Commun. Comput. Phys., 11 (2012), pp. 1261-1278.


An Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation

Zhengru Zhang 1, Zhonghua Qiao 2*

1 Laboratory of Mathematics and Complex Systems, Ministry of Education; School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China.
2 Institute for Computational Mathematics and Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong.

Received 30 August 2010; Accepted (in revised version) 14 April 2011
Available online 29 November 2011
doi:10.4208/cicp.300810.140411s

Abstract

This paper studies the numerical simulations for the Cahn-Hilliard equation which describes a phase separation phenomenon. The numerical simulation of the Cahn-Hilliard model needs very long time to reach the steady state, and therefore large time-stepping methods become useful. The main objective of this work is to construct the unconditionally energy stable finite difference scheme so that the large time steps can be used in the numerical simulations. The equation is discretized by the central difference scheme in space and fully implicit second-order scheme in time. The proposed scheme is proved to be unconditionally energy stable and mass-conservative. An error estimate for the numerical solution is also obtained with second order in both space and time. By using this energy stable scheme, an adaptive time-stepping strategy is proposed, which selects time steps adaptively based on the variation of the free energy against time. The numerical experiments are presented to demonstrate the effectiveness of the adaptive time-stepping approach.

AMS subject classifications: 35L64, 65M93, 65M30

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Key words: Adaptive time-stepping, unconditionally energy stable, Cahn-Hilliard equation, mass conservation.

*Corresponding author.
Email: zrzhang@bnu.edu.cn (Z. R. Zhang), zqiao@hkbu.edu.hk (Z. H. Qiao)
 

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