An$H^2$convergence of a second-order convex-splitting, finite difference scheme for the three-dimensional Cahn–Hilliard equation

In this paper we present an unconditionally solvable and energy stable second order numerical scheme for the three-dimensional (3D) Cahn-Hilliard (CH) equation. The scheme is a two-step method based on a second order convex splitting of the physical energy, combined with a centered difference in space. The equation at the implicit time level is nonlinear but represents the gradients of a strictly convex function and is thus uniquely solvable, regardless of time step-size. The nonlinear equation is solved using an efficient nonlinear multigrid method. In addition, a global in time H-h(2). bound for the numerical solution is derived at the discrete level, and this bound is independent on the final time. As a consequence, an unconditional convergence (for the time step s in terms of the spatial grid size h) is established, in a discrete L-s(infinity) (0,T;H-h(2)) norm, for the proposed second order scheme. The results of numerical experiments are presented and confirm the efficiency and accuracy of the scheme.