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

Type
Article

Authors
Pietschmann, Jan-Frederik
Markowich, Peter Alexander
Burger, Martin

KAUST Grant Number
KUK-I1-007-43

Online Publication Date
2011-11-17

Print Publication Date
2011

Date
2011-11-17

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 nonconvex obstacles. © American Institute of Mathematical Sciences.

Citation
Pietschmann J-F, Markowich PA, Burger M (2011) Continuous limit of a crowd motion and herding model: Analysis and numerical simulations. KRM 4: 1025–1047. Available: http://dx.doi.org/10.3934/krm.2011.4.1025.

Acknowledgements
This publication is based on work supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST), by the Leverhulme Trust through the research grant entitled Kinetic and mean field partial differential models for socio-economic processes (PI Peter Markowich) and by the Royal Society through the Wolfson Research Merit Award of Peter Markowich. PM is also grateful to the Humboldt foundation and to the Foundation Sciences Mathematiques de Paris for their support. The authors thank Armin Seyfried (Julich, Wuppertal) for hints on the model and explanations on experimental facts and Andreas Schadschneider for providing further literature and details on Monte-Carlo implementations. Furthermore, the authors thank Marie-Therese Wolfram for a large number of helpful and stimulating discussions related to the numerical schemes used in this work.

Publisher
American Institute of Mathematical Sciences (AIMS)

Journal
Kinetic & Related Models

DOI
10.3934/krm.2011.4.1025

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