Second-order accurate TVD numerical methods for nonlocal nonlinear conservation laws

TL;DR

This paper introduces a second-order accurate, TVD numerical scheme for nonlocal nonlinear conservation laws, ensuring convergence to entropy solutions and analyzing solution regularity, with numerical experiments validating the approach.

Contribution

It develops a second-order reconstruction-based numerical method for nonlocal conservation laws that converges to the entropy solution and explores solution regularity.

Findings

The method is total variation diminishing (TVD) and converges to the entropy solution.

Discontinuities in solutions are stationary under certain conditions.

Numerical experiments confirm the accuracy and shock formation in the model.

Abstract

We present a second-order accurate numerical method for a class of nonlocal nonlinear conservation laws called the "nonlocal pair-interaction model" which was recently introduced by Du, Huang, and LeFloch. Our numerical method uses second-order accurate reconstruction-based schemes for local conservation laws in conjunction with appropriate numerical integration. We show that the resulting method is total variation diminishing (TVD) and converges towards a weak solution. In fact, in contrast to local conservation laws, our second-order reconstruction-based method converges towards the unique entropy solution provided that the nonlocal interaction kernel satisfies a certain growth condition near zero. Furthermore, as the nonlocal horizon parameter in our method approaches zero we recover a well-known second-order method for local conservation laws. In addition, we answer several questions from the paper from Du, Huang, and LeFloch concerning regularity of solutions. In particular, we prove that any discontinuity present in a weak solution must be stationary and that, if the interaction kernel satisfies a certain growth condition, then weak solutions are unique. We present a series of numerical experiments in which we investigate the accuracy of our second-order scheme, demonstrate shock formation in the nonlocal pair-interaction model, and examine how the regularity of the solution depends on the choice of flux function.