# American Institute of Mathematical Sciences

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

## Numerical approximations of Allen-Cahn and Cahn-Hilliard equations

 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.
Citation: Jie Shen, Xiaofeng Yang. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1669-1691. doi: 10.3934/dcds.2010.28.1669
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