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Riemann Problem Resolution and Godunov Scheme for the Aw-Rascle-Zhang Model

Technical Department for Transport, Roads and Bridges (SETRA), F-92220 Bagneux, Cedex, France
INRETS/Gretia, "Les Descartes 2," F-93166 Noisy le Grand, Cedex, France
INRETS/Gretia, "Les Descartes 2," F-93166 Noisy le Grand, Cedex, France
salim.mammar{at}developpement-durable.gouv.fr
lebacque{at}inrets.fr
haj-salem{at}inrets.fr

Recently, Aw and Rascle and Zhang introduced a new second-order macroscopic model, following the theoretical investigations on the Payne-Whitham second-order modelling by Daganzo. The conserved variables in this model are the density and the relative flow. The aim of this paper is to solve the Riemann problem for the Aw-Rascle-Zhang (ARZ) model for all possible initial conditions, using an extended fundamental diagram. The resolution of the Riemann problem provides the user of the model with a set of nontrivial analytical solutions; it is also a prerequisite for the construction of numerical solution schemes. Some examples are given in which analytical solutions of the Riemann problem are discussed and compared to numerical solutions. The ARZ model shows better fit to real data than the embedded Lighthill-Whitham-Richards (LWR) model for the same set of physical parameters.

Key Words: Riemann problem; analytical solutions; numerical solutions; shock waves; rarefaction waves; contact discontinuities; discretisation
History: Received: January 2007; revised: November 2008; accepted: March 2009.