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

Abstract

R ecently, 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.

A Survey on Visual Traffic Simulation: Models, Evaluations, and Applications in Autonomous Driving

“…The resulting model is referred to as the Aw-Rascle-Zhang (ARZ) model, which has been thoroughly studied since [Ras02, GP06, LMHS07, MR07]. Mammar et al [MLS09] showed that the ARZ model fits real-world data better than the LWR model numerically.…”

“…Mammar et al . [MLS09] showed that the ARZ model fits real‐world data better than the LWR model numerically.…”

Prediction of traffic convective instability with spectral analysis of the Aw–Rascle–Zhang model

“…The derivations in [28] and [29] showed how to embed this model in a network of junctions. Godunov discretization scheme and numerical comparison showed in [30] that ARZ model fits real data better than the LWR model. Relaxed approximations with constant wave have also been used to provide numerical solutions in [31].…”

“…Previous studies focused on averaged errors and only displayed predictions at a couple of points along the freeway [30,60]. Here, we present an entire map of the states and conduct model assessment in a holistic manner.…”

On the modelling and management of traffic

“…We stress that the numerical method used here, i.e. wave front tracking, as well as the general framework described below, may be easily extended substituting the LWR equation with the 2-phase models [7,10,14,15,19] or also with the classical Aw-Rascle model [5,42]. It is natural to conjecture that the results below can be extended to several of these more refined models, however the necessary analytical techniques get more involved.…”