Convergence of the non-staggered Nessyahu-Tadmor scheme for coupled systems of one-dimensional nonlocal balance laws

TL;DR

This paper develops a second-order non-staggered central scheme for coupled nonlocal balance laws, proving boundedness and convergence to weak or entropy solutions, with numerical validation.

Contribution

It introduces a novel second-order non-staggered scheme for nonlocal balance laws and establishes convergence properties under various regularity assumptions.

Findings

Boundedness of approximate solutions over time

Weak convergence to solutions under linearity

Strong convergence to entropy solutions with regular kernels

Abstract

We derive a second-order accurate, non-staggered central scheme based on the well-known Nessyahu-Tadmor scheme to approximate solutions of coupled systems of nonlocal balance laws. We show that the approximate solutions stay bounded by an exponential $L^\infty$ bound in time. Under linearity assumptions on the flux and source terms the approximate solutions converge weakly-$*$ to weak solutions of the nonlocal balance laws. Assuming stronger regularity, in particular on the convolution kernel, we show strong convergence towards entropy weak solutions in the nonlinear case. Numerical examples validate our results and demonstrate its applicability to various systems of nonlocal problems.