A SECOND-ORDER ACCURATE STRUCTURE-PRESERVING SCHEME FOR THE CAHN-HILLIARD EQUATION WITH A DYNAMIC BOUNDARY CONDITION

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) proposed by Furihata and Matsuo [14]. 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 cen-tral 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 proposed by Fukao-Yoshikawa-Wada [13] is first-order ac-curate in space. Also, we show the stability, the existence, and the uniqueness of the solution for our proposed scheme. Computation examples demonstrate the effectiveness of our proposed scheme. Especially through computation ex-amples, we confirm that numerical solutions can be stably obtained by our proposed scheme.