Convergence of a Finite Volume Scheme for a System of Interacting Species with Cross-Diffusion

TL;DR

This paper proves the convergence of a positivity-preserving finite volume scheme for a coupled system of non-local PDEs with cross-diffusion, demonstrating numerical accuracy and the emergence of segregated states.

Contribution

It introduces a new convergence proof for a finite volume scheme applied to a coupled cross-diffusion system, ensuring positivity and capturing segregation phenomena.

Findings

Numerical convergence to reference solutions with first order accuracy.

Recovery of segregated stationary states despite regularization.

Observation of mixing when self- or cross-diffusion is strong enough.

Abstract

In this work we present the convergence of a positivity preserving semi-discrete finite volume scheme for a coupled system of two non-local partial differential equations with cross-diffusion. The key to proving the convergence result is to establish positivity in order to obtain a discrete energy estimate to obtain compactness. We numerically observe the convergence to reference solutions with a first order accuracy in space. Moreover we recover segregated stationary states in spite of the regularising effect of the self-diffusion. However, if the self-diffusion or the cross-diffusion is strong enough, mixing occurs while both densities remain continuous.