A Convergent Structure-Preserving Finite-Volume Scheme for the Shigesada--Kawasaki--Teramoto Population System

Abstract

An implicit Euler finite-volume scheme for an n-species population cross-diffusion system of Shigesada-Kawasaki-Teramoto-type in a bounded domain with no-flux boundary conditions is proposed and analyzed. The scheme preserves the formal gradient-flow or entropy structure and preserves the nonnegativity of the population densities. The key idea is to consider a suitable mean of the mobilities in such a way that a discrete chain rule is fulfilled and a discrete analog of the entropy inequality holds. The existence of finite-volume solutions, the convergence of the scheme, and the large-time asymptotics to the constant steady state are proven. Furthermore, numerical experiments in one and two space dimensiona for two and three species are presented. The results are valid for a more general class of cross-diffusion systems satisfying some structural conditions.