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

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