 Tomoya KEMMOCHI.Numerical Analysis of the Allen-Cahn Equation with Coarse Meshes[J].数学研究及应用,2019,39(6):709~717
Numerical Analysis of the Allen-Cahn Equation with Coarse Meshes
Numerical Analysis of the Allen-Cahn Equation with Coarse Meshes

DOI：10.3770/j.issn:2095-2651.2019.06.014

 作者 单位 Tomoya KEMMOCHI Department of Applied Physics, Graduate School of Engineering, Nagoya University, Aichi 464-8603, Japan

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.

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.