Communications in Mathematical Sciences

Volume 14 (2016)

Number 2

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

Pages: 489 – 515



Jing Guo (School of Mathematical Sciences, Soochow University, Suzhou, Jiangsu, China)

Cheng Wang (Department of Mathematics, University of Massachusetts, North Dartmouth, Mass., U.S.A.)

Steven M. Wise (Department of Mathematics, University of Tennessee, Knoxville, Tenn., U.S.A.)

Xingye Yue (School of Mathematical Sciences, Soochow University, Suzhou, Jiangsu, China)


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 $H^2_h$ 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 $L^{\infty}_s (0, T; H^2_h$) norm, for the proposed second order scheme. The results of numerical experiments are presented and confirm the efficiency and accuracy of the scheme.


Cahn–Hilliard equation, finite difference, second-order, energy stability, multigrid, global-in-time $H^2_h$ stability, $L^{\infty}_s (0, T; H^2)$ convergence analysis

2010 Mathematics Subject Classification

35K30, 65M12, 65M55

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