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

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

doi:10.3934/nhm.2016.11.29

Networks and Heterogeneous Media

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

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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: 01 April 2015 Revised: 01 September 2015
Primary: 35L65; Secondary: 90B20.

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.
- Road traffic modeling,
- point constraint,
- Aw-Rascle and Zhang model,
- entropies,
- renormalization,
- admissible solutions,
- numerical experiments
Citation: Boris P. Andreianov, Carlotta Donadello, Ulrich Razafison, Julien Y. Rolland, Massimiliano D. Rosini. Solutions of the Aw-Rascle-Zhang system with point constraints[J]. Networks and Heterogeneous Media, 2016, 11(1): 29-47. doi: 10.3934/nhm.2016.11.29

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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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