Cross-diffusion systems coupled via a moving interface

Clément Cancès; Jean Cauvin-Vila; Claire Chainais-Hillairet; Virginie Ehrlacher

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arXiv · 2407.15457

Cross-diffusion systems coupled via a moving interface

Clément Cancès·Jean Cauvin-Vila·Claire Chainais-Hillairet·Virginie Ehrlacher

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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 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.

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Clément Cancès, Jean Cauvin-Vila, Claire Chainais-Hillairet, Virginie Ehrlacher. 2024-07-22. Cross-diffusion systems coupled via a moving interface. https://arxiv.org/abs/2407.15457

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