Communications in Mathematical Sciences

Volume 15 (2017)

Number 6

On the stabilization size of semi-implicit Fourier-spectral methods for 3D Cahn–Hilliard equations

Pages: 1489 – 1506



Dong Li (Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada; and Department of Mathematics, Hong Kong University of Science and Technology, Kowloon, Hong Kong)

Zhonghua Qiao (Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong)


The stabilized semi-implicit time-stepping method is an efficient algorithm to simulate phased field problems with fourth order dissipation. We consider the 3D Cahn–Hilliard equation and prove unconditional energy stability of the corresponding stabilized semi-implicit Fourier spectral scheme independent of the time step. We do not impose any Lipschitz-type assumption on the nonlinearity. It is shown that the size of the stabilization term depends only on the initial data and the diffusion coefficient. Unconditional Sobolev bounds of the numerical solution are obtained and the corresponding error analysis under nearly optimal regularity assumptions is established.


Cahn–Hilliard, energy stable, large time stepping, semi-implicit

2010 Mathematics Subject Classification

35Q35, 65M15, 65M70

D. Li was supported by an Nserc discovery grant. The research of Z. Qiao is partially supported by the Hong Kong Research Grant Council GRF grants 202112, 15302214 and NSFC/RGC Joint Research Scheme N HKBU204/12.

Received 13 April 2016

Published 27 June 2017