On the Eikonal equation in the pedestrian flow problem

A hierarchy of models for pedestrian flow with fixed speed is numerically investigated. The starting point is a microscopic model based on a stochastic interacting particle system coupled to an eikonal equation. Starting from this model a nonlocal and nonlinear flux-limited maximum-entropy equation for density and mean velocity is derived via a mean field kinetic equation. Finally, associated classical scalar equations for the density are considered for comparison. These models are compared to each other for different test cases showing the superiority of the flux-limited approach, in particular for situations with smaller values of the stochastic noise.

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Discontinuous Galerkin method for the pedestrian flow problem

10.1063/1.5043669 ◽

2018 ◽

Author(s):

J. Felcman ◽

V. Dolejší ◽

P. Kubera

Keyword(s):

Galerkin Method ◽

Discontinuous Galerkin ◽

Discontinuous Galerkin Method ◽

Flow Problem ◽

Pedestrian Flow

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On a numerical flux for the pedestrian flow equations

Journal of Applied Mathematics Statistics and Informatics ◽

10.1515/jamsi-2015-0014 ◽

2015 ◽

Vol 11 (2) ◽

pp. 79-96 ◽

Cited By ~ 3

Author(s):

P. Kubera ◽

J. Felcman

Keyword(s):

Eikonal Equation ◽

Source Term ◽

Operator Splitting ◽

Pedestrian Flow ◽

Hyperbolic Problems ◽

Hyperbolic Problem ◽

Flow Equations ◽

Numerical Flux ◽

Homogeneity Property ◽

The Right

Abstract The pedestrian flow equations are formulated as the hyperbolic problem with a source term, completed by the eikonal equation yielding the desired direction of the pedestrian velocity. The operator splitting consisting of successive discretization of the eikonal equation, ordinary differential equation with the right hand side being the source term and the homogeneous hyperbolic system is proposed. The numerical flux of the Vijayasundaram type is proposed for the finite volume solution of the hyperbolic problem. The Vijayasundaram numerical flux, originally proposed for the hyperbolic problems possessing the homogeneity property is extended for pedestrian flow, where the homogeneity property is lost. The application of the proposed numerical flux is demonstrated on the physically relevant problem.

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A Cellular Automaton Model for a Pedestrian Flow Problem

Mathematical Modelling of Natural Phenomena ◽

10.1051/mmnp/2021002 ◽

2021 ◽

Author(s):

Jiri Felcman ◽

Petr Kubera

Keyword(s):

Cellular Automaton ◽

Eikonal Equation ◽

Flow Model ◽

Time Discretization ◽

Implicit Time ◽

Two Dimensional ◽

Pedestrian Flow ◽

Finite Set ◽

Ca Model ◽

Volume Method

The evacuation phenomena in the two dimensional pedestrian flow model are simulated. The intended direction of the escape of pedestrians in panic situations is governed by the Eikonal equation of the pedestrian flow model. A new two-dimensional Cellular Automaton (CA) model is proposed for the simulation of the pedestrian flow. The solution of the Eikonal equation is used to define the probability matrix whose elements express the probability of a pedestrian moving in finite set of directions. The novelty of this paper lies in the construction of the density dependent probability matrix. The relevant evacuation scenarios are numerically solved. Predictions of the evacuation behavior of pedestrians, for various room geometries with multiple exists, are demonstrated. The mathematical model is numerically justified by comparison of CA approach with the Finite Volume Method for the space discretization and Discontinuous Galerkin Method for the implicit time discretization of pedestrian flow model.

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