求解极坐标系下Allen-Cahn方程的有限差分方法
A Finite Difference Method for the Allen-Cahn Equation in Polar Coordinate System
DOI: 10.12677/AAM.2021.101013, PDF,    科研立项经费支持
作者: 霍俊蓉, 张荣培, 刘 昊：沈阳师范大学，数学与系统科学学院，辽宁 沈阳
关键词: Allen-Cahn方程；极坐标；有限差分；积分因子；Allen-Cahn Equation； Polar Coordinates； Finite Difference； Integration Factor
摘要: 非线性Allen-Cahn方程是材料学中相场模拟模型的一类重要方程，用于描述二元合金在一定温度下进行相位分离的过程，在许多科学领域中有广泛的应用。以往的工作主要在矩形区域上考虑求解，本文研究有效的数值方法在圆形区域上求解该方程。首先将Allen-Cahn方程中拉普拉斯算子写为极坐标的形式，然后采用中心差分方法在空间r方向和θ方向分别进行空间离散，得到非线性常微分方程组，并将网格上的数值解以矩阵形式表示。在时间离散过程中，采用积分因子法结合Krylov子空间的方法进行求解。最后给出数值试验。
Abstract: The nonlinear Allen-Cahn equation is an important phase field simulation model in materials science. It is used to describe the phase separation process of binary alloys at a certain temperature and has been widely used in many scientific fields. The previous work mainly considers the solution on the rectangular region. In this paper, the effective numerical method is used to solve the equation on a circular region. First, the Laplacian operator in Allen-Cahn equation is written as the form of polar coordinates. Then, the central difference method is used to carry out spatial discretization in the space direction r and θ respectively, and the numerical solution on the grid is expressed in matrix form by using large sparse linear system. In the implementation of time discretization, the integral factor method combined with Krylov subspace method was used to solve the problem. Finally, numerical experiments are given.
文章引用：霍俊蓉, 张荣培, 刘昊. 求解极坐标系下Allen-Cahn方程的有限差分方法[J]. 应用数学进展, 2021, 10(1): 109-114. https://doi.org/10.12677/AAM.2021.101013

参考文献

[1] Allen, S.M. and Cahn, J.W. (1979) A Microscopic Theory for Antiphase Boundary Motion and Its Application to Antiphase Domain Coarsening. Acta Metallurgica, 27, 1085-1095. [Google Scholar] [CrossRef] 
[2] Benes, M., Chalupecký, V., Mikula, K., et al. (2004) Geometrical Image Segmentation by the Allen-Cahn Equation. Applied Numerical Mathematics, 51, 187-205. [Google Scholar] [CrossRef] 
[3] Beneš, M. (2003) Diffuse-Interface Treatment of the Anisotropic Mean-Curvature Flow. Applications of Mathematics, 48, 437-453. [Google Scholar] [CrossRef] 
[4] Krill Iii, C.E. and Chen, L.Q. (2002) Computer Simulation of 3-D Grain Growth Using a Phase-Field Model. Acta Materialia, 50, 3059-3075. [Google Scholar] [CrossRef] 
[5] Lee, H.G. and Lee, J.Y. (2014) A Semi-Analytical Fourier Spectral Method for the Allen-Cahn Equation. Computers & Mathematics with Applications, 68, 174-184. [Google Scholar] [CrossRef] 
[6] 张荣培, 刘佳, 王语. Chebyshev谱配置方法求解Allen-Cahn方程[J]. 沈阳师范大学学报(自然科学版), 2017, 35(4): 435-440.
[7] Stoll, M. and Yücel, H. (2018) Symmetric Interior Penalty Galerkin Method for Fractional-in-Space Phase-Field Equations. AIMS Mathematics, 3, 66-95. [Google Scholar] [CrossRef] 
[8] Lee, H.G. and Lee, J.Y. (2015) A Second Order Operator Splitting Method for Allen-Cahn Type Equations with Nonlinear Source Terms. Physica A: Statistical Mechanics and Its Applications, 432, 24-34. [Google Scholar] [CrossRef] 
[9] Zhai, S., Feng, X. and He, Y. (2014) Numerical Simulation of the Three Dimensional Allen-Cahn Equation by the High-Order Compact ADI Method. Computer Physics Communications, 185, 2449-2455. [Google Scholar] [CrossRef] 
[10] Chen, Y.Y., Huang, Y.Q. and Yi, N.Y. (2019) A SCR-Based Error Estimation and Adaptive Finite Element Method for the Allen-Cahn Equation. Computers and Mathematics with Applications, 78, 204-223. [Google Scholar] [CrossRef] 
[11] 翁志峰, 姚泽丰, 赖淑琴. 重心插值配点法直接求解Allen-Cahn方程[J]. 华侨大学学报(自然科学版), 2019, 40(1): 133-140.
[12] Lai, M.C. (2001) A Note on Finite Difference Discretizations for Poisson Equation on a Disk. Numerical Methods for Partial Differential Equations: An International Journal, 17, 199-203. [Google Scholar] [CrossRef] 
[13] Chen, S. and Zhang, Y. (2011) Krylov Implicit Integration Factor Methods for Spatial Discretization on High Dimensional Unstructured Meshes: Application to Discontinuous Galerkin Methods. Journal of Computational Physics, 230, 4336-4352. [Google Scholar] [CrossRef] 

为你推荐