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We propose a structure-preserving finite difference scheme for theCahn-Hilliard equation with a dynamic boundary condition using the discretevariational derivative method (DVDM). In this approach, it is important andessential how to discretize the energy which characterizes the equation. Bymodifying the conventional manner and using an appropriate summation-by-partsformula, we can use a standard central difference operator as an approximationof an outward normal derivative on the discrete boundary condition of thescheme. We show that our proposed scheme is second-order accurate in space,although the previous structure-preserving scheme by Fukao-Yoshikawa-Wada(Commun. Pure Appl. Anal. 16 (2017), 1915-1938) is first-order accurate inspace. Also, we show the stability, the existence, and the uniqueness of thesolution for the proposed scheme. Computation examples demonstrate theeffectiveness of the proposed scheme. Especially through computation examples,we confirm that numerical solutions can be stably obtained by our proposedscheme.

A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a dynamic boundary condition