Cross-diffusion systems coupled via a moving interface
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| author | Cancès, Clément Cauvin-Vila, Jean Chainais-Hillairet, Claire Ehrlacher, Virginie |
| author_facet | Cancès, Clément Cauvin-Vila, Jean Chainais-Hillairet, Claire Ehrlacher, Virginie |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_15457 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cross-diffusion systems coupled via a moving interface Cancès, Clément Cauvin-Vila, Jean Chainais-Hillairet, Claire Ehrlacher, Virginie Analysis of PDEs Numerical Analysis 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. |
| title | Cross-diffusion systems coupled via a moving interface |
| topic | Analysis of PDEs Numerical Analysis |
| url | https://arxiv.org/abs/2407.15457 |