A pedestrian flow model with stochastic velocities: Microscopic and macroscopic approaches

Simone Göttlich; Stephan Knapp; Peter Schillen

doi:10.3934/krm.2018052

Article Contents

2018, Volume 11, Issue 6: 1333-1358. Doi: 10.3934/krm.2018052

This issue Previous Article Solution to the Boltzmann equation in velocity-weighted Chemin-Lerner type spaces Next Article On the convergence to critical scaling profiles in submonolayer deposition models

A pedestrian flow model with stochastic velocities: Microscopic and macroscopic approaches

Department of Mathematics, University of Mannheim, 68131 Mannheim, Germany

* Corresponding author: S. Göttlich
* Corresponding author: S. Göttlich

Received: March 2017

Revised: December 2017

Published online: June 13, 2018

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Abstract

We investigate a stochastic model hierarchy for pedestrian flow. Starting from a microscopic social force model, where the pedestrians switch randomly between the two states stop-or-go, we derive an associated macroscopic model of conservation law type. Therefore we use a kinetic mean-field equation and introduce a new problem-oriented closure function. Numerical experiments are presented to compare the above models and to show their similarities.

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Mathematics Subject Classification: Primary: 90B20, 65Cxx, 35L60.

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Figure 1. Velocity vector at the boundary

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Figure 2. Overview of the deterministic and stochastic model hierarchy equations

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Figure 3. Densities at different times: $u_{ij}^{\text{Mic}, n}$ for the microscopic and $u_{ij}^{\text{Mac}, n}$ for the macroscopic model

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Figure 4. Mass balances at $x = -1$ and $x = 0$

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Figure 5. $L^1$ and $L^2$ error

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Figure 6. Densities for $\lambda_1$ at different times: $u_{ij}^{\text{Mic}, n}$ for the microscopic and $u_{ij}^{\text{Mac}, n}$ for the macroscopic model

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Figure 7. Densities for $\lambda_2$ at different times: $u_{ij}^{\text{Mic}, n}$ for the microscopic and $u_{ij}^{\text{Mac}, n}$ for the macroscopic model

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Figure 8. Mass balances at $x = 1$

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Figure 9. $L^1$ and $L^2$ errors

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Table 1. Numerical error and EOOC for the first example

	$ \mathsf{err} $	EOOC
$\Delta x = {}^{1}\!\!\diagup\!\!{}_{5}\;$	0.3251	-
$\Delta x = {}^{1}\!\!\diagup\!\!{}_{10}\;$	0.1755	0.8897
$\Delta x = {}^{1}\!\!\diagup\!\!{}_{20}\;$	0.0717	1.2919

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Table 2. Numerical error and EOOC for the second example with rate function $\lambda_1$

	$\mathsf{err} $	EOOC
$\displaystyle \Delta x = {}^{1}\!\!\diagup\!\!{}_{5}\;$	0.4457	-
$\displaystyle \Delta x = {}^{1}\!\!\diagup\!\!{}_{10}\;$	0.2215	1.0085
$\displaystyle \Delta x = {}^{1}\!\!\diagup\!\!{}_{20}\;$	0.0889	1.3170

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Table 3. Numerical error and EOOC for the second example with rate function $\lambda_2$

	$ \mathsf{err} $	EOOC
$\displaystyle \Delta x = {}^{1}\!\!\diagup\!\!{}_{5}\;$	0.5203	-
$\displaystyle \Delta x = {}^{1}\!\!\diagup\!\!{}_{10}\;$	0.2873	0.8567
$\displaystyle \Delta x = {}^{1}\!\!\diagup\!\!{}_{20}\;$	0.1153	1.3176

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