A note on the central-upwind scheme for nonlocal conservation laws

TL;DR

This paper derives and analyzes a central-upwind numerical scheme for nonlocal conservation laws, proving convergence and comparing it with other schemes through numerical examples.

Contribution

It provides a detailed derivation of the fully-discrete scheme for nonlocal laws and extends the central-upwind flux to this class, including convergence proofs.

Findings

The first-order scheme converges to the correct solution.

The second-order scheme also converges under certain conditions.

Numerical tests show the scheme's effectiveness compared to Godunov-type methods.

Abstract

The central-upwind flux is a widely used numerical flux function for local conservation laws. It has been investigated by Kurganov and Polizzi (2009) for a specific nonlocal conservation law and can be derived from a fully-discrete second-order scheme. Here, we derive this fully-discrete scheme in detail with a particular focus on the occurring nonlocal terms. In addition, we derive the central-upwind flux for a class of nonlocal conservation laws and use an estimate on the nonlocal speed which fixes the nonlocality at the cell interfaces. We prove that the resulting first-order numerical scheme converges to the correct solution. Under additional assumptions on the analytical flux we present a similar result for a second-order central-upwind scheme. Numerical examples compare the central-upwind schemes to Godunov-type schemes and the fully-discrete scheme.