Cross-diffusion systems coupled via a moving interface

Abstract

We propose and study a one-dimensional model which consists of two cross-diffusion systems coupled via a moving interface. The motivation stems from the modeling of complex diffusion processes in the context of the vapor deposition of thin films. In our model, cross-diffusion of the various chemical species can be, respectively, modeled by a size-exclusion system for the solid phase and the Stefan–Maxwell system for the gaseous phase. The coupling between the two phases is modeled by linear phase transition laws of Butler–Volmer type, resulting in an interface evolution. The continuous properties of the model are investigated, in particular its entropy variational structure and stationary states. We introduce a two-point flux approximation finite-volume scheme. The moving interface is addressed with a moving-mesh approach, where the mesh is locally deformed around the interface. The resulting discrete nonlinear system is shown to admit a solution that preserves the main properties of the continuous system, namely, mass conservation, nonnegativity, volume-filling constraints, decay of the free energy, and asymptotics. In particular, the moving-mesh approach is compatible with the entropy structure of the continuous model. Numerical results illustrate these properties and the dynamics of the model.