Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case

Abstract

The structure-preserving finite difference schemes for the one dimensional Cahn-Hilliard equation with dynamic boundary conditions are studied. A dynamic boundary condition is a sort of transmission condition that includes the time derivative, namely, it is itself a time evolution equation. The Cahn-Hilliard equation with dynamic boundary conditions is well-treated from various viewpoints. The standard type consists of a dynamic boundary condition for the order parameter, and the Neumann boundary condition for the chemical potential. Recently, Goldstein-Miranville-Schimperna proposed a new type of dynamic boundary condition for the Cahn-Hilliard equation. In this article, numerical schemes for the problem with these two kinds of dynamic boundary conditions are introduced. In addition, a mathematical result on the existence of a solution for the scheme with an error estimate is also obtained for the former boundary condition.