter is satisfied for y < 1 at 67 5 ~ ( y ) The . corresponding numerical values are given in Table 1. Thus difference schemes with weights are pstable under the constraint T 5 TO = 6-'~(7) for the time step. with p = exp(-$7)

7

r

1

0.5

0.75

0.9

0.95

0.975

Table 1. 0.99

2.513

0.734

0.230

0.107

0.052

0.020

5.6.3 THREE-LEVEL SCHEMES

Asymptotical stability can be analysed in detail for three-level difference schemes. In order to solve the initial problem (5)-(7) approximately we use the scheme

with the weights 0, u1, and u2. As the first example we consider implicit schemes with weights where ul = 1 and u2 = 0 in (29). If n 2, where n = 20 - 1, the conditions of asymptotical stability result in the inequality

>

A simpler estimate p 5 l / n follows, in particular, from (30). The obtained constraints for the time step 7 5 r0 = ll(n6) are not restrictive. There is no unconditionally asymptotically pstable schemes with p = exp(-67) among three-level symmetrical difference schemes (0 = 112, u1 = u2 = u).

274


5.6.4 PROBLEMS 1. S h o w t h a t f(x) 2 0 for all x 2 0 if

is satisfied. Solution. In order to show this, it is sufficient to consider the expansion of function f ( x ) in the Taylor series, namely

2. Investigate t h e asymptotical stability of t h e complex scheme which c a n b e w r i t t e n i n t h e canonical f o r m (12) w i t h

Solution. The right-hand side of (13) is satisfied for all p > 0, and, taking into account (31), the left-hand side can be reduced to

Taking into account estimate (7) and p = exp(-6r), we obtain the inequality

$ p2 + p 2 exp(p) - 1,

p = 67.

The latter is not satisfied for any p. As in the purely implicit scheme we can take p = exp(-y6~) at ip 1. The function ~ ( y ) 6.r is given in Table 2.

<

>

Table 2. Y

1

0.5

0.75

0.9

0.95

0.975

0.99

1

Thus the considered difference scheme has good asymptotical properties (a quite large time step is admissible even for the parameter y close to 1). Besides, unlike the purely implicit scheme, the scheme (12) and (31) provides the second order of accuracy with respect to T.


5.7 Hyperbolic Heat Equation

5.7.1 DIFFERENTIAL PROBLEM In modelling high-intensive nonstationary processes of heat conduction one uses (see Section 2.1) the hyperbolic heat equation which takes into account that the heat perturbation propagates with a finite speed. The heat equation in an isotropic medium is

with the conventional operator of heat conduction

In (1) V is the relaxation parameter of the heat flow. The closing relations are the initial and boundary conditions

Let us obtain the a priori estimate of the solution to problem (1)-(5). We will tend towards this estimate in investigating the stability of corresponding difference schemes. Let us multiply equation (1) by aulat in a scalar way in H ' = L2(R). Then we obtain

The right-hand side of (6) is estimated as

It follows from (6) and (7) that

276


where, taking into account the uniform boundary conditions (3), the value

determines the norm of the solution to (1)-(5). Applying the Gronwall lemma (see Section 5.1) to (8), we obtain the estimate

If v = 0, the a priori estimate (9) and (10) is reduced to the conventional estimate of stability for the problem of heat conduction (Problem 2 from Section 5.1). 5.7.2 THE STABILITY OF SCHEMES WITH WEIGHTS It is logical to use three-level schemes with weights to solve the problem (1)-(5) approximately. The common three-parametric family of such schemes can be presented as Ob(x)yt + (1- O)b(x)y-i+ V b ( x ) ~ ~ A (a18 (1 - a1 - oz)yaz$) = p

+

+

(11)

at given yo(x) and y,(x), x E w. The scheme (11) is written in the canonical form with

The difference scheme in hand differs from that with weights for the heat equation only by the additional term V T - ~ ~ ( X )inE the operator R, which improves the stability conditions. The operators B, R, and A in (12) are self-adjoint, with A > 0.Let us check the conditions of stability with respect to the initial data for the scheme ( l l ) , namely B ~ O , R > ~ A . (13) It follows from (12) and (13) that the scheme (11) is stable if


277

It follows, in particular, from (14) that the scheme is absolutely stable under the conventional (see Section 5.5) constraints for the weights, namely

The norms of the difference solution are specified according to the general theory of stability for difference schemes (Section 5.4). 5.7.3 S Y M M E T R I C A L SCHEMES

Among schemes with weights we select a oneparametric family of symmetrical difference schemes for which

In this case difference scheme (11) has the approximation error $, = O(T' lhI2). , , . If (16), then the stability condition (14) is like

+

The symmetrical explicit difference scheme ( a = O ) , in particular, is conditionally stable at r2 5 4 V / A , i.e. for T < O(lh1). Let us present the difference analogue of the a priori estimate ( 9 ) and (10) for the symmetrical scheme (11) and (16) at u = 0.25. Denoting

we write the symmetrical difference scheme (11) and (16) for a = 0.25 as

Taking into account the obvious inequalities

6 + u = -2( w T

"

-w),

we multiply the equation (19) by

i, - u =

T . qw +w), 2

lir + w in a scalar way in H and obtain

278


For the right-hand side we use the estimate

Taking into account that the equality

holds for self-adjoint operators Q, from (20) and (21) we obtain

Taking into account ( I S ) ,from (22) we obtain the inequality

where

a = v(o(~g,,)+ ( A F,

q) .

(24)

Inequality (23)is the differenceanalogue to (9), and the norm (24)is consistent with the norm (10) of the differential problem. The estimate obtained ensures the stability with respect to the initial data and right-hand side. Therefore we have shown that the symmetrical scheme converges to the precise solution in quite simple norms. The same holds for the corresponding difference scheme for the conventional heat equation (V = 0 in (11)).

5.7.4 PROBLEMS 1. For t h e hyperbolic h e a t equation (1) a n d (2) c o n s t r u c t a difference scheme providing t h e f o u r t h o r d e r of t i m e approximation. Solution. Expanding the equation at the point ( x ,t,), x E w , we have


Therefore

au

c(x) at+Vc(z)-

aZu+ Lu - f (x, t) at2

It follows from equation (1) that

Therefore (25) allows us to write the difference scheme

Due to the constructions, the difference scheme (26) has the approximation error 0 ( r 4 lhl2).

+

2. Investigate the stability of the difference scheme (26). Solution. Scheme (26) is written in the canonical form with the operators

By taking into account the inequality A can be transformed to

< Ab(x)E, the condition of stability

280


Thus the scheme is stable if the inequality

holds. Hence it follows that the difference scheme in hand is stable for V 5 ll(2A).

5.8 Regularization of Difference Schemes

5.8.1 THE REGULARIZATION PRINCIPLE

Stable difference schemes are constructed and investigated based on some general statements. One of these is the conservativity principle which requires that the conservation laws are satisfied for the discrete analogue of the difference problem. The construction of difference schemes for evolutionary problems may also be based on the regularization principle for difference schemes. The principle has been discussed for iterative methods in Section 4.7. The regularization principle for dzfference schemes can be considered as a common approach for improving the qualities (stability, accuracy, etc.) of difference schemes. It is based on improving the qualities of the simplest difference scheme by perturbing grid operators. The primary difference scheme is written in the canonical form and the operators are perturbed with the general conditions of stability taken into account. The stability conditions formulated as operator inequalities show the ways to improve difference schemes. The regularization principle is a common approach. It consists of the following stages. (1) The simplest difference scheme is written for the initial differential problem. The scheme approximates the problem but is not unconditionally stable. (2) The initial difference scheme is written in the canonical form. (3) Grid operators of the difference scheme are perturbed on the basis of general conditions of stability so that the approximation conditions are not violated. This results in a class of stable regularized difference schemes. Further we use the regularization principle for constructing economical difference schemes in solving many-dimension problems. This approach is also applied in constructing stable difference schemes for ill-posed (inverse) problems for evolutionary equations.


281

5.8.2 REGULARIZATION OF TWO-LEVEL DIFFERENCE SCHEMES We again (see Section 5.3) consider the boundary value problem of heat conduction

where

According to the regularization principle, we take the simplest explicit difference scheme

as the initial one. The operator A in (5) is determined on the set of grid functions being zero on aw, and, for example,

The difference scheme (5)-(7) approximates the initial boundary value problem (1)-(4) within the accuracy O(T + lhI2).It belongs to the class of conditionally stable schemes, namely (see Section 5.5) it is stable at

where A is the constant in the inequality

A

< Ab(x)E.

We construct absolutely stable difference schemes based on the conditional stable scheme (5)-(7). According to the formulated principle, we write (5) in the canonical form of two-level difference schemes, namely

282


For the grid operator B and A we have

The regularization is based on passing from the initial difference scheme to another (perturbed) scheme. Assume that R = R ' > 0 is the regularized grid operator, and denote by a the parameter of regularization (perturbation). In order for the difference scheme (10) to be stable in H A , where A = A' > 0 (in H g , where B = B' > 0), it is necessary and sufficient that the condition

is satisfied. According to the latter, it is logical to associate the regularization with additive perturbation of the operator B. Therefore we write the regularized scheme for (5) as

In order for (13) to conserve the approximation, we should choose the perturbation parameter a = O(T). Let us show that difference scheme (13) is unconditionally stable in H A and H g with the regularizator R = A, if

The proof is based on checking the necessary and sufficient condition (12). For the scheme (13) we have

Taking into account (9), and substituting (15) into (12) with R = A, we obtain

The proved statement follows from this. Note that the regularization (13) with R = A just corresponds to the conventional scheme with weights ( a = U T ) , and condition (14) is the stability condition for the scheme with weights (see (11) in Section 5.5).


283

Logically, the regularizator R can be chosen in various ways. We mention the variant R = A2. The inequality (12) for such a regularizator is transformed

>

r2/(16%), ~g = minc(x), the Hence it follows that for R = A' and a regularized scheme (13) is absolutely stable. An example of such a scheme with b(x) = 1 has been considered earlier (see Problem 1, Section 5.5). We have improved the properties of explicit scheme (5) by perturbing the grid operator B in the two-level difference scheme (10). The operator A could also be perturbed here (see Problem 1). 5.8.3 ENERGETICALLY EQUIVALENT REGULARIZATORS

We have considered regularizators R which are directly connected with the heat conduction operator A. The regularization principle can also use regularizators R which are energetically equivalent to those constructed on the basis of operator A. We have used such an approach earlier in Section 4.7 for constructing iterative methods for solving stationary problems of heat conduction. It is convenient to construct the regularizators on the basis of simplest grid elliptic operators. For the coefficients of grid elliptic operator A determined according to (7) we have 0 < nl 5 a,(x) 5 nz, a E w , where, for example, nl = min k(x) and n2 = maxk(x). Therefore the operator A j s energetically equivalent to the grid Laplace operator which we denote by A, namely

Let us consider the regularized difference s ~ h e m e(13) with regularizators constructed by the Laplace operator. If R = A, then, taking into account (9) and (16), the necessary and sufficient condition (12) is transformed to

Hence it follows that the regularized difference scheme (13) with the regularizator R = A is stable if a 5 n2(7/2 - A-I). This condition is more rigid than (14).

284


Similarly we consider the regularized difference scheme (13) with R = (A)Z. In this case T

a

T

T

B--A=b(x)E+a(A)'--A=b(x)E+-A2--A, 2 2 4 2 and therefore the scheme is stable for a 2 nzr2/(16co). Thus in this case the regularization parameter a is proportional to the squared maximum of heat conductivity coefficient. 5.8.4 REGULARIZATION OF THREE-LEVEL SCHEMES As the initial scheme we take in an asymmetrical explicit scheme (the Richardson scheme, see Section 5 . 5 ) , namely b(x)y.

t

+ Ay =

ip.

(17)

This scheme provides the second order of time and space approximation and is written in the canonical form By.

t

+ T'RR, + Ay = ip

(18)

with the operators

The stability conditions for the scheme (18) are

The scheme (18) and (19) is absolutely unstable because the second inequality in (20) is not satisfied. Therefore we can regularize it by perturbing the grid operator R. Starting from (17) we construct the regularized scheme

The latter is written in the canonical form (18) with

B = b(x)E,

R = aR,

A = A.

(22)

Let us mention some variants in choosing the regularizators. The simplest way is to put R equal to D, where D is the diagonal part of the grid operator A. Taking into account (Section 4.7) the inequality A < 2 0 , for R = D we obtain R - ' 4A = ~ D - $ A > ( ~ - $ ) D .


285

Therefore the stability conditions ( 2 0 )of the regularized difference scheme ( 1 8 ) and ( 2 2 ) are satisfied for a 2 0.5. The specific case of regularized difference scheme ( 2 1 ) with a = 2 corresponds to the Dufort-Frankel difference scheme considered earlier in Section 5.5. Note that a = 0 ( 1 ) , and therefore the regularized difference scheme (21) essentially worsens the approximation. Assume now that R = A in the regularized scheme (21). In this case

for a > 0.25. Such aregularizator corresponds to the conventional symmetrical three-level scheme (see Section 5.5) with a = a?. Under such a regularization the scheme still provides the second order of time and space approximation. We can also mention other ways to construct regularized difference schemes. Thus, for example, as in two-level schemes we can choose the regularizator on the basis of an energetically equivalent operator. 5.8.5 PROBLEMS

1. C o n s t r u c t a regularized difference scheme o n t h e basis of explicit scheme (13) by regularizing t h e o p e r a t o r A. Solution. We consider the scheme

Let us put R = AZ and check the necessary and sufficient conditions. By taking into account (9),the condition that the operator A = A - @A2 is positive results in the lower estimate of the regularization parameter

a

<

1 A min c ( x )

Inequality (12) is transformed to

Hence we obtain the upper estimate for a , namely a

> 8 max c ( z )

'

286


The explicit scheme (23) is conditionally stable. It follows from (24) and (25) that the maximum time step (for optimal a) is constrained by the condition 8 maxc(x) r
Comparing (26) with the stability condition (T 5 2/A) for the explicit scheme (13), we see that the regularized scheme (23) for R = A2 admits a larger time step. 2. Consider t h e regularized scheme (21) w i t h t h e regularizator R = E ( t h e Dufort-Frankel scheme). Solution. In this case the regularized scheme has the canonical form with

For the grid operator A determined by (7) we have

The condition (20) results in the following constraints for the regularization h;'). parameter: a > max k(z) (h;'

+

5.9 N o n l i n e a r N o n s t a t i o n a r y Problems

5.9.1 QUASI-LINEAR HEAT EQUATION

Here we construct difference schemes for problems of nonstationary heat conduction in media whose heat characteristics depend on temperature. We consider the heat equation in the isotropic medium

where

Equationss (1) and (2) are complemented by the conditions


287

Let us mention conditions under which the problem bas a unique solution. The investigation is based on the corresponding statement on the uniqueness of the solution to the corresponding linear problem. Assume that u(x,t) in Q satisfies the parabolic equation

having coefficients continuous in Q. Using the maximum principle, we can show that the solution to problem (3)-(5)is unique. Note that this holds regardless of the sign of the coefficient ao(x,t). Assume that there exist two solutions of problem (1)-(4),namely up(x,t), P = 1,2, such that

under the corresponding initial and boundary conditions. For the difference of solutions w(x) = u2(x)= -ul(x) we obtain (see Section 4.9)the boundary value problem

Here we denote

The linear boundary value problem (7)-(9)belongs to the class of problems (3)-(5)mentioned above. Therefore the trivial solution w(x,t) = 0 of problem

288


(7)-(9) is unique for sufficiently smooth coefficients c ( x , u ) , k ( x , u ) , righthand side f ( x ,t , u ) and solution of problem (1)-(4). We can require that the inequalities

are satisfied. Thus the solution of the nonlinear problem of heat conduction (1)-(4) is unique in the class of bounded nonlinear coefficients. 5.9.2 LINEARIZED DIFFERENCE SCHEMES

Based on the difference schemes constructed for the simplest nonlinear stationary heat equation (Section 4.8), we present some difference schemes for the nonstationary problem. Let us define the nonlinear grid elliptic operator of stationary heat conduction A(u)y on the set of grid functions on w by the relation 7

The initial differential problem (1)-(4) is related to the difference differential problem

Here, for example,

and the coefficients a,(x,u) are given as

Let us present some difference schemes for the problem (12)-(14).First we mention the class of linearized difference schemes which is characterized by

289


the solution a t the new time level being found by solving a linear difference problem. In the simplest scheme the coefficients are taken from the previous time level. We can cite the difference scheme

+

as an example. This scheme, clearly, has the approximation error O(T lhI2). The further development of scheme (15)-(17) is that with quasi-linearized right-hand side, namely

The latter is linear, but has a larger reserve of stability with respect to the nonlinear right-hand side. Linearized schemes can be constructed on the basis of predictor-corrector difference schemes. In this case difference scheme (15) is linearized in the following way. The explicit scheme

is used at the predictor stage for calculating the coefficients and the right-hand side. Therefore the correction stage may correspond to the scheme

The correction stage may also use the linearization scheme (18). Therefore, it is worthwhile to use the scheme

290


instead of (20).Instead of (21) we can also use another variant of the quasilinearization, namely

The schemes presented reveal large possibilities in constructing linearized difference schemes. In applied mathematical modelling one should carry out special investigations in order to choose difference schemes for specific classes of nonlinear boundary value problems. The theoretical considerations in nonlinear problems are often not sufficient. Logically, linearized difference schemes for problem (12)-(14) can be constructed on the basis of threolevel difference schemes. We do not dwell on detailed description and restrict ourselves to the simplest examples. For the approximate solution of problem (12)-(14) we can use threelevel symmetrical difference schemes

with the corresponding initial and boundary conditions. These linearized schemes provides the second order of both time and space approximation. 5.9.3 NONLINEAR DIFFERENCE SCHEMES

Nonlinear difference schemes are widely applied in calculation practice. Here the solution at each time level is found by solving a nonlinear difference problem. Such schemes are constructed similarly to linear schemes for the linear heat equation. Thus, a purely implicit difference scheme for (12)-(14) is like

Yn+1 - Yn b(x,Y ~ + I ) T

+ A(~n+l)~n+l= ip(x,tn+~,~n+~), x E w , n=0,1,...,

(23)

under the additional conditions (16)and (17).Similarly we construct analogue to schemes with weights. As an example we present the nonlinear analogue to


291

the two-level symmetrical difference scheme (the Crank-Nicolson scheme):

Among three-level nonlinear difference schemes we mention the implicit schemes of the form (see Section 5.3)

If 0 = 1.5 this implicit scheme provides the second order of time and space approximation, and the corresponding linear scheme is unconditionally stable. The difference schemes presented are direct analogues to the implicit schemes for the linear boundary value problem of nonstationary heat conduction. 5.9.4 ITERATIVE REALIZATION OF IMPLICIT SCHEMES In order to find the difference solution at a new time level of implicit schemes, one should solve a nonlinear difference problem. One or other iterative processes are used to determine y,+l. These are constructed on the basis of solution methods discussed in Section 4.9 for stationary nonlinear problems of heat conduction. Note some important features of the corresponding nonlinear grid problems. 1. A good initial approximation is available in iterative realization of an implicit difference scheme. Assume that w k be the iterative approximation to solution y,+l at the kth iteration. It is logical to take the solution at the previous time level as the initial approximation, i.e. w k = y,. 2. The grid problem on determining y n + ~includes a small parameter, namely the time step T . It influences essentially the degree of convergence of the iteration process (the less T the faster the corresponding iterative process converges). These features are also characteristic for linear grid problems in realizing implicit schemes and are discussed in detail in Chapter 6. Here we restrict ourselves to only some examples. We consider the nonlinear difference scheme (23). The simplest iterative process is associated with the successive specification of nonlinear coefficients.

292


In this case the new iterative approximation wk+l to yntl is found by solving the boundary value problem for the difference equation

The linearized difference scheme (15) coincides with the iterative scheme (24) if we restrict ourselves to one iteration. Equation (24) can be modified by linearizing its right-hand side and using the equation

In doing so the grid elliptic operator for determining the new approximation remains self-adjoint. The Newton method is widely applied in realizing nonlinear difference schemes for nonstationary problems. In this case the new approximation is found from the difference equation

Here (see Section 4.9) the grid elliptic operator A'(wk) is like

i.e. it is not self-adjoint. The grid problem for equation (25) is solved by using iterative methods. But the problem becomes more difficult because the grid elliptic operator is not self-adjoint. The degree of convergence of iterative process (25) is investigated in the conventional way, but it is rather cumbersome and we omit it.


293

5.9.5 THE ACCURACY OF DIFFERENCE SCHEMES The investigation of nonlinear difference schemes for nonstationary problems of mathematical physics is difficult because there is no general theory of such schemes, The technique presented earlier for linear schemes is quite general and consists in studying the approximation error, stability and convergence. The methods for obtaining estimates in general theory of linear difference schemes cannot be applied to nonlinear schemes. Therefore there are a variety of approaches to investigation of convergence of difference schemes in nonlinear problems. The same situation takes place in the theory of nonlinear equations of mathematical physics. As like for the stationary problems of heat conduction (Section 4.9), we restrict ourselves to the simplest nonlinear problem (1)-(4) where only the right-hand side is nonlinear, i.e. c = c(x) and k = k(x). For this problem we consider the purely implicit scheme (23)which is like

here A is the linear operator

We assume that the right-hand side of (26) satisfies the condition

In order to investigate the uniqueness of the difference solution determined by equation (26) and conditions (16) and (17),let us assume that here exist two solutions yk and yi. For the difference vn(x) = yi(x) - yL(x) we obtain the problem

Assume that the relation min b(x)

72-

u

294


is satisfied for the time step; then the maximum principle is satisfied for the problem (28)-(30) (see Section 5.3), and the latter has only the trivial solution. If constant v is negative in (27) then the solution is unique for any time step. In order to investigate the accuracy of the difference scheme (26), (16) and (17), we formulate the corresponding problem for the error zn(x) = y,(x) - u(x, tn), x E a. Let &(x) be the approximation error, with &(x) = O(T+ lhj2), lhI2 = hf + h;. The solution error satisfies the equation

and uniform boundary and initial conditions (29) and (30), respectively. Assume that v 5 0 in (27); then (see Section 5.3), from the maximum principle we obtain the estimate

Thus we have proved that the purely implicit difference scheme converges if (27) is satisfied with a nonpositive constant v. 5.9.6 PROBLEMS 1. Show that the a priori estimate I l ~ ( ~ , t ) l l c (5q max ) { l l ~ ( ~ ~ t ) l l cIluo(~)llccn)} (r),

+ \v\-l\\f(">t , O ) l \ ~ ( ~ ) holds for the problem (1)-(4) with c = c(x) and k = k(x) if the inequality -a(fx , t , u ) < v < O

au

is satisfied. Solution. Using the expansion

we write equations (1) and (2) as


295

Further we consider separately the problem for the uniform equation (the stability with respect to the boundary and initial conditions) and that with uniform boundary conditions (the stability with respect to the right-hand side). 2. C o n s t r u c t t h e linear s c h e m e of predictor-corrector providing t h e second o r d e r of t i m e a n d s p a c e a p p r o x i m a t i o n for t h e p r o b l e m of h e a t conduction (1)-(4) w i t h c = c(x) a n d k = k(x). Solution. The predictor stage is used for calculating the nonlinear righthand side at the time t , + ~ ~ ,The . corrector corresponds to using the linearized symmetrical scheme

To find

v we can use the scheme

Logically, other schemes can be used to find V. The corrector-predictor scheme presented has the approximation error 0 ( r 2 + lhI2).


5.10.1 GENERAL NOTES 5.1 Linear problems for second-order parabolic equations are considered in all the manuals on equations of mathematical physics 17, 19, 201. The heat equation is used as the model equation. Similar problems arise in describing diffusion processes. The maximum principle for the heat equation has a clear physical interpretation. The uniqueness of solutions of the main boundary value problems can be proved by using the principle in the simplest way [4]. Nonstationary problems of mathematical physics are often considered as differential operator problems in the corresponding spaces. Such problems are studied most completely in (61. We have restricted ourselves to the simplest estimates in Hilbert spaces. 5.2 The general theory of difference schemes for nonstationary problems is presented according to 112-141, which, in particular, introduce the canonical forms of difference schemes for nonstationary problems. An

296

5.3

5.4

5.5

5.6

5.7

5.8

5.9.


important moment in investigating twelevel difference schemes is to find the relation between the stability with respect to initial data and that with respect to the right-hand side. The methods for constructing difference schemes for nonstationary problems are well shown in manuals 11, 9-12, 211. We have presented estimates of error only for sufficiently smooth solutions, More general problems (discontinuous coefficients, generalized solutions etc.) are considered similar to stationary problems. Such material can be found in 112, 131. The accuracy of difference schemes is investigated on the basis of the maximum principle. Usually (see 112-141) this material is presented for problems with a constant coefficient by the time derivative. Such a simplification is not always justified for the applied problems of physics of heat in hand. The general theory of difference schemes is presented according to 11214, 171. The transition to the canonical form allows us to formulate the necessary and sufficient conditions of stability as operator inequalities. This theory is presented most completely in 1161. This book also reviews other methods for investigating the stability of difference schemes and supplies a number of examples. Here we have specified the results of the general theory of stability for the heat equation, The content is close to 112-14,16,17]. The investigation of various difference schemes for nonstationary problems by other methods is given in 12, 9-11, 18, 211. The notion of asymptotical stability is introduced in 1161. The books 113, 161 cite examples of asymptotically stable difference schemes. Thus, in particular, the conventional schemes with weights are studied for the heat equation. Here the asymptotical stability is agreed with the pstability of the difference scheme, with p < 1 consistent with asymptotical properties of the difference differential problems. The stability of three-level difference schemes for the hyperbolical heat equation is investigated on the basis of the general theory of stability. Such difference schemes for the telegraph equation, which is similar to the hyperbolic heat equation, are investigated in 1161. The regularization principle as a general technique for constructing and investigating difference schemes has been proposed by A. A. Samarskii in 1967 and is discussed in 1131. Here it is used for constructing unconditionally stable difference schemes on the basis of initial conditionally stable or unstable difference schemes. Other applications of the regularization principle are reflected in other parts of the book. Nonlinear boundary value problems of heat conduction are considered in many original papers and monographs. The book 1151, in particular, is devoted to studying processes for the secalled modes with peaking. The theory of quasi-linear parabolic problems is reflected in [8]. Difference

297


m e t h o d s for solving nonlinear problems of h e a t conduction a r e considered in [3, 51. We have restricted ourselves b y presenting only some moments in constructing a n d investigating nonlinear difference schemes.

5.10.2 LITERATURE 1. Bakhvalov N. S., Zhidkov N. P. & Kobel'kov G. M. (1987) Numerical Methods [in Russian]. Nauka, Moscow. 2. D'yakonov E. G . (1972) Difference Methods for Solving Boundary Value Problems. Part 2. Nonstotionory Problems [in Russian]. Izd-vo Mosk. un-ta. Moscow. 3. Fedotov E. M. (1987) Difference Schemes for Nonlinear Nonstationary Problems [in Russian]. Izd-va Kazan. un-ta, Kazan'. 4. Friedman A. (1964) Partial Differential Eouotions o f Parabolic Tuue. ~, ". EndewoodCliffs, New Jersey. 5. Karchevskii M. M. & Lvashko A. D. (1976) Schemes for Nonlinear , . Differrnce Problems of Mathematical Physics [in Russian]. Izd-vo Kaz. un-ta. Kazan'. 6. Krein S. G. (1967) Linear Differential Equations i n Bonoch Spnces [in Russian]. Nauka, Moscow. 7. Ladyzhenskaya 0 . A. (1973) Boundary Value Problems of Mathematical Physics [in Russian]. Nauka, Moscow. 8. Ladyzhenskaya 0. A , , Solonnikov V. A. & Ural'tseva N. N. (1973) Linear and Quasilineor Equotions of Parabolic Type [in Russian]. Nauka, Moscow. 9. Marchuk G. I. (1975) Methods of Numerical Mathematics. Springer-Verlag, New York. 10. Richtmyer R. (1957) Difference Methods for Initial-value Problems. Interscience, New York. 11. Richtmyer R . & Morton K. (1967) Difference Methods for Initial Value Problems. Springer, New York. 12. Samarskii A. A. (1971) Introduction to Theory of Difference Schemes [in Russian]. Nauka, Moscow. 13. Samarskii A. A. (1983) Theory of Difference Schemes [in Russian]. Nauka, Moscow. 14. Samarskii A. A. (1987) Introduction to Difference Methods [in Russian]. Nauka, Moscow. 15. Samarskii A. A,, Galaktianov V. A., Kurdyumov S. P. & Mikhailov A. P. (1987) Models with Peaking i n Problems of Quasy-linear Porobolie Equations [in Russian], Nauka, Moscow. 16. Samarskii A. A. & Gulin A. V. (1973) Stability of Difference Schemes [in Russian]. Nauka, Moscow. 17. Samarskii A. A. & Gulin A. V. (1989) Numerical Methods [in Russian]. Nauka, Moscow. 18. Saul'ev V. K. (1960) Integmtion of Equations of Pombolic Type by the Grid Method [in Russian]. Fizmatgiz, Moscow. 19. Tikhonov A. N. & Samarskii A. A. (1972) Equations of Mathematical Physics [in Russian]. Nauka, Moscow. 20. Vladimirov V. S. (1976) Equations of Mathematical Physics [in Russian]. Nauka, Moscow. 21. Wasow W . & Forsythe G. (1960) Finite-difference Methods for Partial Differrntial Equations. John Wiley & Sons, New York.

-

-

Economical Difference Schemes for Nonst ationary Heat Conduction Problems

Two- and three-level difference schemes for the two-dimensional heat equation have been classified in Chapter 5. When implicit difference schemes are used to find the solution on a new time level it is necessary to invert a grid elliptic operator, and various direct and iterative methods of solving grid elliptic problems are employed to this end. Computational costs can significantly exceed in this case the costs involved when the solution is determined by an explicit scheme. For explicit schemes, the number of arithmetic operations per grid node is independent of the total number of the nodes, i.e. the computational costs are asymptotically optimal. Difference schemes that possess this property are called economical difference schemes. However, the magnitude of the time step for explicit difference schemes is heavily restricted by considerations of stability. The computational costs for the implicit difference schemes considered above are in general higher in the case of multidimensional nonstationary problems, however, some of these schemes are unconditionally stable. Therefore, we are faced with a problem of constructing difference schemes that would he as economical as explicit schemes and would retain the unconditional stability inherent in some of the implicit schemes. This chapter deals with this problem of constructing stable economical difference schemes for the heat equation. We consider here how to implement implicit difference schemes for multidimensional nonstationary problems on a computer. When solving grid elliptic

300


problems iteratively on the top time level, the convergence rate depends not only on the number of space steps but also on the time step. The well-known schemes of alternating directions are simple examples of economical difference schemes for the heat equation. It should be noted that it is difficult to extend the method of alternating directions to three-dimensional problems with nonseparable variables. Factorized schemes, in which the operator is represented on the top level as a product of two economical operators, are widely used to construct economical difference schemes. Such schemes can be constructed by using the principle of regularization of difference schemes. The economical schemes we consider here belong to the class of additive difference schemes. A feature of these schemes is that the main grid operator (the heat conduction operator A in our case) can be represented as a sum of several simpler operators, e.g. one-dimensional grid operators. The notion of a sum approximation is used to construct and analyse the general classes of additive difference schemes. Two- and three-level economical difference schemes for the heat equation are analysed on the basis of general results of the theory of additive schemes. Splitting into the sum of one-dimensional grid operators leads us t o locally one-dimensional difference schemes.

6.1 Implementation of Implicit Schemes on a Computer

6.1.1

GRID ELLIPTIC PROBLEM

We shall consider the ordinary two-level scheme with weights for the heat equation (see Section 5.3) as an example, when modelling heat transport in a rectangular domain 0,namely

+

The accuracy of this scheme is O ( r Y lhI2),where v = 2 if o = 0.5 and v = 1 otherwise. Let A be a constant in the inequality

ECONOMICAL DIFFERENCE SCHEMES..

.

301

then the scheme with weights (1)-(3) is stable (see Section 5.5) provided that

To find the solution on a new time level from (1)-(3),we have to solve the grid elliptic problem ( a > 0) for the equation

equipped with the boundary condition (2).The right-hand side in (6)has the form

The grid elliptic problem (2),(6) explicitly involves the time step T and the weight parameter a. Therefore, it is worthwhile analysing the convergence rate of iterative methods for the problem (2),(6) versus these parameters. 6.1.2 EXPLICIT ITERATIVE METHOD

Let us rewrite the elliptic problem in (2), (6)as the operator equation

in a Hilbert space of grid functions defined on Li and vanishing on 8. operator A is represented as

The

Hence, we have

A=A'>O

(9)

for T > 0 and a > 0. In order to solve the problem in (7) approximately, we shall consider, for definiteness, a two-level iterative process with the Chebyshev set of iterative , parameters ~ k namely

Provided that

B=B*>O

(11)

302


and the following two-sided operator inequality holds

we have (see Section 4.6) the following estimate for the number of Chebyshev iterations (9)-(11) necessary to achieve the prescribed relative accuracy E :

n 2 no = ln(2~-')/ln(~;'),

(13)

where

We have (13), (14) and, therefore, when solving the problem (7), (8) iteratively we are most interested in how E depends on T and u. Let us consider the simple example of an iterative process in which the operator B (see Section 4.7) is the diagonal part of the operator A. For the grid operator (8) the diagonal part D = d(x)E has the form

Let us give some estimates for the constants "i,, a. = 1,2, in (12), given the selection of the operator B = D. As previously (see Section 4.7), we have the equality

Similar to the limit case of u = 1, T = co,which was considered in Section 4.7, we can use solutions of one-dimensional three-point problems to estimate yl. We can take 2

71 = m i n x M;' zEw

The grid functions M;'(xz) M;'

+ min

= rnin M;'(xz) =,EM,

o=1

M;'(xl)

=,EM,

and M;'(X~) are defined according to

= max va(s),

a. = 1,2,

=.EM"

where u(a,v~JZm va(x) = 0,

' 4 ~ )v = -d(x) +T

2,

= 0,1,.

>

XEw,

ECONOMICAL DIFFERENCE SCHEMES.. .

303

It is convenient to trace the effect of r and a using the example of a heat conduction problem in a homogeneous medium when w and the grid are both square (h = hl = h2). In this case k(x) = ko, c ( x ) = @ and the left-hand inequality in (12) takes the form

where A is the grid Laplace operator on a square grid. Taking into account the inequality

where 6 = O(1) is the least eigenvalue, we derive from (17) the following expression for yl:

It follows from (18) that

When we use a symmetric scheme, it is logical to take into account the asymptotic of the approximation error and the condition of asymptotic stability (see Section 5.6) and select the time step r = O(h). Then, we would have = O(h) from (19), and the number of iterations in the method considered would thereby depend on the total number of nodes. The estimate (13) yields n 2 no(^) = ~ ( h - ' I 2h ( ~ - I ) ) . (20)

<

If o # 0.5 and T = 0 ( h 2 ) (the approximation error in this case is 0 ( h 2 ) ) , instead of (20), we obtain from (19)

i.e. the number of iterations does not depend explicitly on the number of nodes. Of course, the relative error E of the iterative solution should be adjusted to the approximation error, and, thereby, to the grid parameters. However, in any case this dependence is weak (for E = TP, /3 > 0, nO(c) = O(ln(r-I)) and for this reason the iterative method can be ascribed to the class of economical methods for solving nonstationary problems, given this relation of space and time steps.

304


6.1.3 ITERATIVE METHOD OF ALTERNATING DIRECTIONS

Let us note some peculiarities of using the iterative method of alternating directions to solve the grid elliptic problem (7), (8) on the top time level. In this case it is necessary (Section 4.7) to represent the operator (8) in the form

To this end we make use of the decomposition of the operator A into onedimensional operators, namely

Then, taking into account (8), we represent the operators A,, a = 1 , 2 , in the

where p is a numerical parameter (0 5 5 1) which is to be selected so as to minimize the number of iterations. In order to solve (7) approximately, we use the method of alternating directions with a constant iterative parameter wo:

The rate of convergence of the iterative process in (24) is determined (see Section 4.6) by constants A,, 6,, a = 1,2, that appear in the inequalities

A decrease in the norm of the error is determined by the quantity

The two components A1, Az of the operator A in (22) obey (see Section 4.7) inequalities similar to (25), namely


-

where

6, = 0 ( 1 ) ,

.

305

i,= O(hZ,),

a = 1,2.

(27)

It immediately follows from (24)-(26) that

61 =pAl=B

min b ( x ) T

max b(x)

+a&, a

62=(1-P)7 ,

1

-

min b ( x )

+

max b ( x )

2 2 ,

(28)

+ax2.

(29)

Using (28),(29),we can derive the corresponding expressions for [, = 6,/A,, a = 1,2. As previously, we have [, = O ( h ) , a = 1,2, for T = O ( h ) and [, = 0 ( 1 ) , a = 1,2, for T = 0 ( h 2 ) .The same conclusions are valid for the number of iterations as in the previously considered case of the iterative method with a diagonal grid operator B. For example, an estimate analogous to (20) holds when T = O ( h ) . For the method of alternating directions we are studying, it is useful to note the following fact, which concerns the choice of parameter in the expansion (23). It would be logical to expect that the rate of convergence increases as T + 0 . However, this is true only if 4 = 0 or /3 = 1. For 0 < P < 1 it follows from (28), (29) that min b ( x ) lim (, = 7-0 max b(x)' which implies that for heat conduction problems in which the heat conductivity is not constant the rate of convergence of the iterative method will remain finite as the time step decreases. Therefore, in the expansion (23) we must > z2/LZor B = 1 when &/il <&/i2. select either fl = 0 when

zl/il

6.1.4 ALTERNATING TRIANGULAR METHOD Speaking about iterative methods for solving grid elliptic problems, we should specially discuss an alternating triangular method. The number of iterations for this method is proportional to the square root of the number of nodes in one direction. Let us consider the capabilities of this method in the form it is applied to the solution of the problem on the top level when implicit schemes for the heat equation are implemented. For the sake of simplicity, we take equation ( 7 ) with constant coefficients of the grid operator in (8), i.e.

where A is the grid Laplace operator

306


Under these conditions, the alternating triangular method corresponds to the choice of the operator B in (10) in the form

where

A = A1

+ A2,

A1 = A;

Taking into account (30), we derive

The number of iterations in the alternating triangular method ( l o ) , (31) can be estimated (see Section 4.7) a s

where 6 and A are positive constants that appear in the operator inequalities

To derive these estimates for the operator in (30), we take advantage of the corresponding estimates for the Laplace operator (Section 4.7):

where

s" = 0 ( 1 ) ,

= 0(lhl-2).

(3'3)

Taking into account (30), (32), and (35), we derive the following for the constant 6 in the inequality (34):

Due to (32), (34), and (35), we have


.

307

and, hence, 2%

~ = -T + a k ~ A It follows from (37), (38),and (36) that 17 = & / A= 0 ( l h l - 2 ~ )and , for the number of iterations in the alternating triangular method we therefore derive

Due to the estimate (39),the number of iterations will only be proportional t o h-'I4 for T = O ( h ) .Such a weak dependence on the space grid is a good reason to think about employing alternating triangular methods as well as other fast iterative methods of solving grid equations when implementing implicit schemes for nonstationary problems of mathematical physics. The topic of iterative implementation of implicit schemes is becoming increasingly popular nowadays. 6.1.5 ITERATIVE METHODS WITH ELLIPTIC OPERATOR B

Let us discuss separately the case in which a grid elliptic operator with constant coefficients is used as the operator B in the iterative method ( l o ) , i.e. B=gA+dE, (40) where g > 0 and d 2 0 are constants. Fast direct methods can be used (see Section 4.5) to invert the operator B. For 0 < nl 5 a,(x) 5 nz, x E w , the operator A is spectrally equivalent to the grid Laplace operator, i.e.

<

<

b(x) bz in the difference scheme (1)-(3). Suppose that we have 0 < bl Then, for the operator A defined by ( 8 ) ,taking into account (41),we have

To estimate the rate of convergence of the iterative method ( l o ) , (40) it is necessary to determine the constants y,, a = 1,2, in (12). For instance, we can pose the problem of optimal selection of the parameters g and d in (40). Let us consider the case in which B is selected according to (40) with

d = 0 and we put g = 1, i.e. B = A. On the basis of the estimates (42) and

b~

"

5A

"

< AE

we obtain in (12)

308

It follows that for


E = 3 / 7 2 we have

Thus, there is no way to use the choice B = A for implementation of implicit schemes. A much more fortunate choice might be

i.e. in (40) we put

+

+

bl b2 nl n2 d= 9 = o 7 27. For the case of (43), the inequalities (42) entail the two-sided inequality (12) with 261 2nl (44) I + b2 nl n2 ' % = 2 . 71 =mi.( 2

y,-)+

On the basis of (44) we can infer that the rate of convergence of iterative process (12), (42) is independent of the space and time steps and is solely determined by the difference in heat conductivity and heat capacity. 6.1.6 PROBLEMS

1. Formulate t h e grid elliptic p r o b l e m o n t h e t o p level for t h e case i n which three-level schemes w i t h weights for t h e h e a t e q u a t i o n are used. Solution. For Y,+~ we obtain equation (7) with the operator (see Section 5.3)

In particular, the implementation of symmetric three-level schemes is associated with inverting the grid elliptic equation

For the class of implicit three-level difference schemes we have

Hence, the above arguments about the dependence of the rate of convergence of the iterative process on the time step remain valid.


.

309

2. Give simple e s t i m a t e s of t h e relation between t h e error of a n iterative process of implementing a n implicit s c h e m e a n d t h e approximation error. Solution. Suppose we seek a solution on the new time level iteratively with some absolute error E O , i.e. instead of yn+l we determine Fn+l with E O . Then, instead of equation ( 1 ) we have y,+l =

+

Thus, we will have an additional contribution to the error. To retain the accuracy O(T" + lhI2) of the iterative scheme it is sufficient to demand that E,, = O ( r v + ' ) ,i.e. the solution on the new time level should be determined with a higher accuracy.

6.2 The Method o f A l t e r n a t i n g Directions

6.2.1 LONGITUDINAL-TRANSVERSE DIFFERENCE SCHEMES FOR

THE HEAT EQUATION We consider the fundamental twwdimensional problem of nonstationary heat conduction in an isotropic medium, when the heat transfer is governed by the heat equation C(X)

au

-+Lu = f ( x , t ) ,

at

( x , t )E

Q,

(1)

where

In order to make operator formulations convenient to use, we shall assume homogeneous boundary conditions for equation (1).Suppose, for example, that the boundary and initial conditions have the form

u ( x , ~= ) 0, E r, u(x,O) = uo(x), x E 0.

(3) (4)

After the problem (1)-(4) has been quantized in space, we arrive (see Section 5.3) at the system of equations

310


supplemented with the conditions

The following representation is, for example, used in equation (5) for the operator A: 2

Av =

C A,u, a=1

a = 1,2,

(8)

A,u = - ( with b(x) = c(x) for x E w. Difference schemes in the method of alternating directions for heat equation (I), (2) are based on the representation of the operator A with respect to space variables in the form of the sum of two operators A1 and A2, each of which is one-dimensional. Difference schemes based on the inversion of onedimensional operators belong to the class of economical schemes, because the computational costs per node (three-point Thomas algorithm, band matrices) are independent of the total number of nodes. The classical difference scheme of alternating directions (the PeacemanRackford scheme) for the problem in (5)-(7) consists of two steps. First, given some y,, we determine an auxiliary grid function (which we will denote by ~ , + ~ from j ~ )the equation

+

If we interpret yn+l12 as a solution at the time t,+l/2 = t,, r / 2 , it can be noticed that (9) amounts to finding the solution by a purely implicit scheme with respect to the variable xl (the operator Al) and by an explicit scheme with respect to the variable x2 (the operator Az). It is clear that in this transcription the second step in the method of alternating directions corresponds to the use of the equation

Thus, the second step corresponds to the use of an explicit scheme with respect to the first variable and a purely implicit scheme with respect to the second variable. The implementation of the scheme of alternating directions (9), (10) corresponds to determining yn+,lz and y,+~ from the equations


.

311

The notation (11) indicates that the solution is found by inverting the appropriate one-dimensional grid operator (using the Thomas algorithm) first with respect to one variable (one direction) and then with respect to the other variable (the other direction). For this reason, the scheme of alternating directions (9), (10) is sometimes called the longitudinal-transverse difference scheme for the heat equation. 6.2.2 THE STABILITY OF THE SCHEME OF ALTERNATING DIRECTIONS The stability of the scheme (9), (10) can be analysed using the following useful assertion. Lemma. Let B = B* > 0, A 0 in H. Then for a 2 0.5 the following inequality holds in HD:

>

if D = B-' We have

By virtue of the assumptions made in the lemma, the right-hand side is nonnegative, i.e. the inequality in (12) holds. Assuming D = B-', we derive immediately from equations (11)

Adding (13) to (14) and taking into account (12) for o = 0.5, we obtain

Now we again apply (12) to get

312


On the basis of a grid analogue of the Gronwall lemma, we derive the following estimate of the stability of the scheme of alternating directions (9), (10)with respect to initial data and right-hand side in HD for D = B-':

We specially note that the convergence of the difference scheme (9),(10)does not follow immediately from the estimate (16)because of the presence of the auxiliary grid function y,+,lz. This situation is typical of all the economical schemes we shall consider below and, therefore, deserves special consideration. 6.2.3 THE ACCURACY OF THE SCHEME OF ALTERNATING

DIRECTIONS To analyse the accuracy of the difference scheme of alternating directions (9),(10) we shall write down the corresponding problem for the error z, = y, = u(x,t,), x E w , assuming Z,+~/Z = - iii (the function 'ii will be selected a little bit later). From (9),(10)we immediately derive

For the errors we have

Put in (19),(20)

In this case it follows from (19),(20) that

Moreover, due to (21)we have $;=

%,+I

+ u,

-141 - b(z)U"+1

-fin

T

- AZU"

+ ip,

7 U"+l hz --

2

7

+ 0(T2).

ECONOMICAL DIFFERENCE SCHEMES

Taking into account

we obtain

$2"

+

= $; = 0(r2 lh12).

(22)

Thus, given a special definition of the intermediate solution (see (21)), the difference scheme of alternating directions (9), (10) is second-order accurate in space and time. To analyse the errors, we consider the grid problem (I?), (18). Taking into account (22) and using the estimate (16) for the exact specification of initial conditions, we have

where D = B-'. On the basis of the estimate in (23) we can draw a conclusion that the scheme of alternating directions (9), (10) converges at a rate of 0 ( r 2 + [hi2)in the appropriate norm. 6.8.4 OTHER SCHEMES O F ALTERNATING DIRECTIONS

Speaking about schemes of alternating directions other than that in (9), (lo), we shall mention the Douglas-Rachford difference scheme, which is written in the form

In the first step (24), the heat equation is approximated over the entire time interval, and the second step (25) is only introduced for stability reasons. Therefore, the schemes like that in (24), (25) are being called now stabilizationcorrection schemes. It is easy t o verify that the scheme (24), (25) has an error of O(T + Ih12) and is absolutely stable (because it is an instance of factorized schemes considered in Section 6.3). Difference schemes of alternating directions for problems with alternating coefficients, like the problem in (1)-(4), belong to the class of unconditionally stable. However, construction of such schemes for three-dimensional problems encounters considerable difficulties. In order to explain this, it is sufficient to

314


write the scheme of alternating directions ( 9 ) , (10) as a two-level scheme by eliminating y,+,lz from ( 9 ) and substituting the result into (10). In the case of a three-dimensional heat conduction problem in a homogeneous parallelepiped with first-kind boundary conditions, the corresponding differential-difference problem has the form of the equation

dv dt

-

+ Av = 4 ( x ,t ) ,

x E w,

0
(26)

supplemented with the conditions in ( 6 ) ,(7). In this case we have

When using the method of alternating directions for equations (26), (27), it is crucial that the operators A,, a = 1,2,3, are self-adjoint, nonnegative, and pairwise-commutative, i.e.

Under these conditions, the following scheme is the simplest generalization of the stabilization-correction scheme:

We could also mention analogues of the scheme of alternating directions ( 9 ) , (10) for the heat conduction problems under conditions (26)-(28).

6.2.5 PROBLEMS 1. Write down the scheme of alternating directions for heat equation (I), (2) with the nonhomogeneous boundary condition

u ( x ,t ) = g ( x ,t ) , x E r.

(29)

Solution. At the inner grid nodes we use the difference scheme ( 9 ) , ( l o ) , and the problem is thus to approximate the boundary condition in (29). In order to retain the accuracy, we will meet the boundary conditions exactly so


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315

as t o have homogeneous boundary conditions for the solution error. Taking into account (21), we put at the boundary nodes

In these conditions, the solution error satisfies the above estimates, and the scheme (9), (10) with boundary conditions (30) will converge and will be second-order accurate in time and space. 2. P r o v e t h e stability of t h e scheme of a l t e r n a t i n g directions for t h e h e a t conduction p r o b l e m i n a homogeneous m e d i u m , using t h e g e n e r a l t h e o r y of stability of difference schemes. Solution. In a homogeneous medium, the scheme of alternating directions has the form (see (26))

where

2

and A, = A:, a = 1 , 2 , and AlAz = A d , Eliminating yn+1/2 we obtain

Therefore, the scheme of alternating directions (31), (32) is written in the canonical form of a two-level difference scheme, given

Now we only have to verify the stability condition. Taking into account the commutativity and positiveness of the operators A,, a = 1,2, we have

Thus, the scheme of alternating directions (31), (32) is stable in Ha

316


6.3 Factorized Difference Schemes for the Heat Equation 6.3.1 FACTORIZED SCHEMES

Let us consider whether it is possible to construct the class of economical difference schemes by representing the grid operator on the top level as the product of economical (onodimensional) operators. The schemes thus constructed are called factorized difference schemes. For the problem (1)-(4) in Section 6.2 we construct a two-level difference scheme which has the canonical form

The factorized scheme is obtained when we select the operator B in the form

where B,, a = 1,2, are economical operators. Let us choose the operators B,, a = 1,2, in the form

+

B1 = b(x)E U T A I , Bz = b-I (x)(b(x)E a ~ A 2 ) .

+

(4)

Given such a representation of B, each of the operators B,, a = 1,2, corresponds to the use of the ordinary scheme with weights for the onedimensional heat conduction operator. The computational implementation of the factorized scheme (I), (3) is associated with solving the problems

Some other possibilities of constructing factorized schemes, distinct from (3), (4), are discussed below. In some cases (see, for example, problem 2 in Section 6.2), schemes of alternating directions are written as a factorized scheme (1)-(4).

6.3.2 STABILITY O F FACTORIZED SCHEMES

A factorized scheme has the canonical form of two-level scheme (1) with the operators A and B defined according t o (2) and (3), (4). A straightforward


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317

analysis of the stability in H A on the basis of verifying the necessary and sufficient condition

B>?A 2

(5)

works for the scheme (1)-(4) only if the operators A,, cu = 1,2, commutate and b(x) = const. For this reason we can try to prove the stability of the scheme (1)-(4) in more complex norms. A guideline can be the proof of the stability of the scheme of alternating directions conducted in Section 6.2. Let us consider only stability with respect t o initial data and put 9, = 0 in (1). Taking into account (3), we write equation (I) in the form

Now we add and subtract from the right-hand side of (6) the term (20)-'B2y, to get 2rJ - 1 1 Bzyn+t = -Bzy, - (Bz - 2u~B;~A)y,. (7) 2u 20 Taking into account (2), (4), we derive

+

Now we derive an estimate of the stability of the factorized scheme (1)-(4) in the following form: IIBzY~+IIID 5 IIBZY~~ID, (9) where D = b(x)E. To this end we denote v, = b 1 / 2 ( x ) ~ z yand , left-multiply = b'lz(x)E to get (7) by

where, taking into account operator Q:

(a),

we have the following representation for the

Hence we represent Q as

Q = sls;,

318


where

We can give a more convenient representation for the operators S , from (12), namely S , = ( E CT~[,)-'(E (13) i[ = b - 1 2 ( ) ~ b - 1 2 ( ) a, = l , 2 .

+

+ CTL),

-

Because of ( 2 ) ,the operators A, possess analogous properties, i.e.

Now we shall formulate the following assertion, known as the Kellog lemma. Lemma. Let an operator C 2 0 in H . Then the following inequality holds whatever 2 0 : - P C ) ( E ac)-'115 1. (15)

I~(E

+

+

Let S = (E - p C ) ( E +PC)-' and G = ( E PC). Then (15) is equivalent t o the operator inequality J = E - S'S 2 0. We have

and the inequality in (15) therefore takes place for nonnegative operators C . Due to the lemma, from (13), (14) we derive IISJ 5 1, a = 1,2. Because of ( l l ) ,the norm of the operator Q is also less than unity, and it follows from (10) that I I v , , + ~ ~ ~ 5 IIwnll, which holds for all c 0.5. Thus we arrive at the desired estimate (9) for the difference solution and we have established the unconditional stability of the factorized scheme in (1)-(4).

>

6.3.3 THE PRINCIPLE OF REGULARIZATION FOR CONSTRUCTING

FACTORIZED SCHEMES In order to construct stable factorized schemes we shall use the principle of regularization discussed in Section 5.8. The widest opportunities are opened by this approach when we are constructing unconditionally stable factorized schemes for parabolic problems, when b(x) = const (for definiteness, b(x) = 1). When considering heat conduction problems, such a situation arises in the modelling of processes in homogeneous bodies. Noncommutativity of the appropriate operators is generated by boundary conditions or by the fact that the computational domain 0 is not rectangular. Therefore, the whole


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319

class of problems in which the grid function b(x) is constant is important and essential. In a more general case we employ a simple transformation of the equation dw b(z) .;ii. Av = $(x, t),

+

x E w,

0

to the form in which the factor that multiplies the time derivative equals unity, while preserving the operator's properties of being self-adjoint and nonnegative with respect to space variables. To this end it is sufficient t o ( ~ ,x E w . After this the introduce a new grid function w(x, t) = b 1 / 2 ( ~ ) w t), equation takes on the form

where

A' = b-'/2~b-'iz,

, $'(x, t) = b - ' / 2 ( ~ ) $ ( ~t).

Because of (2), the operator A' possesses the following properties:

We consider a differential-difference problem for equation (16) with the conditions

In order to solve the problem (16)-(18) approximately we shall proceed from the regularized difference scheme (I), where

Let the scheme in ( I ) , (19) be stable, i.e. let the condition (5) be satisfied. On the basis of this scheme we construct a economical scheme (which is also stable) with the factorized operator

Note that we restrict consideration to two terms only for simplicity, and a more general case in which the operator R is decomposed into a greater number of operators can be considered in exactly the same way. Let R be the sum of economical operators

320


Taking into account (19), (21) we construct the factorized operator (20) with

Let us show that if the original scheme ( I ) , (19) is stable and the operators 0, 0 = 1,2) and R p , 0 = 1,2, are self-adjoint, nonnegative (Rp = R; commutative (R1R2= R z R l ) , then the factorized scheme (2), (22) is also stable. Then, the principle of regularization would allow us to construct economical stable difference schemes given a stable inefficient difference scheme. In order to prove the above it is sufficient to verify the inequality (5). The operators R U , 0 = 1,2, are commutative and, hence, we have RlRz = RzRl 2 0 and from (20)-(22) we obtain

>

Then the factorized scheme is stable because (5) is fulfilled. Taking into account strict limitations on the operators Rp, i9 = 1,2, it is convenient to consider the simplest operators R when selecting R p . Let us take advantage of the fact that the operator A is spectrally equivalent to the grid Laplace operator A, i.e. the following estimate is valid:

It is logical to take the operator R = b-l/z.&-l/z as a regularizer (see Section 5.8). Then the operator A' in the problem (16)-(18) and this regularizer will obey the two-sided inequality of spectral equivalence

The analysis of the regularized scheme ( I ) , (19) shows that for stability it is sufficient, for example, that the regularization parameter o satisfies the condition 7 a > nz-. (24) -2 In the following we shall restrict ourselves to the simplest case in which " b(x) = 1. We use the following representation for the operator A:

0

The operators R p = AL1,fl = l r 2 , possess the necessary properties of selfadjointness, nonnegativeness, and commutativity. Therefore, the factorized scheme Yn+l - Yn ( E + d h ) ( E + URZ) +Ay,=ip,, n = 0 , 1 , ... will be absolutely stable provided that the inequality (24) holds.


.

321

Note that there exists another possibility of decomposing (21) in the case considered. Taking into account (23), the stability conditions will be satisfied for a self-adjoint nonnegative opemtor R if in (21) we have

The selection (21), (25) corresponds to the decomposition of the operator R into triangular operators (see Section 4.7). When using the Laplace operator for R, for the operators Rp, 0 = 1 , 2 , we

The implementation of the factorized scheme (24), (25) corresponds t o the use of the so-called 'running-calculation' formulx. It is interesting t o note that such a factorized scheme can also be constructed for a general problem with b ( x ) # const (see problem 1). 6.3.4 THREE-LEVEL FACTORIZED SCHEMES Let us demonstrate what we can achieve using the principle of regularization when constructing threolevel economical factorized schemes for the class of problems (16)-(18). First, we shall give a fairly general consideration t o the original family of difference schemes

which correspond t o putting B = E in the canonical notation of a threelevel scheme. In order to understand the idea underlying the construction of factorized difference schemes on the basis of the principle of regularization, we write the original scheme (27) in the form

A factorized scheme is constructed on the basis of (28) similarly to the case of two-level difference schemes, Analogously to (21), suppose that we have the decomposition R=Rl+'R.z. (29)

322


We write the factorized scheme in the form

where taking into account (28), (29) we put that

We again assume that the operators Rp, P = 1,2, are self-adjoint, nonnegative (Rp = R i 0, ,!= l 1 , 2 ) and commutative (RlR2 = R z R l ) . We take it for granted that the original difference scheme satisfies the stability conditions for the scheme (27) with A* > 0, i.e. (see Section 5.4) the following inequality holds: R > $A. (32)

>

Let us show that under these conditions the factorized scheme (30) is stable. To this end we write the scheme in the canonical form

Straightforward manipulations yield

Hence, the conditions sufficient for the stability of the scheme in (33) are met subject to the assumption in (32), because

As an illustration of this result, let us construct a factorized scheme on the basis of the following regularized (see Section 5.8) scheme for the problem in which b(x) E 1:

Let us select a A as a regularizer of R. The regularized scheme (35) will be stable provided a > 4 4 . Under the same conditions we will have the stability of the factorized scheme (33), (34), where

Three-dimensional problems are considered similarly.


.

323

6.3.5 PROBLEMS 1.Analyse t h e stability of t h e factorized s c h e m e ( I ) , (3), (4) u n d e r t h e condition A=A, A = + A;=Az. (36)

Solution. Taking into account. (3), (4), (36), we have the following for the operator B:

Because the operators AD, P = 1 , 2 (see (36)) are adjoint, it follows from (37) that B 2 ~uTA, (38)

>

and therefore the factorized scheme (1) will be stable provided that u 0.25. Note that we derived the estimate (38) earlier, in Section 4.7, when considering the alternating-triangular iterative method. 2. C o n s t r u c t a factorized scheme o n t h e basis of t h e following implicit three-level scheme, which i s second-order a c c u r a t e i n t i m e a n d s p a c e ( B a k e r a n d Oliphant):

Solution. The scheme (39) has the form (28) with the operator R = (1/3)A. The factorized scheme is constructed in accordance with (30) and (31), and we have 0

Rp=Ap,

A ~ Y = - Y E ~ Z , ,p = 1,2.

The factorized scheme proper takes on the form

0

Taking into account that the operators Ap, 0 = 1 , 2 , are positive, selfadjoint and commutative, we can easily verify the condition of stability for the scheme (4).

324


6.4 Additive Difference Schemes 6.4.1 THE ADDITIVE SPLITTING OF THE HEAT CONDUCTION OPERATOR It should be noted that the problem of constructing economical difference schemes for general three-dimensional heat conduction problems cannot be solved using factorized difference schemes or the schemes of alternating directions. It is possible to prove the stability of these schemes for variable coefficients only in the two-dimensional case. Therefore, it is necessary to consider economical difference schemes for multidimensional problems from a more general standpoint, i.e. it is necessary to reconsider the very notion of the approximation of difference schemes. We shall consider the following general multidimensional heat equation in an isotropic medium:

where

Let us supplement ( I ) , (2) with the conditions

The quantization of (1)-(4) in space brings us to the differential-difference problem

Let A,, a = 1 , 2 , . . . ,m, denote the difference heat conduction operator in the direction x, and let us define this operator as follows:

Then the operator A can be represented as the sum of one-dimensional operators A,, a = 1 , 2 , . . . ,m, namely

A=Al+Az+

. . . +A,.

(9)


.

325

Difference schemes in which the transition from one time level to another is associated with solving a sequence of problems for the operators A,, a = 1 , 2 , . . . ,m, will be called additive difference schemes. The above (Section 6.3) examples of economical difference schemes belong to the class of additive difference schemes. In ( 9 ) we split the operator A into one-dimensional grid operators A,, a = 1 , 2 , . . . ,m . Other operators can also be taken as primitive operators. For example, we have considered schemes with splitting into triangular operators. When considering more general nonstabionary problems, the operators A,, a = 1,2,. . . ,m, can have other meanings. In the general consideration of additive difference schemes, we can forget about what the separate operators A,, a = 1 , 2 , . . . , m, in the expansion ( 9 ) signify specifically.

6.4.2 INTERMEDIATE PROBLEMS

Let us discuss the main approaches to the construction of additive difference schemes. To this end, it is necessary to formulate auxiliary problems with the operators A,, a = 1,2,. . . ,m, in the additive splitting ( 9 ) . Solving these auxiliary problems we shall derive an approximate solution to the original problem (5)-(7).Two major approaches are used to construct additive schemes, namely, additive schemes with fractional time steps and additive schemes with integer steps. Suppose we introduced a time grid w, = {tit = t , = nr, n = 0 , 1 , . . . ,N,NT = T} with a step T > 0. Now we divide every half-open = t,+ar/m. interval (t,, t,+l] into m parts by introducing the points t,+,,, Taking into account the additive representation ( 9 ) of the operator A we write equation (5) in the form

Here mU(x,t ) are arbitrary functions possessing the same smoothness as 4 ( x ,t ) and satisfying the condition

In accordance with ( l o ) ,we shall consecutively solve the following intermediate problems. Auxiliary functions u a ( x ,t ) are defined by the equations

326


b ( x ) du' m dt

--

+ Alu'

- d l ( x , t ) = 0,

t,

< t 5 t,+ll,,

The initial conditions for (12) have the form

%

,

The function v m ( x ,t,) will be considered to be an approximate solution of the problem (5)-(7) at the times t = t,. The intermediate problems (12), (13) are solved on separate parts of the half-open interval (t,, t,+']. For this reason, difference schemes constructed on the basis of (12),(13) can be called additive difle'erence schemes with fmctional steps. Another method of constructing additive schemes is based on the solution of auxiliary problems on the entire half-open time interval (t,, t,+l] (additive diffeerence schemes with integer steps). In order to determine the functions v a ( x ,t ) , we use the equations

du' b ( z )dt

+

- r n 1 ( ~ , t=)

equipped with the initial conditions

o,

t , < t 5 tn+llm,

ECONOMICAL DIFFERENCE SCHEMES.. .

327

When implementing (14), (15), we first solve the equation for ul(x, t) ( a = 1 in (14)), given the initial condition ul(x,tn) = vm(x,tn), and determine ul(x, t,+l), which is later used as the initial condition when determining v2(x, t), and so on. We take vm(x, tn+l) as an approximate solution of the problem (5)-(7) at the time t,+i. It should be noted that the two approaches to constructing additive difference schemes (namely, (12), (13) and (14), (15)) are closely connected. If the coefficients of the grid equation (5) do not depend on time and the right-hand sides $"(z,t) = 0,then the problems (12), (13) and (14), (15) are actually identical. It is this situation that occurs in our case of a heat conductio~~ problem in nonhomogeneous medium (equations ( I ) , (2)), when we are modelling heat conduction without inner sources of heat (f (x, t ) = 0 in (1)). In more general cases it is preferable to employ the second approach, when auxiliary problems are solved over integer time steps. Now we should answer the question of what is the error within which the system of equations (14), (15) allows us to find the approximate solution to the problem in (5)-(7). In order to answer this question we should extend the notion of approximation. 6.4.3 THE NOTION OF SUMMARIZED APPROXIMATION In the above we mentioned that the approximate solution derived from the solution of the group of intermediate problems (14), (15) is close to the solution of the problem (5)-(7). Every separate intermediate problem does not approximate the original problem (does not produce an approximate solution), and only the consecutive solution of all the intermediate problems, when every next problem is being linked to the previous one by means of the initial conditions, makes it possible to obtain an approximate solution. Therefore, in this case they say that the problem (14), (15) approximates (5)-(7) in the total (the summarized approximation). Let us consider the error tn < t 5 tn+l, zl(x, t ) = vl(x, Y )- u(x, t), a = Z,3,. . . , m . t, < t 5 t,+l, zu(x, t ) = vu(x,y) - u(x, t,+l), We formulate the corresponding problems for the error. Substituting ta(x, t), a = 1,2,. . . ,m, into (14), (15) we obtain the equations dzu b ( z ) - + A,zm = ,x(@ ' t ) , x E w , t, < t 5 t n + ~ , dt (16) which are supplemented by the conditions zl(x,t,) = zm(x,tn), n = l , 2 , . . . , a = 2,3,. . . , m , n = 0,1,. . . . zD(x,t n ) = Z ~ - ~ (tn+l)r X, zl(x,O) = 0,

(17)

328


The error of approximation for each intermediate problem (16),(17) is defined by the expressions

$'(x, t ) = -b(x)

dv

- Alv + & ( x , t ) , dt

(18)

It follows from (18) that for every intermediate problem $=(x, t ) = 0 ( 1 ) , i.e. the problems (16), (17) do not approximate (5)-(7). Let us now consider $ ( x , t ) = Q 1 ( x ,t ) + Q 2 ( x ,t ) + . . . + $"'(x, t ) . It follows from (18) that

Taking into account

u ( x ,t,+l) = v ( x ,t )

+O(T),

a = 2,3,. . . ,m,

t , < t . . . t,+l,

we obtain

where 6,' have

is Kronecker's delta. For the solutions of the problem (5)-(7) we

x "%

,=I

o

$"(x, t ) =

x "'

o=l

x "'

P ( x ,t ) -

dv A,v - b ( x )- = 0 , dt o=l

and therefore for the summarized approximation error we get

We will say that (14), (15) is a first-order summarized approximation to (5)-(7). In order to estimate the accuracy, we shall take that the following conditions are satisfied: IIA.APvII 5 M, a J = l , 2 , . . . ,m. An additional analysis shows that given such additional conditions the estimate of the error is lIvm(z,t,) - v ( x , t,)ll = O ( T ) ,i.e. by solving intermediate problems (14), (15) we can derive an approximate solution of the original differential-difference problem (5)-(7), the solution being first-order accurate in time.


.

329

It is interesting to note that if 4(x, t) = 0, 4*(x, t) = 0, a = 1,2,. . . ,m, b(x) = const and the operators A,, a = 1,2,. . . , m are pairwise commutative (A,Ap = A?A,), then (14), (15) produces the exact solution to the problem (5)-(7) (see problem I ) , that is um(x, t,) = u(x, t,). In order to improve the accuracy of the approximate solution of the problem (5)-(7) on the basis of the solution of intermediate problems, we can use the following idea of symmetrization (bicyclic organization). The problem (14), (15) can be schematically represented by the chain A1 + Az + ...A,, which gives a visual image of the sequence of intermediate problems solved consecutively. The symmetrized chain has the form 0.5111 -+ 0.5Az + . . . 0.5A,, 0.511, + 0.5Am+1 + . . . -+ 0.5Al. -+

This chain corresponds to our replacing (9) with the following additive representation of the operator A:

In this case we have the estimate llum(x, t ) - x(x, t,)ll = 0 ( r 2 ) under some additional restrictions on smoothness.

6.4.4 ADDITIVE DIFFERENCE SCHEMES Additive difference schemes can be obtained by approximating the intermediate problems (14), (15) in time. Without giving general formulations, we shall construct the simplest two-level difference schemes. Let y,+,/, be the approximate solution uu(x,t) at the time t = t,+,. Let us employ a twolevel scheme with weights for every intermediate problem. Then, taking into account the initial conditions (15) and using the additive difference scheme with weights u,, a = 1 , 2 , . . . ,m, we obtain

The two-level additive difference scheme can be written in the following canonical form:

330


The above additive scheme (21) has the canonical form (22) for

The analysis of convergence of additive schemes is essentially peculiar in the sense that the summarized approximation takes place for additive difference schemes. Therefore, it is necessary to derive such special estimates of the stability of additive difference schemes with respect to the right-hand side that would entail the convergence of a difference scheme, given that the summarized approximation condition is satisfied. Now we give some general results that concern this topic. 6.4.5 A PRIOR1 ESTIMATES FOR ADDITIVE DIFFERENCE

SCHEMES Let us consider the additive difference scheme

Let us represent the approximation error in the form

where $; satisfies the conditions

We shall take that the additive scheme (23) possesses the property of summarized approximation, i.e.

in some norm 11 . 112h. In order to derive the desired a priori estimate for the additive scheme, which would take into account (24), (25), we assume that for (23) the usual estimate of stability with respect to the right-hand side holds, i.e.

.


331

Taking into account (25) it follows from (28) that

Therefore, for all n = 1 , 2 , . . . we have q, = 0, 2 , = 0, and

Hence, we obtain the following problem for On+,l,:

where the right-hand side has the form

Using the estimate in (27) for this problem, we immediately derive

Taking into account (30), we rewrite this estimate in the final form as

+

T

max

O
CC

1

m

Amp

a=1 & 1, I

C

1=p+l

B-'$;: '

lI1h

332


This estimate for the difference problem (23) indicates that under the conditions of summarized approximation (24)-(26) the additive difference scheme (22) converges, provided that some additional conditions are satisfied,

Let us recall that the estimate (31) of the error of the additive difference scheme has been derived under assumptions about the stability of this scheme in the ordinary sense (the inequality in (27)). In such a context, we can say that the stability and the summarized approximation of an additive difference scheme imply its convergence. Other a priori estimates for two-level additive difference schemes (22) are currently available. These estimates make it possible, for example, to weaken additional conditions like that in (32). 6.4.6 PROBLEMS

1. P r o v e t h a t t h e intermediate problems (14), (15) yield t h e e x a c t solution to t h e p r o b l e m

Solution. In this case we have

Now let A:, A;, wf( z 1 ) , w ~ ( x 2 )i, = 1 , 2 , . . . ,Nl

- 1. j

= 1 , 2 , . . . , NZ - 1 be

the eigenvalues and the eigenfunctions of the operator A,, a = 1,2 (see Section 4.5). In accordance with (14), (15), we first solve the problem

The solution has the form

ECONOMICAL DIFFERENCE SCHEMES

Now we consider the problem

The solution is

x exp ( - (At

+ X ~ ) T ) W (~x 1 ) w ~ ( x 2 )

This solution coincides with the solution to the original problem (33), (34) at the time t = T . Thus, the solution of the auxiliary problems yields the exact solution at the times t = t,. 2. W r i t e t h e scheme of alternating directions (see Section 6.2) as a two-level additive difference scheme. Solution. We consider the scheme

Using the expansion

we write the additive scheme (see (21))

Straightforward calculations yield

$n=$~+$~+~~+$~=~(~2+lh12),

334


i.e. the scheme of alternating directions represented a s a factorized scheme is second-order accurate (in the sense of summarized approximation) in time and space.

6.5 Locally One-dimensional Difference Schemes

6.5.1 LOCALLY ONE-DIMENSIONAL SCHEMES FOR THE HEAT

EQUATION Here we shall analyse the convergence of additive difference schemes constructed in Section 6.4. In order to approximately solve the problem (1)(4) from Section 6.4, we use the additive difference scheme

Here we have adopted that the operator A, defines the heat conduction in the direction x,, i.e.

with A=A1+A2+

...+A,.

(3)

Each of the operators A,, a = 1,2,. . . ,m in the additive representation (3) is one-dimensional and, therefore, additive difference schemes (1)-(3) are called locally one-dimensional difference schemes. It has been shown in Section 6.4 that we can analyse the convergence of additive difference schemes by deriving appropriate estimates of stability and by studying the summarized approximation. Let us make a couple of remarks about derivation of a priori estimates of stability of additive differenceschemes using the example of locally one-dimensional scheme (1)-(3) and restricting consideration, for the sake of simplicity, to the stability with respect to initial data in some Hilbert space H D , generated by an operator D = D* > 0. We need to derive the a priori estimate of pstability

with some constant p > 0 for the difference problem (1) with vp, = 0, a = 1 , 2 , . . . , m. There are two options we can use to achieve this. The first


.

335

is to eliminate the intermediate grid functions y,+,/,, a = 1 , 2 , . . . ,m - 1, from the difference equation (1). It is clear that for problems with general, noncommutative operators A,, a = 1 , 2 , . . . ,m, using this option will be technically difficult and nonconstructive. The other approach to the analysis of stability of additive difference schemes is based on deriving a priori estimates for all intermediate grid problems. Suppose, for example, that we are able to derive for every a = 1 , 2 , . . . ,m a priori stability estimates of the form

with some p,, a = 1 , 2 , .. . , m . When deriving the estimates (5), the additive scheme (1) is considered as an ordinary two-level difference scheme for every fixed a = 1 , 2 , .. . ,m. Then we can derive from (5) the desired estimate (4) with m P=

npm.

(6)

0=l

For instance, using the estimates (4), (6), we can analyse the asymptotic stability of locally one-dimensional schemes for the heat equation. Let us stress one important point. For every a = 1 , 2 , .. . ,m the stability estimates (5) have to be derived in the same norm if we do not consider a more general situation with the consecutive embedding of separate spaces. We can achieve this for the scheme (1) if we consider stability in norms that are not associated with the operators A,, a = 1 , 2 , .. . , m . 6.5.2 THE CONVERGENCE OF A LOCALLY ONE-DIMENSIONAL

SCHEME In order to investigate the stability of the additive difference scheme ( I ) , we shall use the approach in which we derive a priori estimates like that in (5) for every fixed a = 1 , 2 , .. . ,m. In this case we have an ordinary scheme with weights whose stability has been studied in some spaces H D in Section 5.5. However, the estimates we provided earlier have been derived in norms that are associated with the operators A. For this reason, we shall give some additional estimates for an ordinary scheme with weights. Let us consider a scheme with weights that is written in the canonical form

with the operators

Now we derive an estimate of stability of the scheme (7), (8) in H.

336


The difference scheme ( 7 ) is written in the form

where the transition operator is

Let us find conditions under which the norm of the operator S does not exceed unity (the stability of scheme with p = 1). To this end it is necessary t o verify 0 which takes the following form if we allow the inequality J = S'S - E for (10):

<

In order to derive an equivalent inequality, we left- and right-multiply the original inequality by ( E + U T A )to arrive at the inequality

It follows from (11) that llSll Section 5.5)

<

1 subject to the usual condition (see

a > -1 - - 1 2 AT' where A is the constant in the estimate A 5 AE.

(12)

(13)

Given (12), we derive from ( 9 ) the estimate

for the scheme with weights ( 7 ) ,(8). Now let us consider a wider class of schemes with weights, when in equation ( 7 ) we have

In order to derive the corresponding estimate, we derive a new function u, = D 1 / 2 y ,and multiply equation ( 7 ) , (15) by 0 - ' I 2 to obtain


.

337

The differencescheme in (16),(17)belongs to the class of schemes with weights ( 7 ) , (a), which we have already considered. The stability estimate in this case looks like

Il~n+lllD5

I I Y ~ ~ ~+D TlIB-lPnll~.

(18)

Thus, the problem ( 7 ) , (15) is stable in H D subject to the condition (12) in which (13) is replaced by the estimate

A

< AD.

(19)

Now we are able to derive an estimate of stability of the locally onedimensional scheme ( 1 ) . Taking into account

we have a problem like that in ( 7 ) ,(15) for every fixed a = 1,2,. . . , m and on the basis of the estimate (18) we conclude

A simpler estimate can be used for the right-hand side, namely

. into account (20), We have IIB;'w,uII~ = (DB;',B;'v) = ( G u , ~ )Taking we shall prove the inequality G = B;'DB;' 5 D-'. To this end, we leftand right-multiply it by B, to get

Under our assumptions the last inequality evidently holds The estimate (21) holds (see (12), ( 1 9 ) )for

where A, is the constant in the inequality A, 5 A,D, a = 1,2,. . . ,m. We shall assume these conditions to be satisfied.

338


Summing (21) for all a = 1,2,. . . , m, we arrive at the following estimate of stability of the additive scheme (1):

The estimate (22) is the sought-for estimate of stability of the locally onedimensional scheme (1). In order to study the convergence of the scheme (1) we write the following problem for the error z+,/, = yn,+, - u(x, t n + ~ ) ,x E w :

Here u(x, t) is the exact solution of the problem

where

For the approximation error we obtain the expressions

Hence, the approximation error can be represented in the form


.

where for $: we have

for

m

C fm(x,t)= f(">t). o=1

For the solutions of the problem (24)-(26) we have

For the second term in ( 2 7 ) we derive

i.e. the locally one-dimensional scheme ( 1 ) possesses the property of summarized approximation. Now we take advantage of the estimate ( 3 1 ) from Section 6.4. On the basis of the a priori stability estimate ( 2 2 ) we derive

where

For problems with fairly smooth solutions, the estimates ( 3 0 ) , ( 3 1 ) , and the summarized approximation conditions (27)-(29) entail

i.e. the locally onedimensional scheme ( 1 ) converges and is first-order accurate in time and second-order accurate in space in H D , D = b ( x ) E .

340


6.5.3 CONVERGENCE IN A HOMOGENEOUS NORM

Let us prove the convergence of the locally one-dimensional scheme (1) in a homogeneous norm. In this case the estimate of stability is also obtained by deriving estimates for fixed a = 1 , 2 , . . . ,m and using the appropriate results from Section 6.4. Therefore, the technique of analysing the accuracy remains the same as when deriving the estimate (32). The unconditional uniform convergence has been shown (see Section 5.3) for a purely implicit scheme. Therefore, when investigating the uniform convergence of locally one-dimensional schemes we limit our consideration to the scheme (1) with a, = 1, a = 1,2,. . . , m , i.e. we consider the scheme

The estimate of stability for the scheme (33) is derived on the basis of the maximum principle in the usual way (see Section 5.3). For every fixed a = 1,2, .. . , m we obtain

Summing with respect to for one time level:

ct

from 1 to m we arrive at the following estimate

The desired estimate of stability of the locally one-dimensional scheme (1) with respect to the right-hand side and initial conditions follows from the last inequality, namely

The analysis of convergence can be conducted using the notion of summarized approximation and the stability estimates (34) by the above procedure. This analysis yields the following estimate of the error of the locally o n e dimensional scheme (33): ( l y n + ~ ( s-) u ( x ,t,+l)Ilc(,) _< M(T + lhI2).


.

341

6.5.4 ADDITIVELY AVERAGED DIFFERENCE SCHEMES The additive difference schemes we have previously considered are based on a strict timing of the calculations, that is to find y,+,/, we need to know Y,+(,-~)/,. If we mean that additive difference schemes will be used for constructing parallel algorithms, it seems more attractive to have locally one-dimensional schemes that permit asynchronous calculations. Let us give some brief remarks concerning parallel implementation. Without giving general formulations, we shall construct an asynchronous locally one-dimensional scheme for the model heat conduction problem (1)-(4) by the difference from Section 6.4. We define auxiliary grid functions h+olm equations

These functions are then used to find the solution on a new time level:

On every time level we solve m one-dimensional problems and for this reason these schemes are ascribed t o the class of locally one-dimensional additive a = 1 , 2 , . . . , m , can be difference schemes. Note that the solutions determined independently (asynchronous calculations). Then we average these solutions (see (36)). Because of this procedure, the schemes like (35), (36) are called additiwely averaged difference schemes. It is easy t o derive (see problem 2) the estimate (22) of stability of the additively averaged difference scheme (35), (36) with respect to the righthand side and initial data. Now let us consider the appropriate problem for the error:

c,,+,lm,

-zn+a/m = Yn+o/m 2,

We have from (35) and (36)

- e(x. tn+,),

= y, - u(x, t,),

x Ew

342


For the approximation error we can write

Hence, the additively averaged difference scheme has summarized approximation, i.e. conditions (27)-(29) are satisfied. Having established the results for stability and summarized approximation, we can analyse the convergence of the scheme ( 3 5 ) , ( 3 6 ) in the usual way (see Section 6.4). We put

Here ij,+,/, is determined from the conditions

Summing these equations, we obtain

Because of ( 2 8 ) and ( 3 9 ) , we have qntl = qn = . . . = qo = 0 . It follows from ( 4 0 ) that ii,+,/, = m ~ b - ' ( x ) $ , " ,a = 1 , 2 , . . . , m - 1 . Now let us formulate the problem for O,+,l,. Taking into account (37)-(39), we obtain

-

-

For $: we have

* $: = $; - ~ m u ~ ~ ~ b - l ( z ) &

(43) Using the estimate ( 2 2 ) for the problem in (41)-(43) and taking into account that z, = B,, we obtain


.

343

The estimate (44) ensures the convergence of the additively averaged scheme (35), (36) with the first order in time and the second order in space for a fairly smooth solution of the problem (1)-(4) from Section 6.4. 6.5.5 PROBLEMS 1. For t h e m o d e l p r o b l e m w e have considered w r i t e d o w n a locally one-dimensional difference s c h e m e which is second-order a c c u r a t e i n time. Solution. According to Section 6.4, we use the bicyclic organization of computations, which corresponds to a symmetric splitting of the following form: 2m

In order to derive a locally one-dimensional difference scheme, every intermediate problem is approximated by a symmetric scheme. This leads to the difference scheme

The right-hand side is assumed to satisfy the usual relation

2. Derive t h e following e s t i m a t e of stability

of t h e additively averaged difference scheme (35), (36).

344


Solution. For every fixed a = 1,2,. . . ,m and under the conditions previously imposed on the weights a, we have (see (21))

It immediately follows from (36) that

Taking into account (46), this inequality yields the stability estimate (45), which coincides with the estimate in (22) for the usual locally one-dimensional scheme (1).


6.6.1 GENERAL REMARKS 6.1 Usual schemes with weights for multidimensional equations have been thoroughly studied theoretically and are widely used in practice. The peculiarities of implementing implicit schemes that we have considered here have been dealt with in many works. Our analysis is based on the general theory of iterative methods (111.The modification of the method of alternating directions described in this chapter has been mentioned in [13]. 6.2 Methods of alternating directions are described in every textbook on difference methods [2, 4-8, 10, 12, 14, 151. They originate from the research of Peaceman, Rachford, and Douglas (1955). It should be noted that the unconditional stability of these methods in problems with variable coefficients can only be established for the two-dimensional case. For three-dimensional problems it is necessary that separate operators be commutative. 6.3 Construction of factorized difference schemes has been considered in many works (see [l, 3, 8, 151 and the bibliography therein). We followed [6, 71 when discussing the principle of regularization as a general approach to the construction of factorized schemes. 6.4 The notion of summarized approximation and the general approach to the construction of additive difference schemes for evolutionary problems with fractional steps were suggested by A. A. Samarskii in 1962. The additive splitting with integer steps was also suggested by A. A. Samarskii (1965). The theory was then developed in [3, 6, 7, 9, 151.


.

345

6.5 Here we elaborate t h e general results in t h e theory of schemes of summarized approximation for approximate solution of a multidimensional heat conduction problem in a parallelepiped. We used [6, 7, 151 when preparing this section. T h e uniform convergence of a locally one-dimensional scheme for parabolic equations in irregular domains has been studied by A. A. Samarskii (1963). D. V. Gordeziani has been considering additively averaged schemes since 1965.

6.6.2 LITERATURE 1.D'yakonov E. G. (1972) Difference Methods for Solving Boundary Value Pmblems.

Issue 2: Nonstotionory Problems [in Russian]. Moscow State University, Moscow. 2. Marchuk G. I. (1975) Methods of Numerical Mathematics Springer-Verlag, New York. 3. Marchuk G. I. (1988) Splitting Methods [in Russian]. Nauka, Moscow. 4. Richtmyer R. (1957) Diffwenee Methods for Initial Value Problems. Interscience, New York. 5. Richtmver R. & Morton K. 119721 Methods for Initial Value Problems. ~ , Difference Interscience, New York. 6. Samarskii A. A. (1971) Introduction to the Theory of Difference Schemes [in Russian]. Nauka, Moscow. 7. Samarskii A. A. (1983) Theow. o.f Difference Schemes lin Russianl. Nauka. Moscow. 8. Samarskii A. A. (1987) Introduction to Numerical Methods [in Russian]. Nauka, Moscow. 9. Samarskii A. A. & Gulin A. V. (1973) Stability of Difference Schemes [in Russian]. Nauka, Moscow. 10. Samarskii A. A. & Gulin A. V. (1989) Numerical Methods [in Russian]. Nauka, Moscow. 11. Samarskii A. A. & Nikolaev E. S. (1989) Numerical Methods for Grid Equations. Birkhauser-Verlag, Basel. 12. Saul'ev V. K. (1960) Integration of Equations of Parabolic Type by the Grid Method [in Russian]. Fizrnatgiz, Moscow. 13. Vabishchevich P. N. (19871 Numerical Methods for Solvino Free-boundaru Prob~, lems [in Russian]. Moscow State University, Moscow. 14. Wazow W. & Forsythe G. (1960) Finite-difference Methods for Partial Differential Equations. John Wiley & Sons, New York. 15. Yanenko N. N. (1967) The Method of Fmetionol Steps. Springer-Verlag, New Yark.

Heat Conduction Problems with Phase Transit ions

An important class of nonlinear heat exchange problems is connected with phase transformations (see Section 2.3). We shall consider here solid-liquid transitions. In order t o simulate melting/crystallization processes in pure substances, we use the classical Stefan problem, in which we prescribe a constant temperature at the interface. More general models admit that a space crystallization zone is formed in which temperature is equal to the crystallization temperature. Models with nonconstant phase transition temperature are also used. Two main approaches to the numerical solution of problems with phase transformations are employed. First of all, we should mention methods in which the interface is explicitly separated. These methods are sometimes called variable domain methods. The second class is formed by methods in which the interface is not explicitly separated, and these are called fixed domain methods. In the methods of the first group the position of the interface is calculated on every time level. In this case, the free boundary is numerically defined by the position of appropriate nodes. This is achieved by introducing new independent dynamic variables or adjusted dynamic grids in terms of the original variables. In one-dimensional problems the time step is often varied in order to adjust the grid to the position of the interface. Such an approach t o the use of variable time steps (catching of the front into a node of space grid) has been well-known and widely employed for a long time. In some cases it is possible to vary the space step instead. However, all of these methods are poorly suited to multidimensional problems. Speaking about variable domain methods, we should mention a front straightening method in which a fixed-structure dynamic grid whose nodes

348


are bound to the phase interface is nsed. When discussing these approaches, the guideline is to formulate the problem in terms of new independent variables in which the computational domain is regular. For simplicity we consider a single-phase Stefan problem. For multidimensional problems with phase transformations, variable domain methods often encounter difficulties with algorithms of implementation and large computational costs. For this reason fixed domain methods have become popular for approximate solution of these problems. A generalized formulation of the classical Stefan problem is nsed to this end. Appropriate numerical methods for solving the Stefan problem are constructed on the basis of methods for solving quasi-linear heat conduction problems. The enthalpy formulation of the Stefan problem, in which enthalpy rather than temperature is an independent variable, is also used in these problems. Economical difference schemes are used to solve multidimensional problems. Progress in the theoretical research of problems with a free (unknown) boundary has recently been achieved by considering these problems as variational. For the single-phase Stefan problem, a new independent variable (Baiocchi transform) is introduced, which makes it possible to simplify the problem. A fixed domain method for the Stefan problem can also be constructed on the basis of the penalty method applied to an appropriate variational inequality. The quasi-stationary Stefan problem is described by the stationary heat equation. We shall mention below the peculiarities of numerical methods used to solve such problems. For instance, fixed domain methods can be constructed by separating out a singularity of the solution on the phase interface. We shall specially stress the features of mathematical models of phase transformations in alloys. Models with an equilibrium two-phase zone will be formulated. In some cases it is necessary to take into account diffusion of impurities in a liquid or even in a solid phase. The nonlinearity of problems with phase transitions is, first of all, accounted for by the presence of an unknown (free) phase interface. Moreover, similar to ordinary heat conduction problems, the nonlinearity can be due to the dependence of thermal parameters on temperature. In order t o highlight the features of problems with phase transformations, we shall take liquid and solid phases to be homogeneous media with constant thermal parameters.

7.1 Variable Domain Methods 7.1.1 THE MODEL SINGLE-PHASE STEFAN PROBLEM

Let us exemplify the numerical solution of Stefan-like problems by fixed domain methods with the simplest single-phase Stefan problem. Consider an

HEAT CONDUCTION PROBLEMS WITH PHASE TRANSITIONS

349

interval Cl = (0, l ) , which is partitioned by a point x = q(t) (the interface position), q(0) > 0, into two subdomains

We shall assume the phase transition temperature t o be zero (u* = 0). Then in the solid phase, which occupies the domain W ,we have u(x, t) = 0, and in the liquid phase, which occupies the domain R+, we have u(x, t ) > 0. In order to determine the temperature in the liquid phase we consider the heat equation (a homogeneous medium)

Let us equip equation (1) with the initial condition

Suppose the left-hand endpoint of the interval is maintained at a constant temperature, i.e. 0 < t 5 T. u(0, t ) = g(t) > 0, (3) The following conditions are satisfied at the phase interface (see Section 2.3):

Let us recall (Section 2.3) that the constant X is connected with the enthalpy of phase transition. By virtue of the above assumptions about boundary and initial conditions in the single-phase Stefan problem (1)-(5), the speed of the phase front v, = dqldt is positive, i.e. the domain occupied by the liquid phase is expanding. A monotonic rise of the function q(t) follows from the maximum principle (see Section 5.1) for parabolic problems. 7.1.2 CATCHING OF THE FRONT INTO A SPACE GRID NODE Note some simple computational algorithms for solving the one-dimensional problem (1)-(5) that take advantage of a monotonic expansion of the liquidphase domain. We are considering variable domain methods, therefore, a node of the computational grid is associated with the interface. In the domain we introduce a uniform grid with the step h:

350


Let w be the set of the inner nodes of the grid. We shall use the nonuniform time grid

with variable step 7, > 0. We shall select the time step rn+l,n = 0,1,. . . ,N - 1 so that the interface shifts exactly by one space step for the time interval from t, to t,+l. This approach is known a s the method of catching of the front into a space grid node. We assume that the initial domain R+(0) is covered by a part of the grid w (this part will be denoted by w$ C w). We denote the adjusted grid at any time t , in the domain Rt(t,) by w,: where w: c w:+~. We use the notation x = ~ ( t , )= i,h for the boundary node of this dynamic grid. Now we relate to equation (1) the implicit difference scheme

where in indexless notation we have

The approximation of the initial condition (2) yields

The boundary conditions (3), (4) lead to the conditions

Now we only have to approximate the condition in (5). Taking into account the assumption that the front is shifted by one space step (by one node), we

In order to retain the second-order accuracy with respect to h (see Section 4.2), we approximate the condition in (5) as follows:


351

The difference problem (6)-(10) approximates the differential problem (1)-(5) with an error O(T h2). The solution on the new time level y,+, and the time step T , + ~ now stand for unknowns in the problem (6), (8)-(10). Given some r,,+l, the difference can be found from the first boundary value problem (6), (a), (9) or from the mixed boundary value problem (6), (8), (10). The same iterative processes are used in order to solve the nonlinear algebraic problem (6)-(10) on every new time level t = t,+l. The simplest of these processes is connected with an iterative refinement of the time step T,+~. Suppose that an initial approximation ~ j l +is~given (for example, it is logical to put T;+, = r,). Given r;+,, the corresponding approximation for y,+l can be found by solving the linear difference problem

+

The three-point Thomas algorithm is used for this purpose. The difference relation (10) is used to refine the time step. In the simplest case we have

If necessary, an iterative process with a relaxation parameter can be used instead of these successive approximations. The space grid is fixed in this version of the front catching method. For problems as simple as the problem (1)-(5), we can also employ a fixed time grid, in which case the space step is adjusted. In the latter case, it is possible to go with noniterative procedures of passing from one time level t o another. 7.1.3 THE FRONT STRAIGHTENING METHOD

If we are faced with the problem (1)-(S), it is logical to replace x by a new independent variable ,$ such that in terms of the new variables the problem has to be solved in a fixed domain. For the problem (1)-(5) the simplest variable change of the kind has the form

The new independent variable ,$ varies within a finite interval from 0 (at the left-hand endpoint x = 0) to 1 (at the interface we have x = q(t)). When

352


considering heat conduction problems with phase transformations, approaches with such variable changes are referred to as front straightening methods, because in this case the interface coincides with a fixed coordinate line. Now let us formulate the Stefan problem in the new independent variables (t,t ) . Let .(x> t ) = 46 t ) . Then, taking into account (13),we obtain

Hence, equation ( 1 ) becomes

The initial condition (2) yields

~ ( c0 ,) = uo( E ( I ) ( ~ ) ) ) ,

(15)

while the boundary conditions ( 3 ) ,( 4 ) are transformed to become

The resulting boundary value problem (14)-(18) for v(<,t ) and q ( t ) is nonlinear. However, unlike the original problem (1)-(5), it is considered in the fixed computational domain 0 < ( < 1. It is necessary to mention that equation (14) contains a term with the first derivative with respect to the variable <.This term accounts for a heat transfer due to the motion of the computational domain. We shall use difference methods in order to solve (14)-(18) numerically in terms of the new variables. Equation (14) is approximated on the uniform grid ~ { g ( =g = i h , i = 1 . N , N h = 1) as follows:

where 0 = q 2 ( t )

T

Y,+I(€ + h ) - Y,+I(< - h ) 27 2h +Ay,+l = 0 , E w, n = 0 , 1 , . . . , @,+I - 0,

Y ~ +-I yn

@nil

-

<

(19)


353

The difference equation (19) is equipped with the conditions

c

yo(<) = uo(
It follows from (14) and (18) that for

(20) (21) (22)

< = 1 we have the equation

Therefore, an approximation to the boundary condition which is second-order accurate in h on the solutions of equation (14) has the form yn+i(<) - yn+l(E - h) + @"+I- 8, h 27

The implementation of the difference scheme (19)-(23) is connected with defining the grid functions Y,,+~ and @,+I from a nonlinear system of equations on the new time level t = tn+l. Difference methods are usually used t o this end. For example, similar to the front catching method, we can use the following we find y2+l as the solution iterative process (see ( l l ) , (12)). Given a k of the problem

Then, in order to refine Qntl, we use the relation in (23). For instance, the simplest linearization yields

354


Naturally, we can use other formulae: for example, we can determine the new from the appropriate quadratic equation. approximation;::8

7.1.4 FRONT STRAIGHTENING I N A TWO-DIMENSIONAL PROBLEM In order to analyse what the front straightening method has to offer in multidimensional problems, we consider the following model singlophase Stefan problem. Let the rectangle (Fig. 7.1)

be partitioned by the free boundary S = S ( t ) into two parts: R+(t) (the liquid phase) and W ( t ) (the solid phase). We shall assume that the phase interface S is described by the equation xl = q(x2,t), 0 x2 5 l z , and during the whole time period considered it preserves its structure (the function q(x2, t) is single-valued for every fixed t-the free boundary without self-intersections, 0 < q(x2,t) < 11). The single-phase Stefan problem for a homogeneous medium is described by the equation

<

supplemented with the initial condition

On the fixed part of the boundary y = y(t) = aR n aR+(t) we set

Fig. 7.1.


355

On the free boundary we have (see Section 2.3)

Here n is the outward normal and V,, is the speed at which the free boundary moves along the normal to S. Under these conditions we can straighten the front with respect to the variable X I by using the transformation $1 = E1q(xZ,t). Then the problem (24)-(28) is written in the new independent variables (G,C2) with

Putting u(x, t ) = v(E, t ) and taking into account (29), we obtain

In order to write the Laplace operator in the new variables, we shall use general ideas of tensor analysis instead of direct transformations. Let goo, a,/3 = 1,2, be the components of the metric tensor, J be the Jacobian of the transformation (in our case we have J = gllgzz - glzgzl). We denote by a,/3 = 1,2, the components of the associate tensor (the inverse of the metric tensor). Then, in an arbitrary reference frame, we have for the Laplace operator the following:

In order to define of the transformation used, we first calculate the components of the metric tensor. We write (29) in the form X I = Elq((2, t), x2 = t2to obtain

Hence we have

356


and we arrive at the following expression for the Jacobian of the transformation: J = q-1. (33) The substitution of (30), (31) into (24) yields the desired equation

in which the coefficients are defined by (32), (33). Equation (34) is considered in the fixed regular domain

We then correspondingly transform the initial and boundary conditions (25)-(27). As usual, the conditions (28) on the free boundary (at (1 = 1) deserve special discussion. Taking into account the boundary condition (27), we derive from (28)

where vl is the component of the front velocity along the axis from the equation au 1 av from the equation of the free boundary

XI = q(x2, t),

XI.

It follows

and from (35) we derive

Note that the condition (35) on the free boundary exactly coincides with a similar condition for the one-dimensional problem (see (18)). Now the problem is considered in the new variables exactly in the same way as in the one-dimensional case. The iterative refinement of the unknown boundary on every time level is now associated with refining a function rather than a single parameter. For this reason, we may find it useful to take advantage of simplified noniterative schemes, which correspond, for example, to using a single iteration in the above iterative procedures for the onedimensional problem.


357

7.1.5 GENERAL TRANSFORMATION OF INDEPENDENT VARIABLES

More general transformations of independent variables can be used for multidimensional problems instead of the above stretching with respect t o a single variable. For the problem (24)-(28), front straightening can be achieved by the general dynamic transformation

New variables are chosen so that the free boundary is fixed. For instance, the original irregular domain W ( t )is mapped at every moment in time onto the regular one

The above transformation (29) can serve as an example of such a continuous transformation. Two-phase Stefan problems are considered similarly. In this case both domains W ( t )and W ( t ) (the solid-phase domain) are being transformed. It is clear that in this way we shall encounter new difficulties, which are however beyond the scope of our present consideration. The quantization with respect to new variables actually corresponds to the use of irregular free-boundary-consistent grids adapting to the phase interface. Therefore, we can attempt the coustruction of a computational frontstraightening algorithm in this way, without specifying a continuous variable transformation or formulating the differential problem in new variables. In a sense, such constructions may prove to be unnecessary. When constructing grids in multidimensional Stefan-like problems, we should note the following. The grid should be dynamic, adaptive, at least on every time level. Given this requirement, we should reject as unsuitable many popular techniques of constructing the grids. In particular, we should

Fig. 7.2.

358


reject orthogonal grids. Construction of orthogonal grids is a sophisticated problem which can be solved only for fairly simple classes of domains, though even in this case the computational costs are appreciable. When constructing two-dimensional grids, the most economical are algebraic methods. An example is the use of a fixed grid with respect t o the variable x2 in the problem (24)-(28) and of a locally one-dimensional grid with respect to the variable xl (Fig. 7.2). In fact, this procedure is equivalent to using the algebraic relations in (29). Of course, such an approach can be considered in more general situations. Other available methods of constructing grids are associated, in one way or another, with solving some auxiliary problems for partial differential equations and are therefore less suited to the fast construction of adaptive dynamic grids. Methods of 'elliptic' generation of computational grids have become widespread in computation practice. Clearly, the computational costs of solving appropriate boundary value problems may significantly exceed the costs of switching from one time level to another. Methods that involve generation of consistent dynamic grids are especially important when we demand a high accuracy of the results of a numerical solution. Adaptive grids are used in this case. The main singularities of the solutions of Stefan-like problems are localized near the phase interface, where the derivatives of the solution are discontinuous. For this reason, in order to improve the accuracy it is logical to use grids that dynamically condense near the phase interface. 7.1.6 PROBLEMS

1. For t h e m o d e l single-phase Stefan p r o b l e m (1)-(5), consider t h e case i n which t h e front i s caught into a n o d e of a n irregular s p a c e grid. Solution. We shall use an irregular space grid. A new node in this grid will be defined from the condition that the phase interface falls to this new node. Let the grid consist of the nodes x, = h,, i = 1,2,. . . ,in+lat a time t = t,+, and let the position of the boundary grid node x,, m = be unknown (we do not know the step h,). Let us relate to equation (1) the difference equation

+

Equation (38) is supplemented with the initial and boundary conditions


359

In order to approximate the condition in ( 5 ) ,we use the simplest difference

In order t o solve the nonlinear problem (38)-(40) for yn+l and h,, we shall use the Thomas algorithm (see Section 4.5). We shall use the notation u, = yn+l(x,), i = 0,1,. . . , m . The difference problem (38)-(40) is written in the form of the tridiagonal system of linear equations

Cove - BOVI= Fn, -A,ui-1 +C,u;-Biui+l = F , , C,,v, = F,, -A,-lv,-l

a=

1 , 2,..., m - 1 ,

+

whose solution is sought in the form

For the coefficients of the Thoma* algorithm we use the following formulae:

a1 = C ; ' B ~ , a,+l = (C, - Aiai)-'Bi, P1=C;'Fo,

a = 0 , 1 , . . . , m - 1,

P,+l=(C,-A,a,)-'(F,+A,A),

i = O , l , ...,m

In our difference problem (38)-(40) we have

Hence, we can calculate all the coefficients a;, Pi up to i = m - 1. Taking into account the boundary conditions (39),(40),we obtain

Using the expression for P,,

we arrive at the following cubic equation for h,:

360


An additional analysis shows that this equation has one real and two complexconjugate roots. After we have found the grid step, we restore the solution of the difference problem by the reverse Thomas algorithm. It should be stressed that the front-catching algorithm described is noniterative, although the problem (38)-(40)is nonlinear. 2. Write heat equation (24) in terms of new independent variables (&,t2) for a given dynamic transformation of variables (37). Solution. We put u ( x ,t ) = v(C,t ) and get

au au

- = - + - - + -at, at at at

av at2 av at a t ~

at,

From (37) we derive

and, therefore,

at1 = -1 at

ax2 axl axl axz

- - - --

1( a t

at2

at

ac2

Here

is the Jacobian of the transformation. Now we write the Laplace operator in terms of new variables, using the general representation (31).For the transformation (37)we have

Therefore, for the components of the metric tensor we derive


361

Straightforward manipulations yield for the associate metric tensor the following: g12 = - JpgI2, gZZ= 5 - 2 9 1 1 . (44) 9" = J-2 922 1

Substituting (31),(41)-(44)into equation (24) yields the sought-for equation

Here the coefficients of the metric tensor of the transformation (37)are defined by the formulae (43).

7.2 Fixed Domain Methods

7.2.1 TWO-PHASE STEFAN PROBLEM

In Section 2.3 it was mentioned that the classical Stefan problem admits an extended formulation as a single nonlinear heat equation. Given this formulation, we satisfy the necessary conditions on the phase interface. This fact makes it possible to construct computational algorithms for approximate solution of problems with phase transformations without separating the phase boundary explicitly. These methods will be referred to as fied domain methods. We consider the model two-phase Stefan problem in the rectangle R. The free boundary S = S ( t ) divides R into two subdomains W ( t ) and W ( t ) (Fig. 7.1). The heat equation holds in every suhdomain. As usual, we shall assume that thermal parameters of the solid and liquid phases are constant and shall use '+' and '-' indices to refer to one or the other subdomains. Let us write the heat equation in every subdomain:

At the initial moment in time we prescribe some temperature distribution

Let y* = + ( t ) = aR n aRi(t) and

u i ( z , t ) = g(x,t ) , x E yf,

O

< t _< T.

(3)

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The phase transition temperature is taken to be zero and the free boundary S = S(t) is therefore defined by

Two conditions must be satisfied on the free boundary (see Section 2.3): these are the temperature continuity and the law of heat conservation: [u] = 0,

=

x E S(t),

v

x.,),

where X is the enthalpy of the phase transition and V, is the speed at which the free boundary propagates along the normal to S(t). This two-phase Stefan problem can be written as a single common heat equation in the whole domain R. Let 6(u) be the Dirac delta-function. Then, instead of equation (1) and the junction conditions (4)-(6), we can consider the single heat equation

The heat conductivity and specific heat capacity are discontinuous in this equation and have the form

According to (2) and (3), equation (7) is supplemented with the conditions

The peculiarity of the Stefan problem reveals itself in the fact that there is a term with a 6-function in the left-hand side of equation (7). The release and sink of heat, which accompanies a phase change, corresponds to a lumped heat capacity on the interface. The boundary value problem (7)-(9) itself is not very much different from the above quasi-linear heat conduction problems (see Section 5.9). This fact allows us to proceed to the construction of required difference schemes.


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7.2.Z A DIFFERENCE SCHEME WITH SMOOTHED COEFFICIENTS The simplest approach to the approximate solution of the Stefan problem formulated as (7)-(9) is to smooth the coefficients in equation (7), i.e. to switch to the usual heat conduction problem. The heat capacity c(u) and the term M ( u ) appear in equation (7) in the same way. Let us replace the function 6(u) by a function 6(u, A) which is nonzero only within the smoothing interval [-A, A] and let us introduce an effective smoothed heat capacity as

+

E(u) = ~ ( u ) 6(u, A).

(10)

If necessary, we also smooth the heat conductivity (k(u) is replaced by i ( u ) ) and, instead of equation ( 7 ) , we solve the equation with smoothed coefficients

Various approximate formulae for 6(u, A) which are constructed from the assumption of the heat balance on the interval [-A, A] have been widely used in computational practice. The simplest of the kind is to put

As the second example, we can mention the parabolic approximation, when

The condition

5 -n

is evidently fulfilled for this approximation. Numerical experiments provide evidence that the error of the approximate solution only weakly depends on which specific function is selected as an approximation to the 6-function, for instance, on whether we select (12) or (13). The choice of the smoothing parameter A is more significant. Naturally, the choice depends on the grid used and is most often determined empirically from auxiliary computational experiments.

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In fixed domain methods the difference scheme is constructed on the basis of equation (11) under the assumption that the coefficients of this equation are smooth enough. The phase interface itself is not separated explicitly and, therefore, does not appear in the construction of the difference scheme. If necessary, the free boundary is identified as zero isotherm (see ( 4 ) ) after the solution has been found. The difference methods that have been considered previously are employed to solve the problem ( X ) , ( 9 ) , ( 1 1 ) .In accordance with Section 5.9, we apply, for example, the purely implicit difference scheme

Here b(y) = E(Y), and

2

Nv)y =

Cu

v ) ~

o=l

Y

=-

(

x E w.

The boundary and initial conditions ( X ) , (9) become

For definiteness we put

The implementation of the nonlinear difference scheme (14)-(19) is based on iterative methods. The simplest of these is connected to the iterative refinement of the coefficients. The new approximation is determined from the solution of the following linear difference problem (see Section 5.9): " , + +I b(wk) Yn

+ A(wk)wk+l= 0,

w k + l ( s )= g ( s , L + ~ ) ,z E aw,

XEw,

(20)

n = 0,1,.. . .

(21)


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As a rule, a few iterations (20), (21) suffice to ensure the proper smoothness. Instead of the iterative process (20), (21), we can use the Newton method, which leads us to a nonself-adjoint grid elliptic problem for the new approximation. Among practitioners, a linearized purely implicit difference scheme has become popular in which the coefficients are taken from the previous time level. For the problem (a), (9), ( l l ) , at the inner grid nodes we use the equation

supplemented by the boundary conditions (18). When solving problems with phase transformations, the noniterative difference scheme (la), (22) is often preferable rather than the purely implicit scheme (14)-(19) if we want to minimize the total computational costs. 7.2.3. ECONOMICAL SCHEMES

In order to solve multidimensional Stefan-like problems it is logical to use economical difference schemes. For general three-dimensional problems, stable difference schemes can be constructed on the basis of additive locally one-dimensional difference schemes or using the summarized approximation schemes (see Section 6.5). For the two-dimensional problem (8), (9), (11) we can also use factorized difference schemes (see Section 6.3) (difference schemes of alternating directions). Let us give as an example a locally one-dimensional difference scheme for the problem (8), (9), ( l l ) , using as the basis a purely implicit scheme for intermediate problems. In accordance with Section 6.5, we define the approximate solution in two stages. First, taking into account the additive representation (16), we shall find the difference solution y,+,j2 as a solution to the difference problem

Then, using the solution of the grid boundary value problem

we find the solution yntl on the new time level. The implementation of the summarized approximation scheme (23)-(26) is connected with solving a series of one-dimensional nonlinear grid equations

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for yntl12 and y,+,. To this end, we can use an iterative process like (20), (21) or the Newton method. The corresponding systems of linear equations are solved by the Thomas algorithm. The linearized difference scheme (18), (22) will be related to a difference scheme with the equations

and the boundary conditions (24), (26). Instead of (28), the following equation is frequently used

in a hope to increase the stability margin of the linearized difference scheme. We could consider other versions of difference schemes for numerical solution of multidimensional Stefan problems which are based on the use of economical additive schemes described in Chapter 6. For example, we could mention the schemes of alternating directions, which make it possible to calculate the stage in which the temperature regime is stabilized.

7.2.4 ENTHALPY FORMULATION OF THE STEFAN PROBLEM When solving problems in which solid-liquid phase transformations are involved, the enthalpy w(x, t ) rather than the temperature u(x, t ) is often used as an unknown function. For the problem (7)-(9), we define a new function (the enthalpy) by the relation

Introducing enthalpy corresponds (see Section 3.3) to the use of the Goodman transform. In our case of homogeneous liquid and solid phases, in accordance

It follows from (30) that the new unknown function w has a discontinuity on the interface (for u = 0). The heat equation (7) with allowance for (30) takes the form


Fig. 7.3.

In order to construct fixed-domain computational algorithms it is necessary to smooth the discontinuous function w(u). Once more we denote the smoothing parameter by A and introduce the piecewise-linear continuous function (Fig. 7.3)

where w(A) = c+A + X and w(-A) = -c-A. Instead of (32), we can use more intricate approximation formulae. The exact equation (31) is replaced by the approximate equation

The resulting problem is then solved numerically. If the function WA(U)is used as an unknown, one speaks about approximate solution of the Stefan problem in enthalpy formulation. Eliminating wA(u) from (32), (33), we arrive at a smoothing scheme in the above temperature formulation (problem 2). It immediately follows from (32) that for determining u(x, t) by a given wA(u) we have the formula

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If we take this into account, equation (33) becomes

where n ( w ~ = ) k(u(wa)). If the functional transform (34) is given, we can easily formulate initial and boundary conditions for equation (35). It should he noted that simple functional relations (32), (34) between u(x, t) and WA(X,t ) take place for prescribed constant heat capacities. In a more general case of c = ~ ( u )the , transition to enthalpy formulation is not that easy. In order to solve (35) numerically, we can use the difference schemes described in the above when we were considering fixed domain methods in temperature formulation. To identify the free boundary, we use the condition w(0) =

w(-A)

+ w(A) = const,

2

which corresponds to (4). 7.2.5 COMBINED ALGORITHMS We defined fixed domain methods to be those that are not associated with explicit separation of the free interface. When the coefficients are smoothed in the generalized formulation of the Stefan problem (a), (9), ( l l ) , we can use simple formulae like (15), (19) in order to calculate the coefficients in the difference problem (14)-(19), i.e. the discrete problem is constructed by direct approximation (see Section 4.2). More elaborate procedures can be used when constructing difference problems; in particular, we can use a balance method (integro-interpolational method). The balance method can be used to construct a difference scheme immediately for the problem (7)-(9) with explicit separation of the interface. The application of the integro-interpolational method is associated with the use of quadrature formulae when calculating the coefficients of the appropriate difference problem. In order to improve the accuracy it is necessary to take into account discontinuities in the coefficients of the differential equation. For example, this is the case when integrating equation (11) with a discontinuous heat capacity. For this reason the phase interface is not separated explicitly. When using approximation formulae (12), (13), it is necessary to separate the lines of discontinuity (isotherms u f A = const) of heat capacity. The same remark applies to the enthalpy formulation of a problem with phase transformations. These difference methods for solving the problem with smoothed coefficients (8), (9),(11) in the part that concerns the construction of a discrete problem


369

are closely adjacent to variable domain methods. Therefore, we can specify a class of combined methods in which the ideas from both methods are used together. 7.2.6 PROBLEMS 1. F o r t h e p r o b l e m (a), (9), ( l l ) , c o n s t r u c t a linearized analogue of t h e Douglas-Rachford difference scheme. Solution. According to Section 6.2, an analogue of this scheme would be

The implementation of the correction (the second stage in the scheme (36), (37)) can be done on the basis of the difference equation

This equation is distinct from (37) only in that the coefficients are calculated by the solution y,+l/2 rather than the time level t = t,. 2. C o n s t r u c t a linearized difference s c h e m e i n t e m p e r a t u r e formulation o n t h e basis of t h e s m o o t h i n g (32). Solution. For the problem (32), (33) with smoothed coefficients we use the difference scheme

The linearization of this scheme yields the scheme (22) in which

Taking into account (32), we obtain

Such a linearized scheme corresponds to the case in which we use the approximation formula (12) for the &function.

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7.3 Transformation of Dependent Variables 7.3.1 SINGLE-PHASE STEFAN PROBLEM

In Section 7.1 we considered such transformations of independent variables that the original domain with free boundary would become a fixed domain in the new variables. The second option of transforming the original problem is to introduce a new unknown instead of u ( x ,t ) . We can mention the introduction of the enthalpy w ( x ,t ) instead of u ( x ,t ) in Section 7.2 as an example of such a transformation. Here we shall discuss yet another opportunity to introduce a new variable. We consider the model single-phase Stefan problem in a rectangle 0 (the same problem as in Section 7.1). In the liquid phase the following heat equation holds:

We supplement this equation with the simplest initial and boundary conditions

where y+ = y+(t) = an n a n + ( t ) and R+(O) = 0. In the solid phase the temperature is constant and equal to zero, i.e. U ( X ,t

) = O,

2:

E

n - ( t ) = n\Kf(t),

o < t 5 T.

(4)

In particular, on the fixed part of the boundary of the domain W ( t )we have

On the free boundary S ( t ) = an+n X - we have the usual conditions

where Vn is the velocity of motion of the interface. We shall assume that the interface S ( t ) is defined by the equation t = q ( x ) , i.e. s ( t ) = { ( ~ , t ) [ t = v ( x )X, E R ) . (8)


For definiteness, we set n t ( t ) = {(x,t)lt > v(x), x E n ) , n - ( t ) = {(x,t)lt < ~ ( x ) , x E n}. In this case the boundary condition (7) (problem (1)) takes the form gradugradq = - A ,

t = q(x),

0
< T.

(9)

A feature of the problem (1)-(7) is that the solution is continuous over the whole domain R, while the first derivatives are discontinuous when passing through the free boundary. The new unknown function is introduced so that its first derivatives are also continuous. In order to find such a smoother approximate solution we can use simpler computational algorithms. 7.3.2 DUVAUT TRANSFORMATION We take the function

as a new unknown. A transformation similar to (10) was first proposed by Baiocchi for the problem of filtration through a soil dam with a free (unknown) wetting boundary. In the theory of heat Stefan problems, such a transformation has been used by Duvaut (Duvaut transformation). Let us reformulate the single phase problem (1)-(6), (9) for the new unknown w(x, t). First of all, straightforward differentiation of (10) yields

i.e. the temperature u ( x , t ) has the meaning of the rate of change of the function w(x, t) in time. For the derivatives with respect to space variables we have

Taking into account condition (6) on the free boundary, defined by (8), we derive

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Similarly, we have

Taking into account (9), equations ( I ) , and relations ( l l ) ,these representations of the second derivatives yield

Thus, the function

W ( X , t ) satisfies

the equation

Due to ( 4 ) , ( l o ) ,in the solid phase we have W ( X ,t ) = 0 ,

x E R-(t).

(13)

On the free boundary the relations ( 6 ) , (10) yield

w ( x ,t ) = 0 ,

x E S(t),

(14)

while the second condition in (7) is reduced to the form

Because of ( 2 ) ,the initial condition is of the form

x E a.

w(x,O) = 0 ,

(16)

Now we only have to formulate boundary conditions on account ( 3 ) , ( 5 ) ,we write them in the form U(X,t

) = g ( ~t ), ,

E

an =

-+, u

7-.

Because of ( l l ) ,this boundary condition reduces to W ( X ,t

)=

i O

g(x, 6 ) dB,

x E 80.

aS2. Taking into


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Thus, we arrive at the problem with free boundary (12)-(17) for the new unknown function w(x, t ) . The problem (12)-(17) differs from the original problem (I)-(?) for the temperature because this time the solution itself as well as its first derivatives are continuous over the whole domain 0. 7.3.3

PENALTY METHOD

In the subdomain R-(t) we can assume (see (13)) that the following homogeneous equation holds:

Combining (12) and (18) and taking into account the homogeneous conditions (14), (15) on the free boundary, we can write a single general equation for the whole computational domain, namely

where the discontinuous right-hand side has the form

Equations (19), (20) are supplemented by the boundary conditions (17) and the initial condition (16). We can attempt the construction of uniform fixed domain computational algorithms on the basis of the problem (16), (17), (19), (20). However, the discontinuous right-hand side in (20) significantly complicates the search for an approximate solution for w ( x ,t). In theoretical research (unique solvability, smoothness of the free boundary), formulation of the problem (16), (17), (19), (20) as a variational inequality has been successfully used. A penalty method is used for approximate solution of the variational problem. Without giving the variational formulation of the problem concerned, we shall only provide a problem for the approximate solution derived by the penalty method. Let w,(x, t ) denote an approximate solution found by the penalty method for the problem (16), (17), (19), (20), where E > 0 is a penalty parameter. The following problem is formulated for w,(z, t). In the domain 0 we solve

374


the following equation with homogeneous right-hand side:

where 13,(u)is the penalty operator

The initial and boundary conditions for (21), (22) have the usual form (see (16))( 1 7 ) )

w,(x,O) = 0 , W < ( X ,t ) =

x E R,

/ g ( z , R) do,

(23) x6

aa

(24)

0

Now let us consider equations (21), (22) for the approximate solution. Let the level line w,(x) = 0 define an approximation to the free boundary, i.e.

Then we have

n:(t) = { ( ~ , t ) l ~ , (>~ O, ,t ) x E 01,

In order to verify that this definition of the subdomains R;(t) and @(t) is feasible, we shall considet the boundary value problems in separate subdomains. In the subdomain R f ( t ) we have an equation (see (21), (22)) that coincides with the original equation (12) and the corresponding boundary conditions. Therefore, in @ ( t ) we indeed have w,(x,t) > 0. For the subdomain R; ( t ) we derive 2

a2w, 1 ~ at - ~ , : i y W xr ~= - t- ,o~ c t .i r a=1

Due to the maximum principle (Section 5.1), the positiveness of A, and homogeneous first-kind boundary conditions on 8 0 - ( t ) , we have w,(x, t ) < 0. As the penalty parameter E tends to zero, the function w,(x, t ) , defined by (21)-(24),produces an approximate solution of the problem (16), (17), (19), (20), i.e. lim llw,(x, t ) - w ( x ,t)ll = 0 ,

E-0

x E R,

0

< t 5 T.

Only the coefficient LC(w,) is discontinuous in the problem (21)-(24). An approximate free boundary is determined according to (25).


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7.3.4 DIFFERENCE SCHEMES OF THE PENALTY METHOD Let us give some difference schemes for approximate solution of the problem (16), (17), (19, (20). The problem (21)-(24) is used for this purpose. For small values of E we relate the differential problem of the penalty method to a purely implicit difference scheme. In the usual notation we approximate equation (21) on a uniform rectangular grid by the difference equation

with the additional conditions y d x ) = 0, E W, Y , + ~ ( X= ) g ( x , t n + ~ ) , x E aw,

n = 0 , ~. .. .

(27) (28)

Here, as usual, 2

Amy, Y

AY =

=-

xE

W.

m=1 The implementation of the nonlinear difference scheme can be based on the iterative process

with an initial approximation u0 = y,. A new approximation is determined from an inner iterative process which is constructed on the basis of the results in Section 4.7, taking into account the peculiarities of the grid elliptic problem (29), (30) (a diagonal nonnegative grid operator &(wk)E). Instead of the nonlinear scheme (26)-(28) we can use the linearized scheme which corresponds to a single iteration of the process (29), (30). Economical additive schemes for the problem (21)-(25) are constructed in the usual way. We take for example the following additive locally onedimensional scheme (Section 6.5):

with the appropriate initial and boundary conditions. Other summarized approximation difference schemes are constructed analogously.

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7.3.5 PROBLEMS 1. Transform the condition (7) on an interface S ( t ) defined by the relation t = q ( x ) to the condition (9). Solution. Taking into account (6), we derive from ( 7 )

where V, is the component of the velocity of the free boundary along the direction z, a = 1,2. For the point on the interface we have

It immediately follows from the equation of the boundary t = q ( x ) that

We multiply (31) by a q / a x , and sum over a = 1,2. Then, taking into account (32),(33), we derive

This is the sought-for relation (9) on the interface. 2. Reformulate the third-kind boundary condition

for the function w ( x ,t ) defined by (10). Solution. We have from (10)

Differentiating this equality with allowance for (11) and (34),we derive


Integrating this result over t, we obtain

Thus, we arrive a t the third-kind boundary condition for the new function W(X> t).

7.4 Quasi-Stationary Stefan Problem

7.4.1 TWO-DIMENSIONAL MODEL PROBLEM

A quasi-stationary Stefan problem is understood to be (see Section 2.3) the problem about stationary temperature fields in cylindrical bodies when the interface is moving while retaining its shape along an element. These problems are important in microelectronics (growing of single crystals), metallurgy (continuous casting), welding, etc. Let us consider a model two-dimensional problem about an infinite plate passing by a heat source. Let the direction of motion of the plate be along the axis x2 and let the velocity of the plate be Vo.In the immobile reference frame the heat equation has the form

for x = (XI,XZ),0 < X I < 11, -m < x2 < ca.Here the heat conductivity and the specific heat capacity are assumed constant within each phase, i.e.

Let us formulate the boundary conditions on the edges of the plate. We assume that the lower part of the plate is heat-insulated, that is

while the upper part experiences a heat flow (modelling of the heat source):

378


Fig. 7.4.

with g(x) # 0 only in a small neighbonrhood of the point x2 = 0 (a local heat source). Far from the heat source the plate temperature is constant, i.e.

For definiteness, we shall assume that the power of the heat source is sufficient for the total melting of the plate (a tentative pattern of isotherms is presented in Fig. 7.4). In this quasi-stationary Stefan problem (1)-(4), the computational domain is infinite. In order to solve this problem approximately, we single out a large enough rectangle

and consider eqnation (1) in 0 after supplementing this equation with the following conditions on the side boundaries:

Let us note some possible approaches to the approximate solution of the problem (1)-(3), (5).

7.4.2 FIXED DOMAIN METHOD Similar to the approximate solution of nonstationary problems (Section 7.2), we can use algorithms with and without explicit separation of the phase interface (variable and fixed domain methods) for numerical solution of the quasi-stationary problem (1)-(3), (5). What we need is to take into account, whenever possible, peculiarities of the quasi-stationary Stefan problems, which reveal themselves in the fact that the original eqnation is elliptic rather than parabolic.


379

Note that the elliptic operator of the quasi-stationary problem is not self-adjoint. We have not actually discussed the topics of constructing difference schemes, analysing the convergence, and the methods for solving the appropriate grid problems for this type of problem. Some of the related questions are briefly explored below in the part that concerns the problems of heat and mass transport. Here we restrict ourselves to considering the approaches t o a numerical solution on a level of differential formulations. The simplest approach to the approximate solution of the problem (1)-(3), (5) is connected with the construction of iterative fixed domain methods on the basis of smoothing coefficients in equation (1). In accordance with Section 7.2, we pass from equation (1) to the equation

where E(u) = ~ ( u+) 6(u, A) if we use the ordinary approximation formula for

For the numerical solution of the problem (2), (3), ( 5 ) , (6) we can use the simplest iterative process of successive refinement of coefficients. In each iteration we solve the linear boundary value problem

-

=o,

k(vk)

-

k(uk)

x1=11,

auk+'

7 - g(x) 2 0,

uk+'(x) = -1,

21

= 0,

1x21 = 12.

Thus, we are solving a linear boundary value pn3blem for an elliptic equation with convective terms. 7.4.3 SEPARATING THE INTERFACE In the problem (1)-(3), (5) the interface S has a simple structure (see Fig. 7.4). Therefore, we can attempt to use methods with explicit separation of the interface (variable domain methods) for these problems.

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Let us write the heat equation in each phase separately, i.e.

We explicitly separate the junction conditions on the phase interface S, for which we have s = x E n, U(X)= 0). (8)

{XI

The conditions on S have the form [u]=O,

XES,

(9)

In order to simplify the problem (Z), (3), (5), (?)-(lo) it is convenient to use the Kirchhoff transform (see Section 3.3)

In this case equation (7) leads to

where q(v) = Voc(u(u))/k(u(u)). On the free boundary S = { X ~ X E Q , u(x)=o)

(12)

the junction conditions (9), (10) take the form [v]=O,

[El

XES,

= -AVO cos(n,x0),

(13)

x E S.

(14)

The boundary conditions in (2), (3), (5) are correspondingly transformed. The problem for equation (1) is characterized by a singularity in the solution, which manifests itself in nonhomogeneous junction conditions (13), (14) on the unknown free boundary (12). The methods for solving problems with a free boundary are based on those of iterative refinement of the free boundary. Therefore, we can say that, given an approximation to the free


381

boundary during every next iteration, we have to consider the problem with nonhomogeneous junction conditions (14). Similar approaches can also be used when constructing fixed domain difference schemes for the approximate solution of nonstationary problems. Because the problems are nonstationary, we can employ noniterative versions of this approach as well (which essentially means that we go with just one iteration). Let us describe the iterative process of successive refinement of the free boundary on a differential level. It is logical to use the approach in which a new approximation is determined from the solution of the linear equation

when Sk = { X ~ XE 0 ,

vk(x) = 0).

(I6)

The junction conditions (13), (14) yield

where $k(x) = - X V ~ C O S ( ~ ~ ,Thus, Z Z ) . the jumps in the normal derivatives are calculated from the known approximation to the free boundary. Various approaches can be used in order to solve the problem (15)-(18) numerically. The simplest approach is t o construct a difference scheme specifically for this problem. In order to use the integro-interpolational method, it is helpful to include nonhomogeneous junction conditions (17), (18) in the equation. In accordance with Section 2.2, the united equation in R will have the form

where bs, is the surface &function. For an approximate solution of the problem (15)-(18), various versions of the additive separation of the singularity can be used. To this end, we define the solution of the problem (15)-(18) in the form

We select such a function z(x), smooth in R: and R;, that the junction conditions for w(x) on S k are homogeneous. Hence, we should construct such

382


a function z(x) that

Taking into account (20)-(22), for w(x) we obtain the equation

which does not have any singularities all over the domain n. Now the whole problem reduces to constructing such a smooth function z(x) that would satisfy the conditions in (Zl), (22). It is also necessary that the computational costs involved in computing this function be at least comparable to those involved in solving the boundary value problem for w(x). Without attempting any detailed description of possible approaches to constructing the function ~ ( x in ) this problem, we shall only mention the additive separation by a method of harmonic potentials. Taking into account the properties of simplelayer potential, we put

where G(x, y) is the fundamental solution of the twedimensional Laplace operator: 1 2 -112 . G(x,Y) = - ln ((XI y d 2 + (Q - YZ) ) 2r In this case the junction conditions (21), (22) are satisfied. When implementing the method of potentials numerically, it is necessary to calculate a potential in all nodes of the computational grid. A straightforward application of the formula (23) brings about considerable computational costs. We can construct economical algorithms for calculating the values of the potential in the whole computational domain on the basis of the solution of boundary value problems for the Poisson equation.

-

7.4.4 THE SINGLE-PHASE PROBLEM

Subject to some conditions, we can separate the single-phase quasistationary problem. This is the case if we can take the temperature in the liquid phase to be constant. The equalization of the temperature can be accounted for by a greater heat conductivity or by an intensive convective


383

mixing. If the condition of complete melting of the plate is satisfied (Fig. 7.4), we can consider separately the singlophase problem for x2 > 0 and the problem for x2 < 0 (only the sign of velocity is changed). For definiteness, we shall consider the problem for x2 > 0, i.e. in the domain

D- = { X ~ XE R-,

x2 > 0).

The heat equation has the form

where q = Kc-/k-. On the free boundary S defined by (12) we have

Taking into account that n is the outward normal, the second condition on the free boundary has the form

In order to complete the formulation of the problem, we supplement (24)-(26) by the boundary conditions

Let us briefly dwell on possible approaches to the approximate solution of the quasi-stationary single-phase Stefan problem (24)-(29). 7.4.5 INTRODUCING A NEW VARIABLE

Similar to the nonstationary single-phase problem, in order to find an approximate solution of the problem (24)-(29) we can introduce a new variable w(x) instead of u(x) using the Duvaut transformation. To this end, we consider (24)-(29) as a nonstationary problem (in the reference frame associated with the moving plate) and use the Duvaut transformation for the general nonstationary single-pbase problem.

384


Let the equation for the free boundary have the form xz = q(xl), i.e. S=

{XIXZ E

XI), x E a),

and, therefore (see Fig. 7.4)

a+ = {XI

xz

< XI), x E

0-= {xlxz > XI),

a},

x E 0).

The new variable will be (see Section 7.3) a new function w(x) defined in terms of u(x) by the relation

From (30), by straightforward differentiation of the integral with variable integration limits we derive

Similarly,

The differentiation of (31), (32) yields

Taking into account equation (24) and the boundary condition in (25), we transform this relation to the form

Taking into account the boundary condition in (26), we have


385

Therefore, w(x) satisfies the equation

The following homogeneous conditions are satisfied on the boundary S :

The boundary condition (29) is easiest to reformulate. Taking into account

If we take into account (31), the boundary condition (27) yields

Similarly, on the top boundary of the plate we have

The boundary condition (3) depends on the point at which the free boundary S intersects the top boundary of the plate (on the point 2 2 = ~ ( 0 ) A) similar situation also takes place when we are considering a quasi-stationary singlephase Stefan problem with nonhomogeneous conditions of the first kind. The problem (33)-(38) is further considered by the scheme from Section 7.3. In particular, it is convenient to use the penalty method (problem 1) when constructing computational fixed domain algorithms for the approximate solution of the problem for the new function w(z). The solution of the appropriate nonlinear elliptic problem can be found by an iterative method, similar to those that have been considered previously for the nonstationary Stefan problem. 7.4.6 INVERSION OF VARIABLES

In order to solve the quasi-stationary single-phase Stefan problem (24)-(29) approximately, u variable inversion method can be used. The method is based

386


on the passage to new coordinates, where the solution to the problem itself serves as an independent variable. In the problem (24)-(29) we pass from the variables X I ,x 2 , in terms of which the computational domain is known, to the new independent variables

in terms of which the free boundary S is fixed. In this method of front straightening, the function 2 2 = x z ( x 1 , v ) is an unknown, i.e. the unknown variable becomes dependent and vice versa. The transformation in ( 3 9 ) corresponds to the use of the von Mises variables. For the Jacobian of the transformation we have

The coefficients of the metric tensor (see problem 2 in Section 7.1) will have the form 2 g11=

(2)'.(2)' + (2) ,

=1

axl axl axz axz ah aEz ah at2

g12=g21=--+--=--

axz ax2 a x l av '

(41)

Taking into account

av

1

-1

and the corresponding expression for the Laplace operator in the new coordinates, we switch from equation ( 2 4 ) to the following equation for the function xz(x1, v ) :

Substitution of ( 4 0 ) , ( 4 1 ) into this equation yields the sought-for equation


387

This nonlinear equation is considered in the fixed rectangular domain

D 2= {(xi,v)10 < x1 < 11,

-Ilk- < u < 0).

Thus, a nonlinearity of the original problem, accounted for by the presence of a free boundary, reveals itself in a nonlinearity of equation (42) when we pass to the problem in a fixed domain. By virtue of the boundary condition in (25), the free boundary S is defined by the relation xz = xz(xl, 0). In order to formulate boundary conditions for u = 0, we use the condition

which follows from (25), (26). This fact makes it possible to employ the following second-kind boundary condition in terms of the new variables:

The other boundary conditions can be formulated similarly (problem 2). In order to solve the nonlinear boundary value problem for equation (42) with mixed derivatives, we can use iterative methods with successive refinement of the coefficients, like those we considered above in the fixed domain methods for solving Stefan-like problems. 7.4.7 PROBLEMS

1. Formulate t h e b o u n d a r y value problem for t h e a p p r o x i m a t e solution of t h e problem w i t h free b o u n d a r y (33)-(38) o n t h e basis of t h e penalty m e t h o d (see Section 7.3). Solution. We seek an approximate solution w,(x) in the rectangle 0 < X I < 11, 0 < x2 < l z as a solution of the equation

The following conditions are satisfied on the boundaries:

388


In addition to the nonlinearity of the penalty operator, the nonlinearity of this problem is also accounted for by the boundary condition at x l = 0. 2. Reformulate t h e b o u n d a r y conditions (27)-(29) i n t h e v o n Mises variables (for e q u a t i o n (42)). Solution. The condition in (29) is most simply written in the new variables: x2(xl,u) = 12, u = -l/kK. In order to transform the conditions (27), (28), we use the relation av -a=x -2J - . axl axl Hence, (27) yields the boundary condition

For (28) we have

The last condition is nonlinear

7.5 M o d e l l i n g o f Phase T r a n s i t i o n s in Binary A l l o y s

7.5.1 A TWO-PHASE ZONE

Solidification is characterized by concurrent heat processes in solid and liquid phases and kinetic processes on the surfaces of growing crystals. Therefore, these processes are defined by two time scales. The first scale is associated with the melting/crystallization rate and with the velocity of motion of the interface. The second time scale is accounted for by the rate of the phase transition itself, i.e. by its kinetics. The usual assumption that underlies the above Stefan problem is connected with the fact that the rate at which a new phase is formed significantly exceeds the velocity of motion of the interface. Therefore, when considering phase changes we presume that phase changes occur instantaneously and there exists a sharply expressed boundary of the phase transition (the interface). When modelling fast melting/crystallization processes it is often impossible to disregard a finite rate of growth of a new phase. Therefore, when a liquid is crystallized the phase change process takes a finite time, for which the interface shifts by an appreciable distance. Between the solidification onset boundary (the boundary of the liquid phase) and the solidification termination


Fig. 7.5.

boundary (the boundary of the solid phase) there is a domain in which the phase change proper takes place at the constant phase transition temperature. This domain, in which the two phases coexist, is called the two-phase zone. In order t o describe this zone it is logical to introduce a new unknown, namely, the fraction of the solid phase. Thus, given a large velocity of motion of the melting/crystallization boundary, the phase transformation occurs in some space domain (the t w e phase zone). Slow motions of the phase transition boundary make the t w e phase zone become thinner and, as a limit case, we have an expressed phase transition boundary, i.e. the classical Stefan problem. The questions of how to allow for a finite rate of the phase change and the very kinetics of the growth of the new phase are beyond the scope of present consideration. A similar situation takes place when considering melting/crystallization of simple binary alloys (two components, one admixture). In this case new time scales appear, namely, there is a time scale of diffusion of the admixture (equalization of the concentration of the admixture) in the solid and liquid phases. This appreciably widens the spectrum of possible directions in which phase changes may occur. Here we shall discuss some typical modes of melting/crystallization. At a constant temperature, a binary alloy can be in three different states. We shallassume that the temperature of a phase transition in a pure substance (C = 0) equals zero and for small concentrations of the admixture the solidus and liquidus temperatures are straight lines (Fig. 7.5), with the liquidus temperature falling as the concentration of the admixture increases. At a temperature u = ua = const the alloy can be in a solid state (for C < C.,I) or in a liquid state (for C > Cliq), depending on the concentration. For C.,I < C < Cli,) there exists an intermediate equilibrium state when the solid and liquid phases coexist. This twephase zone appears because the alloy is multicomponent.

390


7.5.2 CRYSTALLIZATION WITHOUT REDISTRIBUTION

OF ADMIXTURE Let us consider the simplest model of crystallization in a binary alloy, which is based on the assumption that the time it takes the crystallization to be complete is too small for the admixture to be appreciably redistributed over the material. Within the frames of the equilibrium model, this assumption corresponds (see Fig. 7.5) to a phase change occurring in a temperature interval (u.,,l,uliq), where u.,l is the equilibrium solidus temperature (the crystallization onset) and uli, is the liquidus temperature (the melting onset and the crystallization termination). The release of heat of crystallization is taken into account by introducing an effective heat capacity by the formula

where $(u) is the volume fraction of the solid phase, defined from the equilibrium phase diagram of a particular alloy. It is natural to assume the function $(u) to he continuous and to satisfy the conditions $(u) = 1 for u uar $(u) = 0 for u 2 U I , and 0 < $(u) < 1 for U. < u < UI. In the approximation we shall use, the temperature field arising during phase transitions of a binary alloy is described by the usual heat equation

<

in which the effective heat capacity is introduced according to (1). The approximate solution of the problem for equation (2) can be based on the difference schemes given previously in Section 7.2. Equation (2) differs from the equation with a smoothed specific heat capacity (which is used in fixed domain methods) only by the way in which the effective coefficient is specified (specially smoothed), i.e. by the selection of the smoothing interval ((u,,~,u~~,) instead of A), and by a special approximation relation (-d$/du instead of 6(u, A)). Equation (2) can be approximated, for example, by the purely implicit scheme l 0, b(yntl) Yn+l - yn + N ~ n t d y n t = 7

considered in Section 7.2. Some other difference schemes are also given in this section. Therefore, we need not dwell on these here.


391

7.5.3 THERMODIFFUSION STEFAN PROBLEM The model problem is now the problem with phase transformations and diffusion of an admixture in a rectangular domain 0.This problem is similar to the model two-phase problem from Section 7.2. We shall assume that the velocity of motion of the interface during melting/crystallization of a binary alloy is small and the diffusion of the admixture is predominant. In this case there is an expressed phase transition boundary (a narrow two-phase zone) S = S ( t ) which separates liquid ( R + ( t ) )and solid ( R - ( t ) ) phases. In the frames of the equilibrium approximation, the phase transition boundary is defined by the liquidus temperature for the concentration C+ of the admixture in the liquid phase or by the solidus temperature for the concentration C - of the admixture in the solid phase. The concentrations of the admixture in liquid and solid phases are related by

where k is a distribution coefficient. In the simplest case we are considering (Fig. 7.5), the liquidus equation has the form uli, = -mC+, where the constant m defines the slope, and, therefore, u ( x ,t ) = -mC+(x, t ) , x E S ( t ) , 0 < t 5 T. (4) The conditions in (3), ( 4 ) account for the peculiarity of this problem with phase transitions. The boundary value problems in separate phases are formulated as usual. In each subdomain we have the heat equation

supplemented by appropriate initial and boundary conditions

u*(x,o) = U O ( X ) , u f ( 2 ,t ) = g(x, t ) ,

x E

n*(o),

x E y*,

0 < t 5 T,

where once more we have y* = y*(t) = a R n a R f ( t ) . On the phase transition boundary we have the usual conditions

[u]= 0,

[,g]

= -A,

x E S(t), x E S,)

392


The only difference is that when specifying condition (4) on the free boundary we have instead of the ordinary Stefan condition

Now let us formulate the problem for the distribution of the admixture. We shall assume that the coefficients of diffusion of the admixture are constant for every subdomain, therefore, the diffusion equation has the form

We assume that the admixture cannot leave the material and, therefore, use the boundary condition.

ac*

D*-(x,t)=0, an

XET*,

0
(11)

In addition to the relation in (3), the following condition (see Section 2.3) is satisfied on the interface:

This condition is actually the law of conservation of mass. Additionally we prescribe some initial condition, namely

The themodiffusion Stefan problem is to determine the temperature field from the solution of the problem (5)-(9) and the concentration field from the solution of the problem (3), (10)-(13) under the condition that the interface is defined by (4). The diffusion problem is characterized by somewhat unusual junction conditions (3) and (12) while the heat problem is characterized by a variable temperature on the interface (condition (4)).

7.5.4 NUMERICAL SOLUTION O F THERMODIFFUSION PROBLEM It is impossible to give an extended formulation of the problem (3)(13) as a system of two equations (temperature and diffusion) in the whole computational domain Q. For this reason it is impossible to construct fixed domain methods, similar to those considered in Section 7.2 for the purely heat Stefan problem, for the solution of the thermodiffusion Stefan problem. Therefore, we have to use various versions of computational algorithms


393

based on the separation of the interface. A fairly detailed description of the approaches available is given in Section 7.1. In computational practice one-dimensional thermodiffusion Stefan problems are most widely used. In particular, these refer to the versions of the approach with separation of the interface when front straightening methods are used. It should be noted that the thermodiffusiou problem is usually considered in the two-phase formulation. There are a lot of possible simplifications, of which we shall quote only a few. When modelling redistribution of the admixture it is usually possible t o neglect diffusion in the solid phase because D+ >> D-. Thus, we shall have a single-phase problem when modelling the diffusion of the admixture (see problem 1). Approximations of the thermodiffusion Stefan problem in which the concentration in the liquid phase is equalized (the effective diffusivity D+ >> 1) deserve special notice. This equalization can be accounted for by an intense convective mixing. In this case the phase transition temperature (see condition (4)) is constant all over the boundary, but varies accordingly in time. This situation is typical of single-phase approximations of the heat problem. ~

7.5.5 PROBLEMS 1. Formulate t h e problem of diffusion of a n a d m i x t u r e i n t h e liquid phase assuming t h a t t h e a d m i x t u r e is insoluble in t h e solid phase. Solution. The conditions of the problem actually say that D- = 0. The diffusion equation in the liquid phase has the form (see (10))

The boundary conditions (11) yield

Modelling of the conditions on the interface depends on whether the crystallization process (V* < 0) or the melting process (V,, > 0) actually takes place. The most interesting case is when crystallization takes place on a part of the boundary S and melting occurs on the rest of the boundary. In this case we deal with the change of boundary conditions. During the melting process the concentration field in the solid phase (the function C-(x)) is prescribed and, therefore, on this part of the boundary we have from (3) the first-kind boundary condition

394


The boundary condition for the solidification follows from the junction condition (12) with allowance for D- = 0. This leads us to the third-kind boundary condition

on the interface. 2. Formulate conditions for defining t h e melting t e m p e r a t u r e for t h e case in which t h e r e is n o diffusion of t h e a d m i x t u r e in t h e solid phase a n d t h e a d m i x t u r e is completely mixed in t h e liquid phase. Solution. In our case the phase transition temperature is constant over the interface a t every moment in time, i.e.

Balance relations are used in order to determine the function C + = C+(t). Let C-(x) denote the concentration of the admixture in the solid phase. When a solid region is melted down, the admixture that has been frozen in this region changes the concentration in the liquid phase. Hence, for C+(t) we derive

where V, > 0 is the velocity of motion of the interface, and rnessR+(t) S Rf (t)dx.


7.6.1

GENERAL REMARKS

7.1 Variable domain methods have been widely used for a long time. Similar approaches are also used when solving more general problems with a free boundary. Research in this direction has been conducted since the 1950s. Because of a lack of space in this book, we cannot cite here original works or decide on questions of priority. We restrict ourselves to the citation of general works. A general discussion of the computational algorithms for solving Stefan-like problems is provided in [7],while variable domain methods are under consideration in [8]. 7.2 Fixed domain methods based on a smoothing of coefficients have been considered ever since the works by A. A. Samarskii et al. (1965), in which multidimensional problems were discussed.


395

7.3 T h e Duvaut transformation for t h e Stefan problem has been described in

many books on variational inequalities (see e.g. [3,4]). Numerical analysis of single- and t w e p h a s e problems on t h e basis of such a n approach has been made by K. A. Ichikewa (1979). T h e penalty method is widely used in theoretical and numerical investigation of variational inequalities [5]. 7.4 T h e solvability of quasi-stationary problems is discussed in [2, 61. T h e additive separation of a singularity when solving such problems numerically on the basis of t h e theory of potentials has been suggested by P. N.Vabishchevich (1983). T h e inversion of variables in elliptic problems is discussed in [S]. 7.5 T h e problems of modelling of multicomponent alloys have been dealt with in many works. Various models of t h e t w e p h a s e zone a r e discussed in [I], where results of analytical and numerical research are given.

7.6.2 LITERATURE 1. Avdonin N. A. 11980) Processes lin , , Mathematical Description o.f Crvstollizotion -

Russian]. Zinatne, Riga.

2. Danilvk I. I. (1985) Mot. Nnuk 40, Issue 5 (245), . . On the Stefan Problem,. Uspekhi . . . 133-185. 3. Duvaut G. & Lions J. L. (1972) Les inequations en mechoniques et physique.

Dunod, Paris. 4. Fridman A. (1982) Variational Principles and Free-boundary. John Wiley & Sons,

New York. 5.Glowinski R., Lions 1. J. & Tr&molikres(1976)Anolyse nume'rique des ine'quntions variotionnelles. Dunod, Paris. 6.Meirmanov A. M. (1986) The Stefan Problem [in Russian]. Nauka, Novosibirsk. 7. Rubinsbtein L. I. (1967) The Stefan Pmblem [in Russian]. Zvaigzne, Riga. 8.Vabishchevich P. N.(1987)Numekcol Methodsfor Problems with o Free Boundary

[in Russian]. Moscow State University, Moscow.

Index

algorithm forward-backward algorithm 155 marching algorithm 158 matrix Thomas algorithm 157 Thomas algorithm 155 approximation approximation of the boundary condition on the solutions 105 approximation on the solutions 119 Boussinesq approximation 41 b o u n d a r y conditions 22 approximation of the boundary condition on the solutions 105 Dirichlet boundary conditions 23 error of approximation of the boundary condition 100 first-kind boundary conditions 23 mixed boundary conditions 27 Nenmann boundary conditions 23 second-kind boundary conditions 23 third-kind boundary condition 24 canonical f o r m canonicd form of a difference equation 122 canonical form of a three-level iterative method 167 canonical form of a two-level iterative method 163 canonical form of the three-level difference scheme 232 canonical form of the two-level difference scheme 232 Cholesky decomposition 154 coefficient coefficient of linear expansion 50 coefficient of thermal expansion 41 heat transfer coefficient 24 Lamk coefficients 50 Poisson coefficient 52 complex s c h e m e 274 c o m p u t a t i o n a l experiment 6


condition boundary conditions 22 connected grid 123 conservative difference scheme 109 consistent grid 191 continuity equation 37 convection 15 decomposition m e t h o d 197 density 16 diagnostic computational experiment 11 difference equation canonical form of a difference equation 122 difference problem 97 difference scheme 100 additive difference schemes 325 additive difference schemes with fractional steps 326 additive difference schemes with integer steps 326 additively averaged difference schemes 341 asymptotically stable difference scheme 267 canonical form of the three-level difference scheme 232 canonical form of the two-level difference scheme 232 conservative difference scheme 109 Crank-Nicolson difference scheme 241 difference scheme of alternating directions 310 difference scheme with advancing 241 difference scheme with weights 241 Dauglas-Rachford difference scheme 313 Dufort-F'rankel difference scheme 266 economical difference scheme 299 exact difference scheme 119 factorized difference scheme 316 homogeneous difference scheme 113 locally one-dimensional difference scheme 334 Peaceman-Rackford difference scheme 310 Peaceman-Rackford scheme) 310 predictor-corrector difference scheme 289 purely implicit difference scheme 241 regularization principle for difference schemes 280 Richardson difference scheme 266 stability of the difference scheme 100 stability of the difference scheme with respect to the initial data 233 stability of the difference scheme with respect to the right-hand side 234

INDEX

stabilization-correction difference scheme 313 stable difference scheme 233 symmetrical difference scheme three-level difference scheme 231 three-level difference scheme with weights 245 two-level difference scheme 231 dimensionless p a r a m e t e r 60 direct p r o b l e m 28 Dirichlet b o u n d a r y conditions 23 Douglas-Rachford difference scheme 313 Dufort-Frankel difference scheme 266 D u v a u t transformation 371 economical difference scheme 299 eigenfunction 63 eigenvalue 63 energetically equivalent o p e r a t o r s 136 e n e r g y of t h e o p e r a t o r 135 e n e r g y s p a c e 136 equation boundary layer equation 43 Burgers equation 71 canonical farm of a difference equation 122 continuity equation 37 error of approximation of equation 100 heat equation 16 hyperbolic heat equation 20 maximum principle for parabolic equations 226 maximum principle for second-order elliptic equations 90 Navier-Stokes equations 37 optimization problem for the heat equation 28 error error of approximation of equation 100 error of approximation of the boundary condition 100 e x a c t difference scheme 119 experiment computational experiment 6 diagnostic computational experiment 11 optimizing computational experiment 11 searching experiment 11 explicit s c h e m e 233 factorized difference scheme 316 fast Fourier t r a n s f o r m 159

400


finite element method 99 first difference Green formula 139 first-kind boundary conditions 23 fixed domain method 347 formula first difference Green formula 139 second difference Green formula 140 forward-backward algorithm 155 Fourier law 16 free convection 41 Friedrichs inequality 93 front straightening method 352 function eigenfunction 63 Green function 65 majorant function 128 stream function 39 Gauss method 153 Goodman transform 70 Green function 65 grid 97 connected grid 123 consistent grid 191 method of catching of the front into a space grid node 350 projective-grid methods 99 Gronwall difference lemma 235 Gronwall lemma 227 heat heat conduction 15 heat conductivity 16 heat equation 16 heat transfer coefficient 24 hyperbolic heat equation 20 optimization problem for the heat equation 28 regular regime of heat transfer 77 specific heat capacity 16 heat equation 16 heat transfer coefficient 24 homogeneous difference scheme 113 hyperbolic heat equation 20 ideal contact 25 ill-posed problem 28

INDEX

implicit scheme 233 incompressible tluid 37 inequality Bunyakowskii inequality 136 Cauchy-Schwarz inequality 136 initial conditions 22 integral t r a n s f o r m 67 integro-interpolation m e t h o d 111 inverse p r o b l e m 28 iteration m e t h o d Jacobi iteration method 173 iterative m e t h o d 162 alternating triangle iterative methods 180 canonical form of a three-level iterative method 167 canonical form of a two-level iterative method 163 Chebyshev iterative method 164 iterative method of capacity matrix 199 iterative method of conjugate directions 168 one-step (tw-level) iterative method 162 regularization principle far iterative methods 172 steady-state iterative method 164 three-layer iterative method 162 triangular iterative method 177 two-level iterative method 162 two-step (three-level) iterative method 162 two-step iterative method 213 variational iterative method 166 J a c o b i iteration m e t h o d 173 junction conditions 24 kinematic viscosity 39 Kirchhoff t r a n s f o r m 70 law Fourier law 16 Stefan-Boltzmann law 46 L a m 6 coefficients 50 Laplace o p e r a t o r 17 lemma Gronwall difference lemma 235 Gronwall lemma 227 liquidus t e m p e r a t u r e 34 locally one-dimensional difference scheme 334 LU-decomposition 153


m a j o r a n t function 128 marching algorithm 158 mathematical model 2 m a t r i x T h o m a s algorithm 157 m a x i m u m principle maximum principle for parabolic equations 226 maximum principle for second-order elliptic equations 90 method alternating Schwarz method 201 alternating triangle iterative methods 180 approximate factorization method 183 balance method 111 canonical form of a three-level iterative method 167 canonical form of a twelevel iterative method 163 Chehyshev iterative method 164 decomposition method 197 finite element method 99 fixed domain method 347 front straightening method 352 Gauss method 153 integro-interpolation method 111 iterative method 162 iterative method of capacity matrix 199 iterative method of conjugate directions 168 Jacobi iteration method 173 method of a control volume 111 method of a simple iteration 164 method of a square root 154 method of alternating directions 165 method of catching of the front into a space grid node 350 method of conjugate gradients 168 method of fictitious domains 194 method of lines 240 method of minimal corrections 167 method of separation of variables 63 method of steepest descent 167 Newton method 212 one-step (two-level) iterative method 162 penalty method 373 projective methods 97 projective-grid methods 99 projective-grid methods 99 quasi-linearization method 212

INDEX

reduction method 157 regularization principle for iterative methods 172 Richardson method 164 Seidel method 177 steady-state iterative method 164 symmetrical over-relaxation method 180 three-layer iterative method 162 triangular iterative method 177 twdevel iterative method 162 twwstep (three-level) iterative method 162 two-step iterative method 213 variable domain method 347 variable inversion method 385 variational iterative method 166 mixed b o u n d a r y conditions 27 model mathematical model 2 physical model 2 monophase Stefan problem 32 Navier-Stokes equations 37 negative n o r m 136 N e u m a n n b o u n d a r y conditions 23 Newton m e t h o d 212 nonideal contact 27 number Biot number 60 Kirpichev number 60 Ostrogradskii number 60 Peclet number 62 one-step (two-level) iterative m e t h o d 162 operator adjoint operator 135 bounds of the operator 136 commutative operator 136 compact difference operator 103 operator of transition 234 optimal control problem 29 optimization problem for t h e h e a t equation 28 optimizing computational experiment 11 parameter dimensionless parameter 60 Peaceman-Rackford difference scheme 310

404


penalty method 373 phase transition enthalpy 31 physical model 2 Poincark inequality 94 Poisson coefficient 52 positive deflnite operator 135 positive operator 135 predictor-corrector difference scheme 289 problem difference problem 97 direct problem 28 ill-posed problem 28 inverse problem 28 monophase Stefan problem 32 optimal control problem 29 optimization problem for the heat equation 28 quasi-stationary Stefan problem 33 regularly perturbed problem 77 retrospective problem 23 singularly perturbed problem 78 spectral problem 63 Stefan problem 32 thermodiffusion Stefan problem 392 two-phase Stefan problem 32 well-posed problem 27 projective methods 97 projective-grid methods 99 purely implicit difference scheme 241 quasi-linearization method 212 quasi-stationary Stefan problem 33 radiation 15 reduction method 157 regular regime of heat transfer 77 regularization principle for difference schemes 280 regularization principle for iterative methods 172 regularizer 172 regularly perturbed problem 77 retrospective problem 23 Richardson difference scheme 266 Richardson method 164 scheme complex scheme 274

INDEX

searching experiment 11 second difference Green formula 140 second-kind boundary conditions 23 Seidel method 177 self-adjoint operator 135 self-similar solutions 72 singularly perturbed problem 78 skew-symmetric operator 135 solidus temperature 34 solutions approximation of the boundary condition on the solutions 105 approximation on the solutions 119 self-similar solutions 72 sparse matrix 153 specific heat capacity 16 spectral problem 63 stability of t h e difference scheme 100 stability of t h e difference scheme with respect t o t h e initial data 233 stability of t h e difference scheme with respect t o t h e right-hand side 234 stabilization-correction difference scheme 313 stable difference scheme 233 steady-state iterative method 164 Stefan number 62 Stefan problem 32 Stefan-Boltzmann law 46 Steklov averaging operators 113 stencil 101 stream function 39 stability stability of the difference scheme 100 stability of the differencescheme with respect to the initial data 233 stability of the difference scheme with respect to the right-hand side 234 uniform stability with respect to the initial data 234 summarized approximation 327 symmetrical difference scheme (the Crank-Nicolson scheme) 241 symmetrical over-relaxation method 180 temperature liquidus temperature 34 solidus temperature 34 thermal conductivity 16 thermal diffusivity 17 tbermodiffusion Stefan problem 392

406


third-kind boundary condition 24 Thomas algorithm 155 three-layer iterative method 162 three-level difference scheme 231 three-level difference scheme with weights 245 transform fast Fourier transform 159 Goodman transform 70 integral transform 67 Kirchhoff transform 70 transformation 371 D u ~ u transformation t triangular iterative method 177 two-level difference scheme 231 two-level iterative method 162 two-phase Stefan problem 32 two-step (three-level) iterative method 162 two-step iterative method 213 uniform stability with respect to the initial data 234 variable domain method 347 variable inversion method 385 variational iterative method 166 viscosity 38 well-posed problem 27

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