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

Xinlong Feng; Huailing Song; Tao Tang; Jiang Yang

doi:10.3934/ipi.2013.7.679

Article Contents

2013, Volume 7, Issue 3: 679-695. Doi: 10.3934/ipi.2013.7.679

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

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

Received: October 31, 2012

Revised: May 31, 2013

Published: July 2013

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Abstract

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.

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Mathematics Subject Classification: 65M06, 65M12, 6506, 35k20, 35Q56.

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