In this talk, we consider a Cahn-Hilliard equation with dynamic boundary condition (Fukao Yoshikawa & Wada 2017). For this system, we propose a structure-preserving scheme using the discrete variational derivative method. In this method, how to discretize the energy which characterizes the equation is essential. Modifying the conventional manner and using appropriate summation-by-parts formula, we can use a standard central difference operator as an approximation of an outward normal derivative on the discrete boundary condition of the scheme. Furthermore, we prove the L∞-boundedness, the existence and uniqueness of the solution for the proposed scheme, and the error estimate. Also, we propose structure-preserving schemes for the Allen-Cahn equation with a dynamic boundary condition and the GMS model (Goldstein, Miranville & Schimperna 2011).

 

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