On the Hughes Model and Numerical Aspects

TL;DR

This paper analyzes Hughes' crowd model, establishing theoretical estimates, exploring solution behaviors including shocks and congestion, and introducing a new numerical method with illustrative examples.

Contribution

It provides a comprehensive study of the Hughes model, including theoretical analysis, shock formation insights, and a novel numerical approach.

Findings

Identification of shock formation mechanisms

Demonstration of congestion and model breakdown

Introduction of a new numerical method

Abstract

We study a crowd model proposed by R. Hughes and we describe a numerical approach to solve it. The Hughes model comprises a Fokker-Planck equation coupled with an eikonal equation with Dirichlet or Neumann data. First, we establish a priori estimates for the solutions. Second, we study radial solutions and identify a shock formation mechanism. Third, we illustrate the existence of congestion, the breakdown of the model, and the trend to the equilibrium. Finally, we propose a new numerical method and consider two numerical examples.