Numerical approximations of  Allen-Cahn and Cahn-Hilliard  equations

Jie Shen; Xiaofeng Yang

doi:10.3934/dcds.2010.28.1669

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2010, Volume 28, Issue 4: 1669-1691. Doi: 10.3934/dcds.2010.28.1669

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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
2.
Department of Mathematics, University of South Carolina, Columbia, SC 29208

Received: September 30, 2009

Revised: January 31, 2010

Published: November 2010

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Abstract

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.

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Mathematics Subject Classification: Primary: 65N35, 65M15, 65M70; Secondary: 65Z05.

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