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 development of a theoretical framework of global measure-valued solutions for a class of hyperbolic–parabolic cross-diffusion systems, and its application to the convergence analysis of a fully discrete finite-volume scheme. After introducing an appropriate notion of dissipative measure-valued solutions to the PDE system, a numerical scheme is proposed which is shown to generate, in the continuum limit, a dissipative measure-valued solution. The “parabolic density part” of the limiting measure-valued solution is atomic and converges to its constant state for long times. Furthermore, it is proved that whenever the PDE system possesses a strong solution, the convergence of the approximation scheme holds in the strong sense. 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.