# American Institute of Mathematical Sciences

August  2013, 7(3): 679-695. doi: 10.3934/ipi.2013.7.679

## Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation

 1 College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China 2 Institute of Theoretical and Computational Studies & Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China, China, China

Received  November 2012 Revised  June 2013 Published  September 2013

In this paper, we will investigate the first- and second-order implicit-explicit schemes with parameters for solving the Allen-Cahn equation. It is known that the Allen-Cahn equation satisfies a nonlinear stability property, i.e., the free-energy functional decreases in time. The goal of this paper is to consider implicit-explicit schemes that inherit the nonlinear stability of the continuous model, which will be achieved by properly choosing parameters associated with the implicit-explicit schemes. Theoretical justifications for the nonlinear stability of the schemes will be provided, and the theoretical results will be verified by several numerical examples.
Citation: Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems & Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679
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