Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model

Raimund Bürger; Antonio García; Kenneth H. Karlsen; John D. Towers; Raimund Bürger; Antonio García; Kenneth H. Karlsen; John D. Towers

doi:10.3934/nhm.2008.3.1

Networks and Heterogeneous Media

2008, Volume 3, Issue 1: 1-41. doi: 10.3934/nhm.2008.3.1

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Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model

1.
CI²MA and Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción
2.
Departamento de Ingeniería Metalúrgica, Facultad de Ingeniería y Ciencias Geológicas, Universidad Católica del Norte, Avenida Angamos 0610, Antofagasta
3.
Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo
4.
MiraCosta College, 3333 Manchester Avenue, Cardiff-by-the-Sea, CA 92007-1516

Received: 01 August 2007 Revised: 01 October 2007
Primary: 35L65, 65M06; Secondary: 90B20.

The classical Lighthill-Whitham-Richards (LWR) kinematic traffic model is extended to a unidirectional road on which the maximum density $a(x)$ represents road inhomogeneities, such as variable numbers of lanes, and is allowed to vary discontinuously. The car density $\phi = \phi(x,t)$ is then determined by the following initial value problem for a scalar conservation law with a spatially discontinuous flux:
$\phi_t + (\phi v(\phi/{a(x)})_x = 0, \quad \phi(x,0)=\phi_0(x),\quad x \in \mathbb{R},\quad t\in (0,T),$ (*)

where $v(z)$ is the velocity function. We adapt to (*) a new notion of entropy solutions (Bürger, Karlsen, and Towers [Submitted, 2007]), which involves a Kružkov-type entropy inequality based on a specific flux connection $(A,B)$, and which we interpret in terms of traffic flow. This concept is consistent with both the driver's ride impulse and the desire of drivers to speed up.
We prove that entropy solutions of type $(A,B)$ are unique. This solution concept also leads to simple, transparent, and unified convergence proofs for numerical schemes. Indeed, we adjust to (*) new variants of the Engquist-Osher (EO) scheme (Bürger, Karlsen, and Towers [Submitted, 2007]), and of the Hilliges-Weidlich (HW) scheme analyzed by the authors [ J. Engrg. Math., to appear]. It is proven that the EO and HW schemes and a related Godunov scheme converge to the unique entropy solution of type $(A,B)$ of (*). For the Godunov version, this is the first rigorous convergence and well-posedness result, since no unnecessarily restrictive regularity assumptions are imposed on the solution. Numerical experiments for first-order schemes and formally second-order MUSCL/Runge-Kutta versions are presented.
- Conservation law,
- discontinuous flux,
- traffic model,
- entropy condition,
- difference scheme,
- driver's ride impulse
Citation: Raimund Bürger, Antonio García, Kenneth H. Karlsen, John D. Towers. Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model[J]. Networks and Heterogeneous Media, 2008, 3(1): 1-41. doi: 10.3934/nhm.2008.3.1

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Abstract

The classical Lighthill-Whitham-Richards (LWR) kinematic traffic model is extended to a unidirectional road on which the maximum density $a(x)$ represents road inhomogeneities, such as variable numbers of lanes, and is allowed to vary discontinuously. The car density $\phi = \phi(x,t)$ is then determined by the following initial value problem for a scalar conservation law with a spatially discontinuous flux:
$\phi_t + (\phi v(\phi/{a(x)})_x = 0, \quad \phi(x,0)=\phi_0(x),\quad x \in \mathbb{R},\quad t\in (0,T),$ (*)

where $v(z)$ is the velocity function. We adapt to (*) a new notion of entropy solutions (Bürger, Karlsen, and Towers [Submitted, 2007]), which involves a Kružkov-type entropy inequality based on a specific flux connection $(A,B)$, and which we interpret in terms of traffic flow. This concept is consistent with both the driver's ride impulse and the desire of drivers to speed up.
We prove that entropy solutions of type $(A,B)$ are unique. This solution concept also leads to simple, transparent, and unified convergence proofs for numerical schemes. Indeed, we adjust to (*) new variants of the Engquist-Osher (EO) scheme (Bürger, Karlsen, and Towers [Submitted, 2007]), and of the Hilliges-Weidlich (HW) scheme analyzed by the authors [ J. Engrg. Math., to appear]. It is proven that the EO and HW schemes and a related Godunov scheme converge to the unique entropy solution of type $(A,B)$ of (*). For the Godunov version, this is the first rigorous convergence and well-posedness result, since no unnecessarily restrictive regularity assumptions are imposed on the solution. Numerical experiments for first-order schemes and formally second-order MUSCL/Runge-Kutta versions are presented.
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