A Bound Preserving Energy Stable Scheme for a Nonlocal Cahn–Hilliard Equation

Rainey Lyons; Adrian Muntean; Grigor Nika

doi:10.5802/crmeca.265

Article de recherche

A Bound Preserving Energy Stable Scheme for a Nonlocal Cahn–Hilliard Equation
[Un schéma stable préservant l’énergie pour une équation de Cahn–Hilliard non locale]

Rainey Lyons ¹ ; Adrian Muntean ¹ ; Grigor Nika ¹

¹ Department of Mathematics and Computer Science, Karlstad University, Sweden

Comptes Rendus. Mécanique, Volume 352 (2024), pp. 239-250.

Résumé
Abstract

Nous présentons un schéma numérique basé sur les volumes finis pour une équation de Cahn–Hilliard non locale qui combine des idées de schémas numériques récents pour les équations de flot de gradient et les équations de Cahn–Hilliard non locales. L’équation en question est un cas particulier d’un système d’équations précédemment dérivé et étudié qui décrit la séparation des phases dans les mélanges ternaires. Nous prouvons que le schéma est à la fois stable sur le plan énergétique et qu’il respecte les limites analytiques de la solution. En outre, nous présentons des démonstrations numériques des résultats théoriques en utilisant les potentiels d’énergie libre de Flory–Huggins (FH) et de Ginzburg–Landau (GL).

We present a finite-volume based numerical scheme for a nonlocal Cahn–Hilliard equation which combines ideas from recent numerical schemes for gradient flow equations and nonlocal Cahn–Hilliard equations. The equation of interest is a special case of a previously derived and studied system of equations which describes phase separation in ternary mixtures. We prove the scheme is both energy stable and respects the analytical bounds of the solution. Furthermore, we present numerical demonstrations of the theoretical results using both the Flory–Huggins (FH) and Ginzburg–Landau (GL) free-energy potentials.

Reçu le : 2024-04-05
Accepté le : 2024-08-25
Accepté après révision le : 2024-08-29
Publié le : 2024-11-19

DOI : 10.5802/crmeca.265

Classification : 65M08, 35R09, 35Q70
Keywords: Nonlocal Cahn–Hilliard equation, gradient flow, finite-volume method, bound preserving energy stable schemes
Mot clés : Équation de Cahn–Hilliard non locale, flot de gradient, méthode des volumes finis, schémas stables de préservation de l’énergie

Affiliations des auteurs :

Rainey Lyons ¹ ; Adrian Muntean ¹ ; Grigor Nika ¹

¹ Department of Mathematics and Computer Science, Karlstad University, Sweden

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Rainey Lyons and Adrian Muntean and Grigor Nika},
     title = {A {Bound} {Preserving} {Energy} {Stable} {Scheme} for a {Nonlocal} {Cahn{\textendash}Hilliard} {Equation}},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {239--250},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {352},
     year = {2024},
     doi = {10.5802/crmeca.265},
     language = {en},
}

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Rainey Lyons; Adrian Muntean; Grigor Nika. A Bound Preserving Energy Stable Scheme for a Nonlocal Cahn–Hilliard Equation. Comptes Rendus. Mécanique, Volume 352 (2024), pp. 239-250. doi : 10.5802/crmeca.265. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.265/

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