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 modelling 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 modelled 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 modelled 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.

翻译：本文提出并研究了一个一维模型，该模型由两个通过移动界面耦合的交叉扩散系统组成。其建模动机源于薄膜气相沉积过程中复杂扩散过程的描述。在我们的模型中，不同化学物质的交叉扩散可分别通过固相的尺寸排阻系统和气相的Stefan-Maxwell系统进行建模。两相之间的耦合采用Butler-Volmer型线性相变定律描述，从而形成界面演化。我们研究了该模型的连续性质，特别是其熵变分结构和稳态。我们引入了一种两点通量近似的有限体积格式。移动界面采用动网格方法处理，网格在界面周围进行局部变形。所得离散非线性系统被证明存在保持连续系统主要性质的解，包括：质量守恒、非负性、体积填充约束、自由能衰减及渐近行为。特别地，动网格方法与连续模型的熵结构具有兼容性。数值结果验证了这些性质并展示了模型的动态行为。