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Volume 34, Issue 3
Unconditional Bound-Preserving and Energy-Dissipating Finite-Volume Schemes for the Cahn-Hilliard Equation

Rafael Bailo, José A. Carrillo, Serafim Kalliadasis & Sergio P. Perez

DOI: 10.4208/cicp.OA-2023-0049

Commun. Comput. Phys., 34 (2023), pp. 713-748.

Published online: 2023-10

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  • Abstract

We propose finite-volume schemes for the Cahn-Hilliard equation which unconditionally and discretely preserve the boundedness of the phase field and the dissipation of the free energy. Our numerical framework is applicable to a variety of free-energy potentials, including Ginzburg-Landau and Flory-Huggins, to general wetting boundary conditions, and to degenerate mobilities. Its central thrust is the upwind methodology, which we combine with a semi-implicit formulation for the free-energy terms based on the classical convex-splitting approach. The extension of the schemes to an arbitrary number of dimensions is straightforward thanks to their dimensionally split nature, which allows to efficiently solve higher-dimensional problems with a simple parallelisation. The numerical schemes are validated and tested through a variety of examples, in different dimensions, and with various contact angles between droplets and substrates.

  • Keywords

Cahn-Hilliard equation, diffuse interface theory, gradient flow, finite-volume method, bound preservation, energy dissipation.

  • AMS Subject Headings

65M08, 35Q92, 35Q35, 35Q70

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-34-713, author = {Bailo , RafaelCarrillo , José A.Kalliadasis , Serafim and Perez , Sergio P.}, title = {Unconditional Bound-Preserving and Energy-Dissipating Finite-Volume Schemes for the Cahn-Hilliard Equation}, journal = {Communications in Computational Physics}, year = {2023}, volume = {34}, number = {3}, pages = {713--748}, abstract = {

We propose finite-volume schemes for the Cahn-Hilliard equation which unconditionally and discretely preserve the boundedness of the phase field and the dissipation of the free energy. Our numerical framework is applicable to a variety of free-energy potentials, including Ginzburg-Landau and Flory-Huggins, to general wetting boundary conditions, and to degenerate mobilities. Its central thrust is the upwind methodology, which we combine with a semi-implicit formulation for the free-energy terms based on the classical convex-splitting approach. The extension of the schemes to an arbitrary number of dimensions is straightforward thanks to their dimensionally split nature, which allows to efficiently solve higher-dimensional problems with a simple parallelisation. The numerical schemes are validated and tested through a variety of examples, in different dimensions, and with various contact angles between droplets and substrates.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2023-0049}, url = {http://global-sci.org/intro/article_detail/cicp/22022.html} }
TY - JOUR T1 - Unconditional Bound-Preserving and Energy-Dissipating Finite-Volume Schemes for the Cahn-Hilliard Equation AU - Bailo , Rafael AU - Carrillo , José A. AU - Kalliadasis , Serafim AU - Perez , Sergio P. JO - Communications in Computational Physics VL - 3 SP - 713 EP - 748 PY - 2023 DA - 2023/10 SN - 34 DO - http://doi.org/10.4208/cicp.OA-2023-0049 UR - https://global-sci.org/intro/article_detail/cicp/22022.html KW - Cahn-Hilliard equation, diffuse interface theory, gradient flow, finite-volume method, bound preservation, energy dissipation. AB -

We propose finite-volume schemes for the Cahn-Hilliard equation which unconditionally and discretely preserve the boundedness of the phase field and the dissipation of the free energy. Our numerical framework is applicable to a variety of free-energy potentials, including Ginzburg-Landau and Flory-Huggins, to general wetting boundary conditions, and to degenerate mobilities. Its central thrust is the upwind methodology, which we combine with a semi-implicit formulation for the free-energy terms based on the classical convex-splitting approach. The extension of the schemes to an arbitrary number of dimensions is straightforward thanks to their dimensionally split nature, which allows to efficiently solve higher-dimensional problems with a simple parallelisation. The numerical schemes are validated and tested through a variety of examples, in different dimensions, and with various contact angles between droplets and substrates.

Bailo , RafaelCarrillo , José A.Kalliadasis , Serafim and Perez , Sergio P.. (2023). Unconditional Bound-Preserving and Energy-Dissipating Finite-Volume Schemes for the Cahn-Hilliard Equation. Communications in Computational Physics. 34 (3). 713-748. doi:10.4208/cicp.OA-2023-0049
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