A Class of Stable and Conservative Finite Difference Schemes for the Cahn-Hilliard Equation

In this paper, we propose a class of stable finite difference schemes for the initial-boundary value problem of the Cahn-Hilliard equation. These schemes are proved to inherit the total mass conservation and energy dissipation in the discrete level. The dissipation of the total energy implies boundness of the numerical solutions in the discrete H-1 norm. This in turn implies boundedness of the numerical solutions in the maximum norm and hence the stability of the difference schemes. Unique existence of the numerical solutions is proved by the fixed-point theorem. Convergence rate of the class of finite difference schemes is proved to be O(h(2) + tau(2)) with time step tau and mesh size h. An efficient iterative algorithm for solving these nonlinear schemes is proposed and discussed in detail.