Continuous limit of a crowd motion and herding model: Analysis and numerical simulations

Martin Burger; Peter Alexander Markowich; Jan-Frederik Pietschmann

doi:10.3934/krm.2011.4.1025

Article Contents

2011, Volume 4, Issue 4: 1025-1047. Doi: 10.3934/krm.2011.4.1025

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Continuous limit of a crowd motion and herding model: Analysis and numerical simulations

1.
Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms-Universität (WWU) Münster, Einsteinstr. 62, D-48149 Münster
2.
DAMTP, University of Cambridge, Cambridge CB3 0WA

Received: May 31, 2011

Revised: August 31, 2011

Published: November 2011

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Abstract

In this paper we study the continuum limit of a cellular automaton model used for simulating human crowds with herding behaviour. We derive a system of non-linear partial differential equations resembling the Keller-Segel model for chemotaxis, however with a non-monotone interaction. The latter has interesting consequences on the behaviour of the model's solutions, which we highlight in its analysis. In particular we study the possibility of stationary states, the formation of clusters and explore their connection to congestion.
We also introduce an efficient numerical simulation approach based on an appropriate hybrid discontinuous Galerkin method, which in particular allows flexible treatment of complicated geometries. Extensive numerical studies also provide a better understanding of the strengths and shortcomings of the herding model, in particular we examine trapping effects of crowds behind non-convex obstacles.

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Mathematics Subject Classification: Primary: 35K57, 35Q91.

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