Commun. Comput. Phys., 5 (2009), pp. 821-835.

Application of the Local Discontinuous Galerkin Method for the Allen-Cahn/Cahn-Hilliard System

Yinhua Xia 1, Yan Xu 1, Chi-Wang Shu 2*

1 Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China.
2 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA.

Received 12 September 2007; Accepted (in revised version) 6 February 2008
Available online 1 August 2008


In this paper, we consider the application of the local discontinuous Galerkin method for the Allen-Cahn/Cahn-Hilliard system. The method in this paper extends the local discontinuous Galerkin method in \cite{Xia} to the more general application system which is coupled with the Allen-Cahn and Cahn-Hilliard equations. Similar energy stability result as that in \cite{Xia} is presented. Numerical results for the nonlinear problems which include the Allen-Cahn/Cahn-Hilliard system for one-dimensional and two-dimensional cases demonstrate the accuracy and capability of the numerical method.

AMS subject classifications: 65M60, 35K55

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Key words: Allen-Cahn/Cahn-Hilliard system, local discontinuous Galerkin method, free energy, stability.

*Corresponding author.
Email: (Y. Xia), (Y. Xu), (C.-W. Shu)

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