A first-order hyperbolic reformulation of the Cahn-Hilliard equation
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910559683739648 |
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| author | Dhaouadi, Firas Dumbser, Michael Gavrilyuk, Sergey |
| author_facet | Dhaouadi, Firas Dumbser, Michael Gavrilyuk, Sergey |
| contents | In this paper we present a new first-order hyperbolic reformulation of the Cahn-Hilliard equation. The model is obtained from the combination of augmented Lagrangian techniques proposed earlier by the authors of this paper, with a classical Cattaneo-type relaxation that allows to reformulate diffusion equations as augmented first order hyperbolic systems with stiff relaxation source terms. The proposed system is proven to be hyperbolic and to admit a Lyapunov functional, in accordance with the original equations. A new numerical scheme is proposed to solve the original Cahn-Hilliard equations based on conservative semi-implicit finite differences, while the hyperbolic system was numerically solved by means of a classical second order MUSCL-Hancock-type finite volume scheme. The proposed approach is validated through a set of classical benchmarks such as spinodal decomposition, Ostwald ripening and exact stationary solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03862 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A first-order hyperbolic reformulation of the Cahn-Hilliard equation Dhaouadi, Firas Dumbser, Michael Gavrilyuk, Sergey Numerical Analysis Analysis of PDEs 35K55, 35L65, 35L03, 65M06, 65M08 In this paper we present a new first-order hyperbolic reformulation of the Cahn-Hilliard equation. The model is obtained from the combination of augmented Lagrangian techniques proposed earlier by the authors of this paper, with a classical Cattaneo-type relaxation that allows to reformulate diffusion equations as augmented first order hyperbolic systems with stiff relaxation source terms. The proposed system is proven to be hyperbolic and to admit a Lyapunov functional, in accordance with the original equations. A new numerical scheme is proposed to solve the original Cahn-Hilliard equations based on conservative semi-implicit finite differences, while the hyperbolic system was numerically solved by means of a classical second order MUSCL-Hancock-type finite volume scheme. The proposed approach is validated through a set of classical benchmarks such as spinodal decomposition, Ostwald ripening and exact stationary solutions. |
| title | A first-order hyperbolic reformulation of the Cahn-Hilliard equation |
| topic | Numerical Analysis Analysis of PDEs 35K55, 35L65, 35L03, 65M06, 65M08 |
| url | https://arxiv.org/abs/2408.03862 |