A positivity preserving second-order scheme for multi-dimensional system of non-local conservation laws
Nikhil Manoj, G. D. Veerappa Gowda, Sudarshan Kumar K
TL;DR
This work presents a fully discrete second-order finite-volume scheme for multi-dimensional systems of non-local conservation laws, combining MUSCL-type spatial reconstruction with a two-stage SSP Runge-Kutta time discretization. The scheme employs Lax-Friedrichs fluxes and carefully constructed discrete convolutions to handle nonlocal flux terms of the form f^k(t,x,y,ρ^k,η*ρ) and g^k(t,x,y,ρ^k,ν*ρ). Theoretical contributions include proofs of positivity preservation and \\mathrm{L}^{\\infty} stability under CFL-type conditions, with mass conservation ensured at the discrete level. Numerical experiments in both scalar and system settings demonstrate that the second-order scheme significantly improves accuracy over first-order methods, maintains positivity and stability, and correctly captures the singular limit as the nonlocal horizon vanishes, supporting its robustness and practical applicability.
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.