Fully discrete follow-the-leader approximation of one-dimensional scalar conservation laws with vacuum

TL;DR

This paper introduces a fully discrete particle method for one-dimensional scalar conservation laws that can handle vacuum states, proving convergence to the entropy solution under certain conditions.

Contribution

It develops a novel vacuum-compatible particle approximation scheme with convergence guarantees for scalar conservation laws.

Findings

Converges to the unique entropy weak solution.

Handles vacuum states effectively.

Provides a new discrete approximation framework.

Abstract

We present a fully discrete particle approximation for one-dimensional scalar conservation laws. Under suitable monotonicity assumptions on the macroscopic velocity, we construct a vacuum-compatible family of time-discrete particle equations and show that an appropriate piecewise-constant density reconstruction from the particle setting converges to the unique entropy weak solution of the macroscopic scalar conservation law.