# American Institute of Mathematical Sciences

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September  2017, 16(5): 1915-1938. doi: 10.3934/cpaa.2017093

## 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

Received  October 2016 Revised  April 2017 Published  May 2017

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

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.

Citation: Takeshi Fukao, Shuji Yoshikawa, Saori Wada. Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1915-1938. doi: 10.3934/cpaa.2017093
##### References:

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##### References:
Relationship between $x$ and $k$
Discrete quantities $A_{1,d}^{(n)}$ and $M_d({{\mathit{\boldsymbol{U}}}^{\left( n \right)}})$
Discrete quantities $A_{2,d}^{(n)}$ and $M_{2,d}^{(n)}$
Case 1 for $\gamma =0.005$
Case 1 for $\gamma =0.001$
Case 1 for $\gamma =0.0005$
Case 2 for $\gamma =0.005$
Case 2 for $\gamma =0.001$
Case 2 for $\gamma =0.0005$
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