A discrete Hughes model for pedestrian flow on graphs

Fabio Camilli; Adriano Festa; Silvia Tozza

doi:10.3934/nhm.2017004

Article Contents

2017, Volume 12, Issue 1: 93-112. Doi: 10.3934/nhm.2017004

This issue Previous Article Modelling heterogeneity and an open-mindedness social norm in opinion dynamics Next Article A mathematical framework for delay analysis in single source networks

A discrete Hughes model for pedestrian flow on graphs

1.
Dip. di Scienze di Base e Applicate per l'Ingegneria, "Sapienza" Università di Roma, via Scarpa 16, 00161 Roma, Italy
2.
RICAM, Austrian Academy of Sciences (ÖAW), Altenbergerstr. 69, 4040 Linz, Austria
3.
Dip. di Matematica, "Sapienza" Università di Roma, P.le Aldo Moro 5, 00185 Roma, Italy

* Corresponding author:Fabio Camilli
* Corresponding author:Fabio Camilli

Received: March 31, 2016

Revised: November 30, 2016

Published: May 2017

AF is supported the Austrian Academy of Sciences ÖAW via the New Frontiers Group NST-001.

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Abstract

In this paper, we introduce a discrete time-finite state model for pedestrian flow on a graph in the spirit of the Hughes dynamic continuum model. The pedestrians, represented by a density function, move on the graph choosing a route to minimize the instantaneous travel cost to the destination. The density is governed by a conservation law whereas the minimization principle is described by a graph eikonal equation. We show that the discrete model is well-posed and the numerical examples reported confirm the validity of the proposed model and its applicability to describe real situations.

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Mathematics Subject Classification: Primary: 90B20; Secondary: 35R02, 65M06.

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Figure 1. Scheme of the network and initial density.

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Figure 2. Test 1 (Dirichlet boundary conditions): density and potential before the first time of interaction.

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Figure 3. Test 1 (Dirichlet boundary conditions): density and potential after the first time of interaction at $(0.2,-0.8)$.

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Figure 4. Test 2 (No-flux boundary conditions): stable configuration obtained for $t>3.5$.

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Figure 5. Test 3 (Dirichlet BCs with diffusion $\epsilon=1$): Density at two different time steps ($t=0.75$ and $t=1.75$).

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Figure 6. The Wuhan Sports Centre (left) and the evacuation network considered in our study (right).

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Figure 7. Initial distribution of density on the graph (up) and drift potential in the initial configuration (down).

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Figure 8. Initial distribution of density on the graph (up) and drift potential in the initial configuration (down).

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