A positivity preserving second-order scheme for multi-dimensional system of non-local conservation laws

TL;DR

This paper introduces a second-order numerical scheme for multi-dimensional non-local conservation laws that preserves positivity and stability, improving accuracy over first-order methods.

Contribution

A fully discrete, second-order scheme using MUSCL reconstruction and Runge-Kutta integration for non-local systems, with proven positivity and stability.

Findings

Second-order scheme outperforms first-order in accuracy.

Scheme preserves positivity and ensures L-infinity stability.

Numerical experiments validate theoretical properties.

Abstract

Non-local systems of conservation laws play a crucial role in modeling flow mechanisms across various scenarios. The well-posedness of such problems is typically established by demonstrating the convergence of robust first-order schemes. However, achieving more accurate solutions necessitates the development of higher-order schemes. In this article, we present a fully discrete, second-order scheme for a general class of non-local conservation law systems in multiple spatial dimensions. The method employs a MUSCL-type spatial reconstruction coupled with Runge-Kutta time integration. The proposed scheme is proven to preserve positivity in all the unknowns and exhibits L-infinity stability. Numerical experiments conducted on both the non-local scalar and system cases illustrate the8 importance of second-order scheme when compared to its first-order counterpart.