This article is concerned with the approximation of hyperbolic-parabolic cross-diffusion systems modeling segregation phenomena for populations by a fully discrete finite-volume scheme. It is proved that the numerical scheme converges to a dissipative measure-valued solution of the PDE system and that, whenever the latter possesses a strong solution, the convergence holds in the strong sense. Furthermore, the ``parabolic density part'' of the limiting measure-valued solution is atomic and converges to its constant state for long times. The results are based on Young measure theory and a weak-strong stability estimate combining Shannon and Rao entropies. The convergence of the numerical scheme is achieved by means of discrete entropy dissipation inequalities and an artificial diffusion, which vanishes in the continuum limit.

翻译：本文研究采用全离散有限体积格式近似描述种群隔离现象的双曲-抛物交叉扩散系统。我们证明了该数值格式收敛于偏微分方程系统的一个耗散测度值解，且当后者存在强解时，该收敛在强意义下成立。此外，极限测度值解中的"抛物密度分量"呈原子态，并在长时间尺度下收敛至其常值状态。该结果基于Young测度理论以及结合Shannon熵与Rao熵的弱-强稳定性估计。数值格式的收敛性通过离散熵耗散不等式与随连续极限消失的人工扩散实现。