Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case

Takeshi Fukao; Shuji Yoshikawa; Saori Wada

doi:10.3934/cpaa.2017093

Article Contents

2017, Volume 16, Issue 5: 1915-1938. Doi: 10.3934/cpaa.2017093

This issue Previous Article Dynamics of some stochastic chemostat models with multiplicative noise Next Article Corrigendum to "On small data scattering of Hartree equations with short-range interaction" [Comm. Pure. Appl. Anal., 15 (2016), 1809-1823]

Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case

1.
Department of Mathematics, Kyoto University of Education, 1 Fujinomori, Fukakusa, Fushimi-ku, Kyoto, 612-8522, Japan
2.
Division of Mathematical Sciences, Department of Integrated Science and Technology, Faculty of Science and Technology, Oita University, 700 Dannoharu, Oita, 870-1192, Japan
3.
Department of Engineering for Production and Environment, Graduate School of Science and Engineering, Ehime University, Bunkyo-cho 3, Matsuyama, Ehime, 790-8577, Japan

* Corresponding author: yoshikawa@oita-u.ac.jp
* Corresponding author: yoshikawa@oita-u.ac.jp

Received: October 2016

Revised: April 2017

Published online: May 2017

The authors are supported by JSPS KAKENHI Grant-in-Aid for Scientific Research(C), Grant Numbers 26400164 for TF and 16K05234 for SY.

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Abstract

The structure-preserving finite difference schemes for the one dimensional Cahn-Hilliard equation with dynamic boundary conditions are studied. A dynamic boundary condition is a sort of transmission condition that includes the time derivative, namely, it is itself a time evolution equation. The Cahn-Hilliard equation with dynamic boundary conditions is well-treated from various viewpoints. The standard type consists of a dynamic boundary condition for the order parameter, and the Neumann boundary condition for the chemical potential. Recently, Goldstein-Miranville-Schimperna proposed a new type of dynamic boundary condition for the Cahn-Hilliard equation. In this article, numerical schemes for the problem with these two kinds of dynamic boundary conditions are introduced. In addition, a mathematical result on the existence of a solution for the scheme with an error estimate is also obtained for the former boundary condition.

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Mathematics Subject Classification: Primary: 65N06; Secondary: 35K35.

Citation:

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Figure 1. Relationship between $x$ and $k$

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Figure 2. Discrete quantities $A_{1,d}^{(n)}$ and $M_d({{\mathit{\boldsymbol{U}}}^{\left( n \right)}})$

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Figure 3. Discrete quantities $A_{2,d}^{(n)}$ and $M_{2,d}^{(n)}$

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Figure 4. Case 1 for $\gamma =0.005$

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Figure 5. Case 1 for $\gamma =0.001$

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Figure 6. Case 1 for $\gamma =0.0005$

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Figure 7. Case 2 for $\gamma =0.005$

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Figure 8. Case 2 for $\gamma =0.001$

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Figure 9. Case 2 for $\gamma =0.0005$

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