Structure-preserving schemes for Cahn–Hilliard equations with dynamic boundary conditions

Two kinds of Cahn–Hilliard equations with dynamical boundary conditions have been proposed by Goldstein–Miranville–Schimperna and Liu–Wu, respectively. These models have characteristic conservation and dissipation laws. From the perspective of numerical computation, the properties often lead us to stable computation. Hence, if the designed schemes retain the properties in a discrete sense, then the schemes are expected to be stable. In this paper, we propose structure-preserving schemes for the two-dimensional setting of both models that retain the conservation and dissipation laws in a discrete sense. Also, we discuss the solvability of the proposed scheme for the model of Goldstein–Miranville–Schimperna. Moreover, computation examples demonstrate the effectiveness of our proposed schemes. Especially through computation examples, we confirm that numerical solutions can be stably obtained by our proposed schemes.