The Lighthill-Whitham-Richards Model Emanuel P. Fontelles T´opi op icos co s de Mecˆ Me cˆanic an ica a Esta Es tatt´ısti ıs tica ca
June, 2016
Introduction Continuity Equation: The continuity equation is a partial differential or difference equation for the macroscopic quantities ρ (density) and V (speed) or Q (flow). Due to the hydrodynamic relation “flow equals density times speed” these two options are equivalent. While the parameterless continuity equation is always valid, we need an additional equation for the flow or speed to complete the model. Since the continuity equation is completely determined by the geometry of the road infrastructure, the macroscopic models differ in their modeling of speed or flow, only.
Continuity Equation:
∂ ρtot ∂ t t
+
∂ (ρtot V ) ∂ x x
=0
8.1 Model Equations
Lighthill-Whitham-Richards (LWR) Model: In 1955 and 1956, Lighthill and Whitham, and independently also Richards, proposed the following static relation to complement the continuity equation: Q (x , t ) = Q e (ρ(x , t ))
or V (x , t ) = V e (ρ(x , t ))
8.1 Model Equation
Inserting the assumption LWR on continuity equation and applying the ∂ Qe dQ (ρ) ∂ρ chain rule = , yield the simplest form of a Lighthill ∂ x d ρ ∂ x Whitham Richards model ∂ρ dQ (ρ) ∂ρ + =0 ∂ t d ρ ∂ x ∂ρ ∂ρ dV e + V e + ρ d ρ =0 ∂ t ∂ x
LWR Model
LWR Model
8.1 Model Equations The schematic example of a static speed-density relation to LWR models.
Figure: Schematic example of a static speed-density relation for LWR models.
Lighthill-Whitham-Richards (LWR) Model Proposed by Lighthill and Whitham (1955) and Richards (1956), the LWR model describes traffic flow on a single one-way road without entrances or exits:
ρt + ρV (ρ)
x
=0
ρ = density V (ρ) = preferred velocity, a given nonincreasing function of ρ,
nonnegative for ρ between 0 and ρm (the“jam” density) Predicts piece-wise smooth density, with transitions between regions approximated by shocks Problem: Doesn’t adequately describe the motion of cars passing through shocks (cars change velocity instantaneously)
8.2 Propagation of Density Variations One of important parameters is the propagation velocity c ˜(ρ), this parameter is related with density variation, and using the traveling-wave ansatz ρ(x , t ) = ρ 0 (x − c˜t )
ρ0 (x ) = ρ(x , 0)
We can derive the propagation velocity c˜ . The partial differential equations are nonlinear wave equations Describes the propagation of kinematic waves Propagation velocity c˜ c˜ (ρ) =
dQ e d ρ
=
d (ρV e (ρ)) d ρ
8.2 Propagation of Density Variations
Figure: Propagation velocity c˜ = Q e (ρ) of density and speed variations in the LWR model in comparison with the local vehicle speed V (ρ). In the fundamental diagram (top), c˜ is given by the slope of the tangent while V is given by the slope of the secant through the origin.
8.3 Shock Waves
8.3.1 Formation Continuous LWR models describes density variations of constant amplitudes but with varying local propagation velocities How to density variations affects the traffic? How to behave the vehicles on a stop-go-wave? (Density Profile)
8.3 Shock Waves 8.3.1 Formation
Figure: Emergence of shock waves due to the density-dependent local propagation velocities in the LWR model.
It is noted a discontinuous transition indicated by the vertical line which is the defining feature of a shock wave or shock front.
8.3 Shock Waves 8.3.2 Derivation of the Propagation Velocity The transitions free → congested and congested → free in the LWR models are unrealistic The propagation of the wave positions, the motion of transition zones to and from extended congested traffic, are described realistically
Figure: A shock front at location x 12 (t ) with constant flow and density within small road sections on either side.
8.3 Shock Waves 8.3.2 Derivation of the Propagation Velocity 12 To derive the propagation velocities, to find the velocity c 12 = dx , we dt will express the rate of change in the number of vehicles, dn , in two dt different ways: From the conservation of vehicles And with the definition of the density dn
= Q 1 − Q 2
dt n = ρ1 x 12 + ρ2 (L − x 12 )
Comparing both expressions for
c 12 =
Q 2
−
Q 1
ρ2 − ρ1
=
→
dn dt
dn dt
12 = (ρ2 − ρ1 ) dx = (ρ2 − ρ1 )c 12 dt
gives us
Q e (ρ1 ) − Q e (ρ2 )
ρ2 − ρ1
Propagation of Shock Waves
8.3 Shock Waves 8.3.3 Vehicle Speed Versus Propagation Velocities The LWR model allows us to extract all relevant velocities directly from the fundamental diagram
Figure: Visualization of how to obtain vehicle speeds and propagation velocities from the fundamental diagram.
8.3 Shock Waves
8.3.3 Vehicle Speed Versus Propagation Velocities 1 The propagation velocity of density variations c ˜ (ρ) = Q e (ρ) is given by the slope of the fundamental diagram. 2 The propagation velocity of shock fronts c is given by the slope of 12 the secant connecting points of the fundamental diagram corresponding to traffic on either side of the front. 3 The vehicle speed V = Q (ρ)/ρ is given by the slope of the secant e e connecting the origin with the corresponding point on the fundamental diagram.
8.4 Numerical Solution This is generally done by finite-difference methods: Space is divided into cells of generally constant length ∆x (although this is not required), and time in the index k increasing in the downstream direction. The equations for the LWR models have the form of a so-called conservation law for which many specialized explicit solution methods are available.
Figure: Cells of the CTM for a simple straight road and definition of the relevant quantities for the supply demand method.
8.4 Numerical Solution
The most common integration method for LWR models is the Godunov scheme Courant-Friedrichs-L´evy condition (CFL condition) for LWR models These discretization errors lead to the phenomenon of numerical diffusion which increases with the cell size
8.5 LWR Models with Triangular Fundamental Diagram
The simplest of the Lighthill-Whitham-Richards models uses a “triangular” fundamental diagram
The continuous version is called section-based model. The discrete version is formulated as an iterated coupled map with time and space discretized into time steps and cells, respectively, and supplemented by a special “supply-demand” update rule. This model is known as cell-transmission model
8.5 LWR Models with Triangular Fundamental Diagram
Figure: Triangular fundamental diagram, as used in the cell transmission model and the section-based model.
8.5 LWR Models with Triangular Fundamental Diagram 8.5.1 Model Parameters
For vehicular traffic, the maximum density corresponds to the inverse of the minimum distance headway l eff , which is the average vehicle length plus the average minimum gap s 0 in stopped traffic: l eff = s 0 + l =
1 ρmax
8.5 LWR Models with Triangular Fundamental Diagram 8.5.1 Model Parameters
For vehicular traffic, the maximum density corresponds to the inverse of the minimum distance headway l eff , which is the average vehicle length plus the average minimum gap s 0 in stopped traffic: l eff = s 0 + l =
1 ρmax
8.5 LWR Models with Triangular Fundamental Diagram
8.5.9 Examples
8.6 Diffusion and Burgers’s Equation
Shock waves are not very realistic in describing traffic flow. Furthermore, the associated discontinuities turn out to be problematic for a numerical solution - at least for non-triangular fundamental diagrams. As a simple phenomenological solution, one may introduce diffusion to the continuity equation by adding a diffusion term D ∂ρ/∂ x 2 with the diffusion constant D > 0:
Questions?
