Solutions of the Aw-Rascle-Zhang system with point constraints

Boris P.  Andreianov; Carlotta Donadello; Ulrich Razafison; Julien Y.  Rolland; Massimiliano D.  Rosini

doi:10.3934/nhm.2016.11.29

Article Contents

2016, Volume 11, Issue 1: 29-47. Doi: 10.3934/nhm.2016.11.29

This issue Previous Article A combined finite volume - finite element scheme for a dispersive shallow water system Next Article Biological and industrial models motivating nonlocal conservation laws: A review of analytic and numerical results

Solutions of the Aw-Rascle-Zhang system with point constraints

1.
LMB Laboratoire de Mathématiques CNRS UMR6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex
2.
Laboratoire de Mathématiques CNRS UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex
3.
LMB Laboratoire de Mathématiques CNRS UMR6623, Université de Franche-Comté, 16 route de Gray, 25030 Besancon Cedex
4.
Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, pl. Marii Curie-Skłodowskiej 1, 20-031 Lublin

Received: March 31, 2015

Revised: August 31, 2015

Published: February 2016

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Abstract

We revisit the entropy formulation and the wave-front tracking construction of physically admissible solutions of the Aw-Rascle and Zhang (ARZ) ``second-order'' model for vehicular traffic. A Kruzhkov-like family of entropies is introduced to select the admissible shocks. This tool allows to define rigorously the appropriate notion of admissible weak solution and to approximate the solutions of the ARZ model with point constraint. Stability of solutions w.r.t. strong convergence is justified. We propose a finite volumes numerical scheme for the constrained ARZ, and we show that it can correctly locate contact discontinuities and take the constraint into account.

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Mathematics Subject Classification: Primary: 35L65; Secondary: 90B20.

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