CFD CODES
( i n n F F o r t r r a n )
INDEX 1. Steady State Heat Conduction Equation (Poisson and Laplace Equations) •
1-D - Finite Difference Method - Dirichlet BC - Solver:
6
TDMA
•
1-D - Finite Difference Method - Dirichlet & Neumann BCs 11
Solver: TDMA
•
1-D - Finite Difference Method - Dirichlet BC - Solver: 16
Gauss-Seidel(SOR)
•
1-D - Finite Difference Method - Dirichlet & Neumann BCs 22
Solver: Gauss-Seidel(SOR)
•
•
1-D - Finite Volume Method - Dirichlet BC - Solver: TDMA
28
First Order Approximation of Flux at Boundaries 1-D - Finite Volume Method - Dirichlet BC - Solver: TDMA
•
5-136
34
Second Order Approximation of Flux at Boundaries 1-D - Finite Volume Method - Dirichlet & Neumann BCs -
40
Solver: TDMA
•
1-D - Finite Volume Method - Neumann BC - Solver: Gauss46
Seidel(SOR)
2
•
2-D - Finite Volume Method - Dirichlet BC - Solvers: PSOR, 52
LSOR, ADISOR •
2-D - Finite Volume Method - Neumann BC - Solvers: PSOR, 69
LSOR, ADISOR •
3-D - Finite Volume Method - Dirichlet BC - Solver: PSOR
•
3-D - Finite Volume Method - Neumann BC - Solver: PSOR
2. Unsteady Heat Conduction Equation (Diffusion Equation) •
1-D - FVM - Explicit Euler Scheme - Dirichlet BC
•
1-D - FVM - General Implicit Scheme - Dirichlet BC -
•
1-D - FDM - Adam-Bashforth time stepping - Dirichlet BC
•
2-D - FVM - Crank-Nicolson Scheme - Dirichlet BC - Solver:
3. Steady Convection-Diffusion Equation
138
159
187-225
1-D - FVM - Central Differencing Scheme - Dirichlet BC 188
Solver: TDMA 1-D - FVM - Central and Upwind Differencing Scheme -
195
Dirichlet BC - Solver: TDMA •
137-186
167
PSOR
•
112
147
Solver: TDMA
•
87
2-D - FVM - Central and Upwind Differencing - Dirichlet BC 206
- Solvers: PSOR, ADISOR
3
4. Unsteady Convection-Diffusion Equation •
1-D - FVM - Euler Implicit - Central Differencing - Dirichlet 227
BC - Solver: TDMA •
1-D - FVM - MacCormack Explicit Method - Central 236
Differencing - Dirichlet BC
•
1-D - FVM - Adam-Bashforth time stepping - Central 244
Differencing - Dirichlet BC
5. Burger's Equations (Hyperbolic) •
226-249
250 - 272
1-D - Linear Advection Equation - FTS, Upwind, LaxFriedrichs, Lax-Wendroff, MacCormack, and Godunov
251
Schemes
•
1-D - Non-linear Advection Equation - FTS, Upwind, LaxFriedrichs, Lax-Wendroff, MacCormack, and Godunov
261
Schemes
6. Navier-Stoke's Equation •
273 - 318
2-D - Navier-Stokes code for incompressible flows. The MAC Method.
4
274
& + $ 3 7 ( 5 Steady State Heat Conduction Equation (Poisson and Laplace Equations)
5
1(a): 1-D FINITE DIFFERENCE METHOD (Drichlet BC – SOLVER:
TDMA)
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1( b): 1-D FINITE DIFFERENCE METHOD (Drichlet & Neumann BC – SOLVER: TDMA)
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1(e): 1-D FINITE VOLUME METHOD (Drichlet BC – SOLVER: TDMA)
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1(f): 1-D FINITE VOLUME METHOD (Drichlet BC – SOLVER: TDMA)
First Order Approximation of Flux at Boundaries 7KHPRGHOHTXDWLRQWREHVROYHGLVRIWKHIRUP 7 G [ ¦
¦
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/ < 7 ,& $ / 6 2 / 8 7 ,2 1
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Q G
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39
1(g): 1-D FINITE VOLUME METHOD (Drichlet & Neumann BC – SOLVER: TDMA)
Second Order Approximation of Flux at Boundaries 7KHPRGHOHTXDWLRQWREHVROYHGLVRIWKHIRUP 7 G [ §
§
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8 1 ,) 2 5 0 * 5 ,’
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45
1(h): 1-D FINITE VOLUME METHOD (Neumann BC – SOLVER: Gauss- Siedel (SOR))
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6 8 % 5 2 8 7 ,1 ( $ 1 $ / < 7 ,& $ / 6 2 / 8 7 ,1 V X E U R X W LQ H $ 1 $ / < 7 Q G [ 7 L Q W H J H Q S D UD P H WH
G L Q Q G
G R X E OH S U H F LV RQ 7 $ 7 Q G 7 $ 1 / Q G ( Q G [ G [ ’ H I LQ LQ J W KH $ Q D O\ WLF D 6 R OX W RQ V R OX W RQ GH SH QG V RQ ERX QG DU \ FR Q G LWLR Q V 7 $ [ [
[ G
[ G G R L Q [
[ G [ 7 $ 1 / L 7 $ [
(
G [
D E V 7 $ 1 / L
7 L
F R Q W LQ X H ULWH U L W H 7 KH $ Q D O\ WLF D 6 R OX W LRQ LV Z U LW H
7 $ 1 / L L Q
IR UP D W [ H ULWH ULWH 7 KH ’ L
I I H U H Q F H % H WZ HH Q WK H $ Q D O\ W LF D DQG 1 X P H U LF D
R O X W L R Q V Z U LW H
( L L Q
UH WX UQ H Q G
49
6 8 % 7 2 8 7 ,1 ( 6 2 5 ) 2 5 7 5 ,’ ,$ * 2 1 $ / 0 $ 7 5 ,; V X E U R X W LQ H 6 2 5 7 ’ Q LW H U P [ W R O Z / ’ 8 & ; L
Q W H J H Q G Q Q P LW H U LW H U P [ S D UD P H WH
QG
G R X E OH S U H F LV RQ / Q G ’ Q G 8 Q G & Q G ; Q G G R X E OH S U H F LV RQ W R OZ U H V G [ R OG [ Q H Z W H U P Q P Q L
UH V LG G ULW ,7 ( 5
5( 6 ,’
LI U H V LG J W W R O D Q G LW H U OW LW H U P [
WK H Q
/ G 8 Q P G G R L Q P [ Q H Z & L
/ L ; L 8 L ; L
; L Z [ Q H Z G
’ L
Z [ R OG
F R Q W LQ X H
& D OF X OD W LR Q R I 5 H V L G X D OV UH V LG G /
G
8 Q P G G R L Q P W H U P & L
/ L ; L ’ L ; L 8 L ; L
U H V L G H V L G W H U P F R Q WLQ X H U H V L G 6 4 5 7 U H V L G Q P
50
L W H L
ULWH IR UP
LWH U U H V LG
W [ L [ H
JR WR H Q G LI L I L W H U P [ H L W H
WK H Q
Z ULWH 7 KH 7 R H UD Q F H KDV Q R W P H W Z
WH U Q X P E H R I
LW H U
D W L R Q V , Q F U H D V L Q J W K H Q X P E H R I L W H U D W R Q V P D \ K H S W R P H H W W K H W R OH U D Q F H H Q G LI UH WX UQ H Q G
51
1(i): 2-D FINITE VOLUME METHOD (Drichlet BC – SOLVERS: PSOR, LSOR, ADIDOR)
7KHPRGHOHTXDWLRQWREHVROYHGLVRIWKHIRUP 7 6 [ \ ©
©
7 + ,6 35 2 * 5 $ 0 6 2 / 9(6
’ 3 2 ,6 6 2 1 ( 4 8 $ 7 ,2 1 : ,7 + ,1 $
5 ( & 7 $ 1 * 8 / $ 5 ’2 0 $ ,1 : ,7 + ’,5 ,& + / ( 7 % 2 8 1 ’ $ 5 < & 2 1 ’ ,7 ,2 1 6 2 1 $ // % 2 8 1 ’ $ 5 ,(6 8 6 ,1 * ) ,1 ,7 ( 9 2 / 8 0 ( 0 ( 7 + 2 ’ 8 1 , ) 2 5 0 * 5 , ’
6 R O Y H 2 S W L R Q V 3 6 2 5 / 6 2 5 $ ’ , 6 2 5
S UR J UD P S J L Q W H J H Q G P G L M P Q P Q S N L W H U P [ F R X Q W S D UD P H WH
Q G P G
G R X E OH S U H F LV RQ 6 ) 6 ) 1 ) : ) ( $ 3 Q G $ : Q G $ ( Q G $ 6 Q G $ 1 Q G & Q G 7 Q G 7 6 P
G 7 1 P G 7 : P G 7 ( P G
G R X E OH S U H F LV RQ [ \ O[ O\ G [ G \ K [ K \ W R OZ Z O Z D F R P P R Q F R H I$ 3 $ 6 $ : $ ( $ 1 F R P P R Q WH P S 7
52
’ H I LQ LQ J 6 R X U F H ) X Q F W LRQ 6 RQ W KH 5 LJ K W 6 H W W L Q J 6 Z 6 [ \
F K
+ D Q G 6 LGH RI WKH 3 R LV V RQ HTXDW R Q
D QJ H W KH 3 R LV V RQ HT XDW LRQ Q WR /D SOD F H HT XD WLR Q
[ \ G G
’ H I LQ LQ J W KH % R X Q G D U \ & R Q G LW LR Q V ) 6 [ [ [ ) 1 [ [ O \ [ O \ [ O \ [ O \ [ ) : \ \ \ ) ( \ \ O[
\ O[ \ O[ \ O[ \ O[
[ G [ G F ) : \ G \ G ULW
U L W ( Q W H U W KH /H QJ WK RI ’ R P D LQ LQ W KH ; U
G LU H F W LR Q
H D G O[ U L W ( Q W H U W KH /H QJ WK RI ’ R P D LQ LQ W KH < UH
G LU H F W LR Q
O\ U L W ( Q W H U W K H 1 X P E H R I & 9 Q W K H ;
UH
G LU H F W LR Q
U L W ( Q W H U W K H 1 X P E H R I & 9 Q W K H <
UH
G LU H F W LR Q
ULW
P Q P Q G [ O [
G P
G \ O \
G Q
53
K [ G \ G [ K \ G [ G \ ( V W D E OLV K QJ W KH % R X Q G D U \ 9 D O X H V 6 R X WK [ G [ G G R L P [ [ G [ 7 6 L ) 6 [
G [
F R Q W LQ X H ( D V W \ G \ G G R L Q G \ 7 ( L ) ( \
G \
F R Q W LQ X H 1 R UWK [ G [ G G R L P G [ 7 1 L
) 1 [ G [
F R Q W LQ X H : H V W \ G \ G G R L Q G \ 7 : L ) : \
G \
F R Q W LQ XH & R Q V W U X F W LR Q R I ( OH P H Q W V LQ W K H & R H I I LF LH Q W 0 D W U L[ I R
54
, Q W H U LR 1 R G H V
G R L P Q $ 6 L K \ $ : L K [ $ ( L K [ $ 1 L K \ $ 3 L
$ : L $ ( L $ 6 L $ 1 L
F R Q W LQ X H 0 R G L I L F D W L R Q R I $ 6 D Q G $ 1 I R 1 R G H V $ G M D F H Q W W R 6 2 8 7 + % R X Q G D U \ G R L P
$ 6 L G G K \ $ 1 L G G K \ $ 3 L
$ : L $ ( L $ 6 L $ 1 L
F R Q W LQ X H 0 R G L I L F D W L R Q R I $ 6 D Q G $ 1 I R 1 R G H V $ G M D F H Q W W R 1 2 5 7 + % R X Q G D U \ G R L P Q
P P Q
$ 6 L G G K \ $ 1 L G G K \ $ 3 L
$ : L $ ( L $ 6 L $ 1 L
F R Q W LQ X H 0 R G L I L F D W L R Q R I $ : D Q G $ ( I R 1 R G H V $ G M D F H Q W W R : ( 6 7 % R X Q G D U \ G R L P P Q
P P
$ : L G G K [ $ ( L G G K [ $ 3 L
$ : L $ (
L $ 6 L $ 1 L
F R Q W LQ X H 0 R G L I L F D W L R Q R I $ : D Q G $ ( I R 1 R G H V $ G M D F H Q W W R ( $ 6 7 % R X Q G D U \
55
G R L P P Q
P P
$ : L G G K [ $ ( L G G K [ $ 3 L
$ : L $ ( L $ 6 L $ 1 L
F R Q WL
Q X H
0 R G L I L F D W L R Q $ 6 $ : $ ( D Q G $ 1 I R & 2 5 1 ( 5 1 R G H V $ 6 G G K \ G G K [ $ ( G G K [ $ 1 G G K \ $ 3
$ : $ ( $ 6 $ 1 G G
$ : P
K \
G G K [ G G K [ G G K \
S P Q
$ : P
$ ( P
$ 6 P
$ 1 P
P
$ 6 S G G K \ G G K [ $ ( S G G
K [
$ 1 S G G K \ $ 3 S
$ : S $ ( S $ 6 S $ 1 S
$ 6 P Q G G K \ $ : P Q G G K [ $ ( P Q G G K [ $ 1 P Q G G K \ $ 3 P Q
$ : P Q $ ( P Q $ 6 P Q
$ 1 P Q
& R Q V W U X F W LR Q R I 5 + 6 9 H F W R U
56
\
G \ G
G R M P Q
P P
G \ [
G [ G N
G R L M N P [ [ G [ &
6 [ \ G [ G \
F R Q WL
Q X H
F R Q WLQXH 0 R G LILF D W LR Q RI 5 + 6 9 H F W R U 6 R X WK G R L P
& L & L
G G K \ 7 6 L
$ 6 L G F R Q WLQ X H ( D V W F R X Q W G R L P P Q
P P
F R X Q W F R X Q W & L
& L
G G K [ 7 ( F R X Q W
$ ( L G F R Q WLQ X H 1 R UWK F R X Q W G R L P Q
P P Q
F R X Q W F R X Q W & L & L
G G K \ 7 1 F R X Q W
57
$ 1 L G F R Q WLQ X H : H V W F R X Q W
G R L P P Q
P P
F R X Q W F R X Q W & L & L
G G K [ 7 : F R X Q W
$ : L G F R Q WLQXH &
G G K \ 7 6
G G K [ 7 :
$ 6 G $ : & P
G & P
G G K \ 7 6 P
G G K [ 7 (
G G P
& P Q
G G K [ 7 ( Q G G K \ 7 1 P
$ ( P Q G $ 1 P Q G S P Q
P
& S
& S
G G K [ 7 : Q
G G K \ 7 1
G $ 1 S G 6 R OX WLRQ RI ( T X D W LR Q V 8 V LQJ 3 6 2 5 / 6 2 5 R $ ’ ,6 2 5 0 H W K R G Z G Z G Z D G W R G
ULW ( Q WH U UH
0 D [ LP X P 1 X P E H R I ,WH U D W R Q V
LWH U P [
58
G R L P Q 7 L G F R Q WLQ X H R S H Q X Q L W I L O H F R X W F & $ // 3 6 2 5 P Q LWH U P [ WR OZ F & $ // /6 2 5 P Q W H U P [ WR O Z
3 R L Q W 6 X F F H V V LY H 2 Y H U
/ LQH 6 X F F H V V Y H 2 Y H U
5 H OD [ D WLR Q 5 H OD [ D W
& $ / / $ ’ , 6 2 5 P Q L W H U P [ W R O Z D $ W H U Q D W L Q J ’ L U H F W L R Q , P S O F L W ULW U L W 7 K H 1 X P H U LF D 6 R OX W LR Q V Z U LW H
7 L L P Q
& D OF X OD W LR Q R I $ Q D O\ W LF D 6 R OX W LR Q D QG * OR E D (
UUR
1 R W H ,I D Q D O\ WLF D V R OX W RQ LV NQ R Z Q WK H F R P P H Q W F D Q E H U H P R Y H G I U R P W K H & $ / / V W D W H P H Q W E H R Z F & $ // $ 1 $ / < 7 P Q G [ G \ I R U P D W [ H V WR S HQG 6 8 % 5 2 8 7 ,1 ( $ 1 $ / < 7 ,& $ / 6 2 / 8 7 ,2 1 V X E U R X W LQ H $ 1 $ / < 7 P Q G [ G \ L Q W H J H Q G L M P Q P Q N S D UD P H WH
Q G
G R X E OH S U H F LV RQ 7 $ 7 Q G 7 $ 1 / Q G ( Q G G R X E OH S U H F LV RQ H SV [ \ G [ G \ P D [ H U U
59
S D UD P H WH
H S V G
F R P P R Q WH P S
7
’ H I LQ LQ J W KH $ Q D O\ WLF D 6 R OX W RQ V R OX W RQ GH SH QG V RQ ERX QG DU \ FR Q G LWLR Q V [ \ [ \ [ \ [ \ [ \ [ \ P Q P Q
\ G \ G G R M P Q
P P
G \ [
G [ G N
G R L M N P [ [ G [ 7 $ 1 / L 7 $ [
G [ \ G \
( L G L I 7 $ 1 / L J W H S V W K H Q ( L D E V 7 $ 1 / L
7 G 7 $ 1 / L
H Q G LI
F R Q WLQ X H
F R Q W LQ X H ULW U L W 7 KH $ Q D O\ WLF D 6 R OX W LRQ LV Z U LW H
7 $ 1 / L P Q
IR UP D W [ H F Z ULWH F Z U LW H 7 KH * OR E D ( U U R U
60
F Z U LW H
( L L P Q
& D O F X O D W L R Q R I 0 D [ L P X P * O R E D ( U U R U G G R L P Q L I ( L J W P D [ H P D [ H
WK H Q
( L
F R Q W LQ X H ULW U L W 7 K H 0 D [ L P X P * O R E D ( U U R L V P D [ H U U UH WX UQ H Q G
6 8 % 5 2 8 7 ,1 ( 3 2 ,1 7 6 2 5 ) 2 5 12 1 $ ’ - $ & ( 1 7 3 ( 1 7 $ ’ ,$ * 2 1 $ / 0 $ 7 5 ,; V X E U R X W LQ H 3 6 2 5 P Q LW H U P [ WR O Z L Q W H J H Q G L P Q P Q L W H U L W H U P [ S D UD P H WH
QG
G R X E OH S U H F LV RQ ’ Q G / / Q G / Q G 8 Q G 8 8 Q G & Q G G R X E OH S U H F LV RQ W R OZ U H V G [ R OG [ Q H Z W H U P F R P P R Q F R H I’ / / / 8 8 8 F R P P R Q F R Q V W& F R P P R Q WH P S ; P Q P Q L
61
; Q G
UH V LG G ULW ,7 ( 5
5( 6 ,’
LI U H V LG J W W R O D Q G LW H U O
WLWH U P [
WK H Q
G R L P Q [ Q H Z & L
/ / L ; L P 8 8 L ; L P
; L Z [ Q H Z G
/ L ; L
8 L ; L
’ L Z [ R OG
F R Q WLQ X H & D OF X OD W LRQ RI 5 H V LG X D OV UH V LG G G R L P Q W H U P & L
/ / L ; L P
L ; L ’ L ; L
8 L ; L 8 8 L ; L P
U H V L G H V L G W H U P F R Q WLQ X H UH V LG
6 4 5 7 U H V LG P Q
L W H L
ULWH IR UP
LWH U U H V LG
W [ L [ H
JR WR H Q G LI L I L W H U H T L W H U P [ W K H Q Z ULWH 7 KH 7 R H UD Q F H KDV Q R W P H W Z D WLR Q
WH U Q X P E H R I LWH U
V , Q F U H D V L Q J W K H Q X P E H R I L W H U D W R Q V P D \ K H S W R P H H W W K H W
R OH U D Q F H H Q G LI UH WX UQ
62
HQG
6 8 % 5 2 8 7 ,1 ( / ,1 ( 62 5 ) 2 5 12 1 $ ’ - $ & ( 1 7 3 ( 1 7 $ ’ ,$ * 2 1 $ / 0 $ 7 5 ,; V X E U R X W LQ H / 6 2 5 P Q LW H U P [ WR O Z L
Q W H J H Q G P G L M N N N P P Q P P LW H U LW H U P [ S D UD P H WH
QG P G
G R X E OH S U H F LV RQ ’ Q G / / Q G / Q G 8 Q G 8 8 Q G & Q G ; Q G / 2 P G ’ ,P G 8 3 P G & 2 P G ; ; P G G R X E OH S U H F LV RQ Z U H V LG W H U P W R O F R P P R Q F R H I’ / / / 8 8 8 F R P P R Q F R Q V W& F R P P R Q WH P S ; P Q P Q P P P L
UH V LG G ULW ,7 ( 5
5( 6 ,’
L I U H V L G J W W R O D Q G L W H U O W W H U P [ W K H Q N G R M P Q
P P P
N N NN G R L M N P NN NN / 2 N N Z / L
63
’ , N N ’ L 8 3 N N Z 8 L & 2 N N Z & L
/ / L ; L P
G
8 8 L ; L P
Z ’ L ; L
F R Q WLQ X H & $ // 7 ’ 0 $ P / 2 ’ ,8 3 & 2 ; ; NN
G R L M N P NN NN ; L ; ; N N
F R Q WLQ X H
F R Q WLQ X H & D OF X OD W LRQ RI 5 H V LG X D OV UH V LG G G R L P Q W H U P & L
/ / L ; L P
/ L ; L ’ L ; L
8 L ; L 8 8 L ; L P
U H V L G H V L G W H U P F R Q WLQ X H UH V LG 6 4 L W H L
ULWH IR UP
5 7 UH V LG P Q LWH U U H V LG
W [ L [ H
JR WR H Q G LI L I L W H U H T L W H U P [ W K H Q Z ULWH 7 KH 7 R H UD Q F H KDV Q R W P H W Z D WLR Q V
WH U Q X P E H R I LWH U
,Q F U H D V LQJ WKH QXP E H R I LWH U D W LR Q V P D \ K H OS WR P H H W WKH W
64
R OH U D Q F H H Q G LI UH WX UQ HQG
6 8 % 5 2 8 7 ,1 ( $ ’ ,6 2 5 ) 2 5 12 1 $ ’ - $ & ( 1 7 3 ( 1 7 $ ’ ,$ * 2 1 $ / 0 $ 7 5 ,; V X E U R X W LQ H $ ’ , 6 2 5 P Q LW H U P [ WR O Z LQ W H J H Q G P G L M N N N P Q P Q P P W H U LW H U P [ S D UD P H WH
QG P G
G R X E OH S U H F LV RQ ’ Q G / / Q G / Q G 8 Q G 8 8 Q G & Q G ; Q G / 2 P G ’ ,P G 8 3 P G & 2 P G ; ; P G G R X E OH S U H F LV RQ U H V LG WH U P Z W R O F R P P R Q F R H I’ / / / 8
8 8
F R P P R Q F R Q V W& F R P P R Q WH P S ; P Q P Q P P P L
UH V LG G ULW ,7 ( 5
5( 6 ,’
L I U H V L G J W W R O D Q G L W H U O W W H U P [ W K H Q N 6 Z HHS E\ 5 R Z V GR
P Q
P P P
65
N N NN G R L M N P NN NN / 2 N N Z / L ’ , N N ’ L 8 3 N N Z 8 L & 2 N N Z & L G
/ / L ; L P
8 8 L ; L P
Z ’ L ; L
F R Q W LQ X H & $ // 7 ’ 0 $ P / 2 ’ ,8 3 & 2 ; ; NN
G R L M N P NN NN ; ; ; N N
F R Q W LQ X H
F R Q WLQ X H 6 Z HHS E\ & R OX P Q V G R M P NN G R L M P Q
P P Q P
NN NN / 2 N N Z / / L ’ , N N ’ L 8 3 N N Z 8 8 L & 2 N N Z & L
/ L ; L 8 L ; L Z ’ L ;
F R Q WLQ X H & $ // 7 ’ 0 $ Q / 2 ’ ,8 3 & 2 ; ;
66
NN G R L M P Q
P P Q P
NN NN ; L ; ; N N
F R Q W
LQ X H
F R Q WLQ X H & D OF X OD W LR Q R I 5 H V L G X D OV UH V LG G G R L P Q W H U P & L
/ / L ; L P
/ L ; L ’ L ; L
8 L ; L 8 8 L ; L P
U H V L G H V L G W H U P F R Q WLQ X H UH V LG 6 4 5 7 UH V LG P Q L W H L
ULWH IR UP
LWH U U H V LG
W [ L [ H
JR WR H Q G LI L I L W H U P [ H L W H
WK H Q
Z ULWH 7 KH 7 R H UD Q F H KDV Q R W P H W Z
WH U Q X P E H R I LWH U
D W L R Q V , Q F U H D V L Q J W K H Q X P E H R I L W H U D W R Q V P D \ K H S W R P H H W W K H W R OH U D Q F H H Q G LI UH WX UQ HQG
67
6 8 % 5 2 8 7 ,1 ( 7 ’ 0 $ V X E UR X W
LQH 7 ’ 0 $ Q / ’ 8 & ;
L Q W H J H Q G L Q S D UD P H WH
Q G
G R X E OH S U H F LV RQ / Q G ’ Q G 8 Q G & Q G ; Q G 3 Q G 4 Q G W H P S / G 8 Q G ) R U Z D U G ( LP LQ D W LR Q G R L Q W H P S ’ L / L 3 L 3 L
8 L WH P S
4 L & L
/ L 4 L WH P S
F R Q W LQ X H % D FN 6 X E V WLWX W G R L Q
; 3 L ; L 4 L F R Q W LQ X H UH WX UQ H Q G
68
1(j): 2-D FINITE VOLUME METHOD (Neumann BC – SOLVERS: PSOR, LSOR, ADISOR)
7KHPRGHOHTXDWLRQWREHVROYHGLVRIWKHIRUP 7 7 6 [ \
7 + ,6 35 2 * 5 $ 0 6 2 / 9(6
’ 3 2 ,6 6 2 1 ( 4 8 $ 7 ,2 1 : ,7 + ,1 $
5 ( & 7 $ 1 * 8 / $ 5 ’2 0 $ ,1 : ,7 + 1( 8 0 $ 11 % 2 8 1 ’ $ 5 < & 2 1 ’ ,7 ,2 1 6 2 1 $ // % 2 8 1 ’ $ 5 ,(6 8 6 ,1 * ) ,1 ,7 ( 9 2 / 8 0 ( 0 ( 7 + 2 ’ 8 1 , ) 2 5 0 * 5 , ’ 6 R O Y H 2 S W
R Q V 3 6 2 5 / 6 2 5 $ ’ , 6 2 5
S UR J UD P S J L Q W H J H Q G P G L M P Q P Q P S P P P P S D UD P H WH
P LWH U P [
Q G P G
G R X E OH S U H F LV RQ 6 ) 4 6 ) 4 1 ) 4 : ) 4 ( $ 3 Q G $ : Q G $ ( Q G $ 6 Q G $ 1 Q G & Q G 7 Q G 4 6 P G 4 1 P G 4 : P G 4 ( P G G R X E OH S U H F LV RQ [ \ O[ O\ G [ G \ K [ K \ W R OZ Z O Z D U K V U K V V X P F R P P R Q F R H I
$ 3 $ 6 $ : $ ( $ 1
F R P P R Q WH P S 7
69
’ H I LQ LQ J 6 R X U F H ) X Q F W LRQ 6 RQ W KH 5 LJ K W 6 H W W L Q J 6 Z
F K D Q J H W K H 3 R L V V R Q H T X D W L R Q Q W R / D S O D F H H T X D W L R Q
6 [ \
[ \ G
’ H I LQ LQ J W KH % R X Q G D U \
& R Q G LW LR Q V
) 4 6 [
[ [
) 4 1 [
[ [ O \ [ O
) 4 : \
\ \
) 4 ( \
\ \ O[ \ O[
ULW Z U LW H UH
+ D Q G 6 LGH RI WKH 3 R LV V RQ HTXDW R Q
( Q W H U WKH /HQJWK RI ’ R P D LQ LQ WKH ;
G LU H F W LR Q
O[ U L W ( Q W H U W KH /H QJ WK RI ’ R P D LQ LQ W KH <
UH
G LU H F W LR Q
O\ ULW ( Q WH U WKH 1 X P E H
UH
R I & 9 LQ W KH ;
G LU H F W LR Q
ULW (
UH
Q WH WKH 1 X P E H
R I & 9 LQ WKH <
G LU H F W LR Q
ULW
P Q P Q P P P
P P P
P S P P P G [ O [
G P
G \ O \
G Q
K [ G \ G [ K \ G [ G \
70
( V W D E OLV K QJ W KH % R X Q G D U \ 9 D O X H V 6 R X WK [ G [ G G R L P G [ 4 6 L ) 4 6 [
G [
F R Q W LQ X H ( D V W \ G \ G G R L P P Q P \ \
G \
4 ( L ) 4 ( \
G \
F R Q W LQ X H 1 R UWK [ G [ G G R L P Q
P P P Q
G [ 4 1 L ) 4 1 [
G [
F R Q W LQ X H : H V W \ G \ G G R L P Q
P P P
G \ 4 : L
) 4 : \ G \
F R Q W LQ XH & R Q V W U X F W LR Q R I ( OH P H Q W V LQ W K H & R H I I LF LH Q W 0 D W U L[ I R G R L P Q $ 6 L K \ $ : L K [
71
, Q W H U LR 1 R G H V
$ ( L K [ $ 1 L K \ $ 3 L
$ : L $ ( L $ 6 L $ 1 L
F R Q WLQ X H 0 R G L I L F D W L R Q R I $ 6 I R 1 R G H V $ G M D F H Q W W R 6 2 8 7 + % R X Q G D U \ G R L P P $ 6 L G $ 3 L
$ : L $ ( L $ 6 L $ 1 L
F R Q W LQ X H 0 R G L I L F D W L R Q R I $ 1 I R 1 R G H V $ G M D F H Q W W R 1 2 5 7 + % R X Q G D U \ G R L
P Q P P P Q
$ 1 L G $ 3 L
$ : L $ ( L $ 6 L $ 1 L
F R Q W LQ X H 0 R G LI LF D W LR Q R I $ : I R 1 R G H V $ G MD F H Q W W R : ( 6 7 % R X Q G D U \ G R L P S P Q
P P
$ : L G $ 3 L
$ : L $ ( L $ 6 L $ 1 L
F R Q W LQ X H 0 R G L I L F D W L R Q R I $ ( I R 1 R G H V $ G M D F H Q W W R ( $ 6 7 % R X Q G D U \ G R L P P Q
P P
$ ( L G $ 3 L
$ : L $ ( L $ 6 L $ 1 L
F R Q W LQ X H 0 R G L I L F D W L R Q $ 6 $ : $ ( D Q G $ 1 I R & 2 5 1 ( 5 1 R G H V
72
$ 6
G
$ Z
G
$ 3
$ : $ ( $ 6 $ 1 G G
$ : P
$ : P Q
P P G
$ 1 P Q
P P G
$ 3 P Q
P P
$ ( P Q
G
$ 1 P Q
G
$ 3 P Q
$ ( P
$ 6 P
$ : P Q P P $
$ 1 P
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137
2(a): 1-D FINITE VOLUME METHOD Explicit Euler Scheme (Drichlet BC)
7KHPRGHOHTXDWLRQWREHVROYHGLVRIWKHIRUP Ï 7 N 7 6 [
7 + ,6 35 2 * 5 $ 0 6 2 / 9(6
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2(b): 1-D FINITE VOLUME METHOD General Implicit Scheme (Drichlet BC- Solver:
TDMA)
7KHPRGHOHTXDWLRQWREHVROYHGLVRIWKHIRUP 7 6 [ Ï 7 N
7 + ,6 35 2 * 5 $ 0 6 2 / 9(6
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