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 twostep 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 H2 h 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 Ls (0,T ;H 2 h) norm, for the proposed second order scheme. The results of numerical experiments are presented and confirm the efficiency and accuracy of the scheme.

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