  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.
