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

Handle URI:
http://hdl.handle.net/10754/597846
Title:
Continuous limit of a crowd motion and herding model: Analysis and numerical simulations
Authors:
Pietschmann, Jan-Frederik; Markowich, Peter Alexander; Burger, Martin
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.
Publisher:
American Institute of Mathematical Sciences (AIMS)
Journal:
Kinetic and Related Models
KAUST Grant Number:
KUK-I1-007-43
Issue Date:
Nov-2011
DOI:
10.3934/krm.2011.4.1025
Type:
Article
ISSN:
1937-5093
Sponsors:
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.
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Full metadata record

DC FieldValue Language
dc.contributor.authorPietschmann, Jan-Frederiken
dc.contributor.authorMarkowich, Peter Alexanderen
dc.contributor.authorBurger, Martinen
dc.date.accessioned2016-02-25T12:57:42Zen
dc.date.available2016-02-25T12:57:42Zen
dc.date.issued2011-11en
dc.identifier.citationPietschmann 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.en
dc.identifier.issn1937-5093en
dc.identifier.doi10.3934/krm.2011.4.1025en
dc.identifier.urihttp://hdl.handle.net/10754/597846en
dc.description.abstractIn 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.en
dc.description.sponsorshipThis 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.en
dc.publisherAmerican Institute of Mathematical Sciences (AIMS)en
dc.subjectAsymptotic analysisen
dc.subjectContinuum modelen
dc.subjectCrowd motionen
dc.subjectHerdingen
dc.titleContinuous limit of a crowd motion and herding model: Analysis and numerical simulationsen
dc.typeArticleen
dc.identifier.journalKinetic and Related Modelsen
dc.contributor.institutionWestfalische Wilhelms-Universitat Munster, Munster, Germanyen
dc.contributor.institutionUniversity of Cambridge, Cambridge, United Kingdomen
dc.contributor.institutionKing Saud University College of Science, Riyadh, Saudi Arabiaen
kaust.grant.numberKUK-I1-007-43en
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