A first-order energy stable scheme for the Allen–Cahn equation with the Allen–Cahn type dynamic boundary condition

Abstract

In this paper we construct and analyze a first-order time marching scheme for the Allen–Cahn equation with the Allen–Cahn type dynamic boundary condition. To hold the mass conservation in the bulk and on the surface respectively, we add two penalty terms to the model. By using the stabilization method, we verify that the scheme is energy stable and is of first-order in time by the error estimates. Finally we present enough numerical results to validate the convergence and the energy stability of such scheme. Moreover, we simulate the shape deformation of a droplet and the moving process of the contact line.