Deterministic particle approximation of the Hughes model in one space dimension

TL;DR

This paper introduces a deterministic particle method to approximate solutions of Hughes' pedestrian movement model in one dimension, providing theoretical existence results and numerical simulations.

Contribution

It presents a new particle-based approach for Hughes' model with rigorous existence proofs and compares it with a Godunov scheme through numerical tests.

Findings

The particle method accurately approximates the model.

Numerical results align with theoretical predictions.

The approach is effective for symmetric and Riemann data.

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 shortly 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.