Numerical Analysis of the Allen-Cahn Equation with Coarse Meshes
Received:August 24, 2019  Revised:October 12, 2019
Key Words: Allen-Cahn equation   finite difference method   asymptotic behavior   maximum principle  
Fund Project:Supported by JSPS KAKENHI (Grant No.19K14590), Japan.
Author NameAffiliation
Tomoya KEMMOCHI Department of Applied Physics, Graduate School of Engineering, Nagoya University, Aichi 464-8603, Japan 
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      In this paper, we consider the finite difference semi-discretization of the Allen-Cahn equation with the diffuse interface parameter $\varepsilon$. While it is natural to make the mesh size parameter $h$ smaller than $\varepsilon$, it is desirable that $h$ is as big as possible in view of computational costs. In fact, when $h$ is bigger than $\varepsilon$ (i.e., the mesh is relatively coarse), it is observed that the numerical solution does not move at all. The purpose of this paper is to clarify the mechanism of this phenomenon. We will prove that the numerical solution converges to that of the ordinary equation without the diffusion term if $h$ is bigger than $\varepsilon$. Numerical examples are presented to support the result.
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