Follow-the-leader approximations of macroscopic models for vehicular and pedestrian flows

TL;DR

This paper reviews and extends follow-the-leader particle approximation methods for macroscopic traffic and pedestrian flow models, proving convergence and demonstrating numerical consistency for various models and boundary conditions.

Contribution

It introduces new convergence proofs for particle schemes approximating the LWR, Hughes, and ARZ models, including boundary value problems and second-order models.

Findings

Proven convergence of particle schemes for the LWR model on the real line.

Strong $L^1$ convergence of schemes for IBVP with Dirichlet data.

Numerical simulations confirm the consistency of the proposed schemes.

Abstract

We review recent results and present new ones on a deterministic follow-the-leader particle approximation of first and second order models for traffic flow and pedestrian movements. We start by constructing the particle scheme for the first order Lighthill-Whitham-Richards (LWR) model for traffic flow. The approximation is performed by a set of ODEs following the position of discretised vehicles seen as moving particles. The convergence of the scheme in the many particle limit towards the unique entropy solution of the LWR equation is proven in the case of the Cauchy problem on the real line. We then extend our approach to the Initial-Boundary Value Problem (IBVP) with time-varying Dirichlet data on a bounded interval. In this case we prove that our scheme is convergent strongly in $L^1$ up to a subsequence. We then review extensions of this approach to the Hughes model for pedestrian movements and to the second order Aw-Rascle-Zhang (ARZ) model for vehicular traffic. Finally, we complement our results with numerical simulations. In particular, the simulations performed on the IBVP and the ARZ model suggest the consistency of the corresponding schemes, which is easy to prove rigorously in some simple cases.