Numerical schemes for a class of nonlocal conservation laws: a general approach

TL;DR

This paper introduces a general numerical approach for approximating solutions to nonlocal conservation laws, combining quadrature-based nonlocal term approximation with flux functions, ensuring convergence to weak entropy solutions.

Contribution

The paper proposes a novel, general framework for numerically solving nonlocal conservation laws with proven convergence guarantees and broad applicability.

Findings

Numerical examples confirm the theoretical convergence results.

The approach is adaptable to various nonlocal problems.

Explicit conditions for flux functions ensure solution accuracy.

Abstract

In this work we present a rather general approach to approximate the solutions of nonlocal conservation laws. In a first step, we approximate the nonlocal term with an appropriate quadrature rule applied to the spatial discretization. Then, we apply a numerical flux function on the reduced problem. We present explicit conditions which such a numerical flux function needs to fulfill. These conditions guarantee the convergence to the weak entropy solution of the considered model class. Numerical examples validate our theoretical results and demonstrate that the approach can be applied to other nonlocal problems.