A First-Order Linear Energy Stable Scheme for the Cahn–Hilliard Equation with Dynamic Boundary Conditions Under the Effect of Hyperbolic Relaxation

Abstract

In this paper we focus on the Cahn–Hilliard equation with dynamic boundary conditions, by adding two hyperbolic relaxation terms to the system. We verify that the energy of the total system is decreasing with time. By adding two stabilization terms, we have constructed a first-order temporal accuracy numerical scheme, which is linear and energy stable. Then we prove that the scheme is of first-order in time by the error estimates. At last we present comprehensive numerical results to validate the the temporal convergence and the energy stability of such scheme. Moreover, we present the differences of the numerical results with and without the hyperbolic terms, which show that the hyperbolic terms can help the total energy decreasing slowly.