# American Institute of Mathematical Sciences

August  2020, 40(8): 4927-4960. doi: 10.3934/dcds.2020206

## A structure-preserving scheme for the Allen–Cahn equation with a dynamic boundary condition

 1 Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University 1-5 Yamadaoka, Suita, Osaka 565-0871, Japan 2 Cybermedia Center, Osaka University, 1-32 Machikaneyama, Toyonaka, Osaka 560-0043, Japan

* Corresponding author: okumura@cas.cmc.osaka-u.ac.jp

Received  October 2019 Revised  January 2020 Published  May 2020

We propose a structure-preserving finite difference scheme for the Allen–Cahn equation with a dynamic boundary condition using the discrete variational derivative method [9]. In this method, how to discretize the energy which characterizes the equation is essential. Modifying the conventional manner and using an appropriate summation-by-parts formula, we can use a central difference operator as an approximation of an outward normal derivative on the boundary condition in the scheme. We show the stability and the existence and uniqueness of the solution for the proposed scheme. Also, we give the error estimate for the scheme. Numerical experiments demonstrate the effectiveness of the proposed scheme. Besides, through numerical experiments, we confirm that the long-time behavior of the solution under a dynamic boundary condition may differ from that under the Neumann boundary condition.

Citation: Makoto Okumura, Daisuke Furihata. A structure-preserving scheme for the Allen–Cahn equation with a dynamic boundary condition. Discrete & Continuous Dynamical Systems - A, 2020, 40 (8) : 4927-4960. doi: 10.3934/dcds.2020206
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##### References:
Numerical solution of (1) with (71) obtained by our scheme
Time development of total energy
Numerical solution of (1) with (71) obtained by our scheme
Time development of total energy
Numerical solution of (1) with (100) obtained by our scheme
Time development of total energy
Numerical solution of (1) with (100) obtained by our scheme
Time development of total energy
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