Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation
Xinlong Feng Huailing Song Tao Tang Jiang Yang
Inverse Problems & Imaging 2013, 7(3): 679-695 doi: 10.3934/ipi.2013.7.679
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.
keywords: Implicit-explicit scheme Allen-Cahn equation energy stability.
The stabilized semi-implicit finite element method for the surface Allen-Cahn equation
Xufeng Xiao Xinlong Feng Jinyun Yuan
Discrete & Continuous Dynamical Systems - B 2017, 22(7): 2857-2877 doi: 10.3934/dcdsb.2017154

Two semi-implicit numerical methods are proposed for solving the surface Allen-Cahn equation which is a general mathematical model to describe phase separation on general surfaces. The spatial discretization is based on surface finite element method while the temporal discretization methods are first-and second-order stabilized semi-implicit schemes to guarantee the energy decay. The stability analysis and error estimate are provided for the stabilized semi-implicit schemes. Furthermore, the first-and second-order operator splitting methods are presented to compare with stabilized semi-implicit schemes. Some numerical experiments including phase separation and mean curvature flow on surfaces are performed to illustrate stability and accuracy of these methods.

keywords: Surface Allen-Cahn equation surface finite element method stabilized semi-implicit scheme operator splitting method error estimate

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