  We propose a structure-preserving finite difference scheme for the
Cahn-Hilliard equation with a dynamic boundary condition using the discrete
variational derivative method (DVDM). In this approach, it is important and
essential how to discretize the energy which characterizes the equation. By
modifying the conventional manner and using an 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. 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 in
space. Also, we show the stability, the existence, and the uniqueness of the
solution for the proposed scheme. Computation examples demonstrate the
effectiveness of the proposed scheme. Especially through computation examples,
we confirm that numerical solutions can be stably obtained by our proposed
scheme.
