A convergent finite-volume scheme for nonlocal cross-diffusion systems for multi-species populations

TL;DR

This paper introduces a convergent finite-volume scheme for nonlocal cross-diffusion systems modeling multi-species populations, ensuring key physical properties and providing theoretical convergence and existence results.

Contribution

It develops a novel finite-volume discretization that preserves entropy and nonnegativity, and proves convergence and existence of solutions for nonlocal cross-diffusion models.

Findings

The scheme preserves nonnegativity, mass, and entropy.

Convergence of the scheme to weak solutions is established.

Numerical experiments illustrate solution behaviors.

Abstract

An implicit Euler finite-volume scheme for a nonlocal cross-diffusion system on the one-dimensional torus, arising in population dynamics, is proposed and analyzed. The kernels are assumed to be in detailed balance and satisfy a weak cross-diffusion condition. The latter condition allows for negative off-diagonal coefficients and for kernels defined by an indicator function. The scheme preserves the nonnegativity of the densities, conservation of mass, and production of the Boltzmann and Rao entropies. The key idea is to ``translate'' the entropy calculations for the continuous equations to the finite-volume scheme, in particular to design discretizations of the mobilities, which guarantee a discrete chain rule even in the presence of nonlocal terms. Based on this idea, the existence of finite-volume solutions and the convergence of the scheme are proven. As a by-product, we deduce the existence of weak solutions to the continuous cross-diffusion system. Finally, we present some numerical experiments illustrating the behavior of the solutions to the nonlocal and associated local models.