An adaptive finite-volume method  for a model of two-phase pedestrian flow

Stefan Berres; Ricardo Ruiz-Baier; Hartmut Schwandt; Elmer M. Tory; Stefan Berres; Ricardo Ruiz-Baier; Hartmut Schwandt; Elmer M. Tory

doi:10.3934/nhm.2011.6.401

Networks and Heterogeneous Media

2011, Volume 6, Issue 3: 401-423. doi: 10.3934/nhm.2011.6.401

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An adaptive finite-volume method for a model of two-phase pedestrian flow

1.
Departamento de Ciencias Matemáticas y Físicas, Universidad Católica de Temuco, Temuco
2.
Modeling and Scientific Computing, MATHISCE, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne
3.
Institut für Mathematik, Fakultät II Mathematik und Naturwissenschaften, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin
4.
Department of Mathematics and Computer Science, Mount Allison University, Sackville, NB E4L 1G6

Received: 01 December 2010 Revised: 01 June 2011
Primary: 92D25, 35L65; Secondary: 65M50.

A flow composed of two populations of pedestrians moving in different directions is modeled by a two-dimen\-sional system of convection-diffusion equations. An efficient simulation of the two-dimensional model is obtained by a finite-volume scheme combined with a fully adaptive multiresolution strategy. Numerical tests show the flow behavior in various settings of initial and boundary conditions, where different species move in countercurrent or perpendicular directions. The equations are characterized as hyperbolic-elliptic degenerate, with an elliptic region in the phase space, which in one space dimension is known to produce oscillation waves. When the initial data are chosen inside the elliptic region, a spatial segregation of the populations leads to pattern formation. The entries of the diffusion-matrix determine the stability of the model and the shape of the patterns.
- Crowd model,
- Multiphase flow,
- System of conservation laws,
- Mixed hyperbolic-elliptic system,
- Elliptic region,
- Fully adaptive multiresolution
Citation: Stefan Berres, Ricardo Ruiz-Baier, Hartmut Schwandt, Elmer M. Tory. An adaptive finite-volume method for a model of two-phase pedestrian flow[J]. Networks and Heterogeneous Media, 2011, 6(3): 401-423. doi: 10.3934/nhm.2011.6.401

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Abstract

A flow composed of two populations of pedestrians moving in different directions is modeled by a two-dimen\-sional system of convection-diffusion equations. An efficient simulation of the two-dimensional model is obtained by a finite-volume scheme combined with a fully adaptive multiresolution strategy. Numerical tests show the flow behavior in various settings of initial and boundary conditions, where different species move in countercurrent or perpendicular directions. The equations are characterized as hyperbolic-elliptic degenerate, with an elliptic region in the phase space, which in one space dimension is known to produce oscillation waves. When the initial data are chosen inside the elliptic region, a spatial segregation of the populations leads to pattern formation. The entries of the diffusion-matrix determine the stability of the model and the shape of the patterns.

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