On existence, stability and many-particle approximation of solutions of 1D Hughes model with linear costs

TL;DR

This paper studies the 1D Hughes pedestrian flow model with linear costs, proving existence and stability of entropy solutions, including non-classical shocks, using a particle approximation scheme and numerical simulations.

Contribution

It introduces a new existence proof for solutions with non-classical shocks in the Hughes model using a particle scheme and local reductions, extending previous results.

Findings

Existence of solutions with non-classical shocks is established.

The particle approximation scheme effectively captures complex behaviors.

Numerical simulations demonstrate the impact of parameters on evacuation times.

Abstract

This paper deals with the one-dimensional formulation of Hughes model for pedestrian flows in the setting of entropy solutions, which authorizes non-classical shocks at the location of the so-called turning curve. We consider linear cost functions, whose slopes $\alpha$ 0 correspond to different crowd behaviours. We prove existence and partial well-posedness results in the framework of entropy solutions. The proofs of existence are based on a a sharply formulated many-particle approximation scheme with careful treatment of interactions of particles with the turning curve, and on local reductions to the well-known Lighthill-Whitham-Richards model. For the special case of BV-regular entropy solutions without non-classical shocks, locally Lipschitz continuous dependence of such solutions on the initial datum $\rho$ and on the cost parameter $\alpha$ is proved. Differently from the stability argument and from existence results available in the literature, our existence result allows for the possible presence of non-classical shocks. First, we explore convergence of the many-particle approximations under the assumption of uniform space variation control. Next, by a local compactness argument that permits to circumvent the possible absence of global BV bounds, we obtain existence of solutions for general measurable data. Finally, we illustrate numerically that the model is able to reproduce typical behaviours in case of evacuation. Special attention is devoted to the impact of the parameter $\alpha$ on the evacuation time.