A deterministic particle approximation for non-linear conservation laws

TL;DR

This paper reviews deterministic particle schemes for approximating one-dimensional conservation laws, including pedestrian and traffic models, with convergence results and numerical validation.

Contribution

It introduces new particle approximation schemes for several conservation law models and proves their convergence in the many particle limit.

Findings

Convergence of particle schemes demonstrated

Numerical simulations confirm scheme consistency

Applicable to pedestrian and vehicular traffic models

Abstract

We review our analytical and numerical results obtained on the microscopic Follow-The-Leader (FTL) many particle approximation of one-dimensional conservation laws. More precisely, we introduce deterministic particle schemes for the Hughes model for pedestrian movements and for two vehicular traffic models, that are the scalar Lighthill-Whitham-Richards model (LWR) and the $2\times2$ system Aw-Rascle-Zhang model (ARZ). Their approximation is performed by a set of ODEs, determining the motion of platoons of possible fractional vehicles or pedestrians seen as particles. Convergence results of the schemes in the many particle limit are stated. The numerical simulations suggest the consistency of the schemes.