Finite Volume Method for a System of Continuity Equations Driven by Nonlocal Interactions

TL;DR

This paper introduces a novel finite volume numerical scheme for a system of nonlocal continuity equations modeling interacting species, extending previous work on aggregation equations with rough velocity fields.

Contribution

The paper develops a new finite volume method specifically designed for nonlocal transport systems with nonlinear, rough velocity fields, and provides convergence analysis and numerical validation.

Findings

Method effectively approximates nonlocal transport equations

Numerical simulations demonstrate accuracy and stability

Analysis confirms convergence properties

Abstract

We present a new finite volume method for computing numerical approximations of a system of nonlocal transport equation modeling interacting species. This method is based on the work [F. Delarue, F. Lagoutire, N. Vauchelet, Convergence analysis of upwind type schemes for the aggregation equation with pointy potential, Ann. Henri. Lebesgue 2019], where the nonlocal continuity equations are 10 treated as conservative transport equations with a nonlocal, nonlinear, rough velocity field. We analyze some properties of the method, and illustrate the results with numerical simulations.