Deterministic particle approximation of the Hughes model in one space dimension

Marco Di Francesco; Simone Fagioli; Massimiliano Daniele Rosini; Giovanni Russo

doi:10.3934/krm.2017009

Article Contents

2017, Volume 10, Issue 1: 215-237. Doi: 10.3934/krm.2017009

This issue Previous Article Self-organized hydrodynamics with density-dependent velocity Next Article A kinetic equation for economic value estimation with irrationality and herding

Deterministic particle approximation of the Hughes model in one space dimension

1.
DISIM, Università degli Studi dell'Aquila, via Vetoio 1 (Coppito), 67100 L'Aquila (AQ), Italy
2.
Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, plac Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland
3.
Dipartimento di Matematica ed Informatica, Università di Catania, Viale Andrea Doria 6,95125 Catania, Italy

* Corresponding author: Marco Di Francesco
* Corresponding author: Marco Di Francesco

Received: February 29, 2016

Revised: June 30, 2016

Published: September 2017

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Abstract

In this paper we present a new approach to the solution to a generalized version of Hughes' models for pedestrian movements based on a follow-the-leader many particle approximation. In particular, we provide a rigorous global existence result under a smallness assumption on the initial data ensuring that the trace of the solution along the turning curve is zero for all positive times. We also focus briefly on the approximation procedure for symmetric data and Riemann type data. Two different numerical approaches are adopted for the simulation of the model, namely the proposed particle method and a Godunov type scheme. Several numerical tests are presented, which are in agreement with the theoretical prediction.

Keywords:

Mathematics Subject Classification: Primary: 35L65; Secondary: 90B20.

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Figure 1. Evolution of $R(t,x)$ with initial data $\bar{\rho}(x)=0.25$ at times $t=0$, $t=0.5$ and $t=1$. In the particle simulations the blu dots represent particles positions, whereas the red line is the discretized density. The magenta vertical line describes the turning point evolution

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Figure 2. Evolution of $R(t,x)$ with initial data $\bar{\rho}(x)=0.6$ at times $t=0$, $t=0.5$ and $t=1$

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Figure 3. Evolution of $R(t,x)$ with initial data $\bar{\rho}(x)$ given in (24)

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Figure 4. Evolution of $R(t,x)$ with initial data $\bar{\rho}(x)$ given in (25)

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Figure 5. Evolution of $R(t,x)$ with initial data $\bar{\rho}(x)$ given in (26)

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Figure 6. Mass transfer across the turning point and non-classical shock with initial data $\bar{\rho}$ given in (26)

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Figure 7. Comparison between the Follow-the-Leader scheme (in red) and the Godunov scheme (in blu)

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Figure 8. Increasing the number of particles, and then the cells integration for the Godunov method, the agreement between the two methods greatly improves. Here we consider the initial datum $\bar{\rho}=0.3\times\,\mathbf{1}_{[-1,0]}+0.7\times \,\mathbf{1}_{(0,1]}$ and we set $N=1000$ with $1500$ time iterations

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