Convergence of a finite volume scheme and dissipative measure-valued--strong stability for a hyperbolic-parabolic cross-diffusion system

Abstract

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.