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December  2010, 28(4): 1669-1691. doi: 10.3934/dcds.2010.28.1669

Numerical approximations of Allen-Cahn and Cahn-Hilliard equations

Jie Shen 1, and  Xiaofeng Yang 2,

1. 

School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China
2. 

Department of Mathematics, University of South Carolina, Columbia, SC 29208, United States

Received  October 2009 Revised  February 2010 Published  June 2010

Stability analyses and error estimates are carried out for a number of commonly used numerical schemes for the Allen-Cahn and Cahn-Hilliard equations. It is shown that all the schemes we considered are either unconditionally energy stable, or conditionally energy stable with reasonable stability conditions in the semi-discretized versions. Error estimates for selected schemes with a spectral-Galerkin approximation are also derived. The stability analyses and error estimates are based on a weak formulation thus the results can be easily extended to other spatial discretizations, such as Galerkin finite element methods, which are based on a weak formulation.
Keywords: Error Analysis, Spectral Method, Cahn-Hilliard, Allen-Cahn, Stability..
Mathematics Subject Classification: Primary: 65N35, 65M15, 65M70; Secondary: 65Z0.
Citation: Jie Shen, Xiaofeng Yang. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1669-1691. doi: 10.3934/dcds.2010.28.1669
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