Structure-preserving schemes for Cahn–Hilliard equations with dynamic boundary conditions

Makoto Okumura; Takeshi Fukao

doi:10.3934/dcdss.2023207

Article Contents

2024, Volume 17, Issue 1: 362-394. Doi: 10.3934/dcdss.2023207

This issue Previous Article On a two-scale phasefield model for topology optimization Next Article Abstract action spaces and their topological and dynamic properties

Structure-preserving schemes for Cahn–Hilliard equations with dynamic boundary conditions

Makoto Okumura^1, , and
Takeshi Fukao^2,

1.
Department of Intelligence and Informatics, Konan University, 6-1 Nishiokamoto, Higashinada-Ku, Kobe-Shi, Hyogo, 658-0073, Japan
2.
Faculty of Advanced Science and Technology, Ryukoku University, 1-5 Yokotani, Seta Oe-Cho, Otsu-Shi, Shiga 520-2194, Japan

^*Corresponding author: Makoto Okumura
^*Corresponding author: Makoto Okumura

Received: June 1, 2023

Revised: October 1, 2023

Early access: November 13, 2023

Published online: November 13, 2023

This work was supported by JSPS KAKENHI Grant Number JP21K03309, JP23K13009.

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Abstract

Two kinds of Cahn–Hilliard equations with dynamical boundary conditions have been proposed by Goldstein–Miranville–Schimperna and Liu–Wu, respectively. These models have characteristic conservation and dissipation laws. From the perspective of numerical computation, the properties often lead us to stable computation. Hence, if the designed schemes retain the properties in a discrete sense, then the schemes are expected to be stable. In this paper, we propose structure-preserving schemes for the two-dimensional setting of both models that retain the conservation and dissipation laws in a discrete sense. Also, we discuss the solvability of the proposed scheme for the model of Goldstein–Miranville–Schimperna. Moreover, computation examples demonstrate the effectiveness of our proposed schemes. Especially through computation examples, we confirm that numerical solutions can be stably obtained by our proposed schemes.

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Mathematics Subject Classification: 65M06.

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Figure 1. The initial data

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Figure 2. Numerical solutions to Scheme 1 for the (GMS) model at time $ t = 0.0125 $ (top left), $ t = 0.025 $ (top right), $ t = 0.0375 $ (bottom left), $ t = 0.05 $ (bottom right)

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Figure 3. Numerical solutions to Scheme 2 for the (LW) model at time $ t = 0.0125 $ (top left), $ t = 0.025 $ (top right), $ t = 0.0375 $ (bottom left), $ t = 0.05 $ (bottom right)

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Figure 4. Time development of $ M_{\rm d}(\boldsymbol{U}^{(n)}) $ obtained by Scheme 1 for the (GMS) model

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Figure 5. Time development of $ E_{\rm d}(\boldsymbol{U}^{(n)}) $ obtained by Scheme 1 for the (GMS) model

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Figure 6. Time development of $ M_{\rm d}^{\rm b}(\boldsymbol{U}^{(n)}) $ obtained by Scheme 2 for the (LW) model

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Figure 7. Time development of $ M_{{\rm d}, 0}^{\rm s}(\boldsymbol{U}^{(n)}) $ obtained by Scheme 2 for the (LW) model

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Figure 8. Time development of $ M_{{\rm d}, K_{2}}^{\rm s}(\boldsymbol{U}^{(n)}) $ obtained by Scheme 2 for the (LW) model

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Figure 9. Time development of $ E_{\rm d}(\boldsymbol{U}^{(n)}) $ obtained by Scheme 2 for the (LW) model

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