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A. Jüngel, A. Zurek:
"A converegent structure-preserving finite-volume scheme for the Shigesada-Kawasaki-Teramota population system";
in: "ASC Report 34/2020", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2020, ISBN: 978-3-902627-13-1, 1 - 32.

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.

Cross-diffusion system, population dynamics, finite-volume method, discreteentropy dissipation, convergence of the scheme, large-time asymptotics.

http://www.asc.tuwien.ac.at/preprint/2020/asc34x2020.pdf