Deterministic particle approximation of scalar conservation laws

TL;DR

This paper demonstrates that a microscopic follow-the-leader particle model can rigorously approximate the unique entropy solution of scalar conservation laws, supported by theoretical proof and numerical simulations.

Contribution

It introduces a novel microscopic particle approach that converges to the entropy solution of scalar conservation laws, bridging microscopic models and macroscopic PDEs.

Findings

Proves convergence of particle model to entropy solution

Provides numerical simulations supporting theoretical results

Establishes a rigorous link between microscopic and macroscopic models

Abstract

In this paper we prove that the unique entropy solution to a scalar nonlinear conservation law with strictly monotone velocity and nonnegative initial condition can be rigorously obtained as the large particle limit of a microscopic follow-the-leader type model, which is interpreted as the discrete Lagrangian approximation of the nonlinear scalar conservation law. The result is complemented with some numerical simulations.