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

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

Martin Burger^*, Peter Alexander Markowich, Jan Frederik Pietschmann

^*Corresponding author for this work

Applied Mathematics and Computational Science

Research output: Contribution to journal › Article › peer-review

38 Scopus citations

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.

Original language	English (US)
Pages (from-to)	1025-1047
Number of pages	23
Journal	Kinetic and Related Models
Volume	4
Issue number	4
DOIs	https://doi.org/10.3934/krm.2011.4.1025
State	Published - Dec 2011

Bibliographical note

KAUST Repository Item: Exported on 2019-02-13
Acknowledged KAUST grant number(s): KUK-I1-007-43
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.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

Keywords

Asymptotic analysis
Continuum model
Crowd motion
Herding

ASJC Scopus subject areas

Numerical Analysis
Modeling and Simulation

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10.3934/krm.2011.4.1025

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title = "Continuous limit of a crowd motion and herding model: Analysis and numerical simulations",

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.",

keywords = "Asymptotic analysis, Continuum model, Crowd motion, Herding",

author = "Martin Burger and Markowich, {Peter Alexander} and Pietschmann, {Jan Frederik}",

note = "KAUST Repository Item: Exported on 2019-02-13 Acknowledged KAUST grant number(s): KUK-I1-007-43 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. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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N2 - 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.

AB - 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.

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

Abstract

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Keywords

ASJC Scopus subject areas

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