General constrained conservation laws. Application to pedestrian flow modeling

Christophe Chalons; Paola Goatin; Nicolas Seguin; Christophe Chalons; Paola Goatin; Nicolas Seguin

doi:10.3934/nhm.2013.8.433

Networks and Heterogeneous Media

2013, Volume 8, Issue 2: 433-463. doi: 10.3934/nhm.2013.8.433

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General constrained conservation laws. Application to pedestrian flow modeling

1.
Univ. Paris Diderot, Sorbonne Paris Cité, Laboratoire Jacques-Louis Lions, UMR 7598, UPMC, CNRS, F-75205 Paris
2.
INRIA Sophia Antipolis - Méditerranée, EPI OPALE, 2004, route des Lucioles - BP 93, 06902 Sophia Antipolis Cedex
3.
UPMC Univ Paris 06 and CNRS UMR 7598, Laboratoire J.-L. Lions, 75005 Paris

Received: 01 July 2012 Revised: 01 November 2012
Primary: 35L65; Secondary: 90B20.

We extend the results on conservation laws with local flux constraint obtained in [2, 12] to general (non-concave) flux functions and non-classical solutions arising in pedestrian flow modeling [15]. We first provide a well-posedness result based on wave-front tracking approximations and the Kružhkov doubling of variable technique for a general conservation law with constrained flux. This provides a sound basis for dealing with non-classical solutions accounting for panic states in the pedestrian flow model introduced by Colombo and Rosini [15]. In particular, flux constraints are used here to model the presence of doors and obstacles. We propose a "front-tracking" finite volume scheme allowing to sharply capture classical and non-classical discontinuities. Numerical simulations illustrating the Braess paradox are presented as validation of the method.
- Constrained scalar conservation laws,
- finite volume schemes,
- non-classical shocks,
- macroscopic pedestrian flow models,
- Braess paradox
Citation: Christophe Chalons, Paola Goatin, Nicolas Seguin. General constrained conservation laws. Application to pedestrian flow modeling[J]. Networks and Heterogeneous Media, 2013, 8(2): 433-463. doi: 10.3934/nhm.2013.8.433

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Abstract

We extend the results on conservation laws with local flux constraint obtained in [2, 12] to general (non-concave) flux functions and non-classical solutions arising in pedestrian flow modeling [15]. We first provide a well-posedness result based on wave-front tracking approximations and the Kružhkov doubling of variable technique for a general conservation law with constrained flux. This provides a sound basis for dealing with non-classical solutions accounting for panic states in the pedestrian flow model introduced by Colombo and Rosini [15]. In particular, flux constraints are used here to model the presence of doors and obstacles. We propose a "front-tracking" finite volume scheme allowing to sharply capture classical and non-classical discontinuities. Numerical simulations illustrating the Braess paradox are presented as validation of the method.

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