Entropy solutions of non-local scalar conservation laws with congestion via deterministic particle method

TL;DR

This paper introduces a deterministic particle scheme for non-local scalar conservation laws with congestion, proving convergence to the entropy solution under broad conditions and demonstrating its effectiveness through numerical simulations, including multi-species scenarios.

Contribution

The paper presents a new particle method for non-local conservation laws with less restrictive assumptions and proves explicit convergence rates to the entropy solution.

Findings

Convergence of the scheme to the entropy solution with explicit rates

Applicability to multi-species models demonstrated through simulations

Method handles discontinuous interaction forces and unbounded mobility

Abstract

We develop deterministic particle schemes to solve non-local scalar conservation laws with congestion. We show that the discrete approximations converge to the unique entropy solution with an explicit rate of convergence under more general assumptions that the existing literature: the velocity fields are less regular (in particular the interaction force can have a discontinuity at the origin) with no prescribed attractive/repulsive regime and the mobility can have unbounded support. We complement our results with some numerical simulations, among which we show the applicability of the schemes to the multi-species setting.