SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 2286-2309, January 2021.
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 analogue of the entropy inequality holds. The existence of finite-volume solutions and the convergence of the scheme are proven. Furthermore, numerical experiments in one and two space dimensions for two and three species are presented. The results are valid for a more general class of cross-diffusion systems satisfying some structural conditions.

中文翻译：

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重定-川崎-寺本种群系统的收敛结构保持有限体积方案

SIAM 数值分析杂志，第 59 卷，第 4 期，第 2286-2309 页，2021 年 1 月。
提出并分析了在无通量边界条件下的有界域中重定-川崎-寺本型的$n$-物种种群交叉扩散系统的隐式欧拉有限体积方案。该方案保留了正式的梯度流或熵结构，并保留了人口密度的非负性。关键思想是以这样一种方式考虑迁移率的合适平均值，即满足离散链规则并且熵不等式的离散类似物成立。证明了有限体积解的存在性和方案的收敛性。此外，还介绍了两个和三个物种的一维和二维空间维度的数值实验。结果对于满足某些结构条件的更一般的交叉扩散系统是有效的。