A Dimensional Splitting Exponential Time Differencing Scheme for Multidimensional Fractional Allen-Cahn Equations

被引:20
作者
Chen, Hao [1 ]
Sun, Hai-Wei [2 ]
机构
[1] Chongqing Normal Univ, Coll Math Sci, Chongqing, Peoples R China
[2] Univ Macau, Dept Math, Macau, Peoples R China
基金
中国国家自然科学基金;
关键词
Fractional Allen-Cahn equation; Discrete maximum principle; Exponential time differencing; Dimensional splitting; Matrix exponential; Toeplitz matrix; 65F10; 65L05; 65N22; 65F15;
D O I
10.1007/s10915-021-01431-0
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
This paper is concerned with numerical methods for solving the multidimensional Allen-Cahn equations with spatial fractional Riesz derivatives. A fully discrete numerical scheme is proposed using a dimensional splitting exponential time differencing approximation for the time integration with finite difference discretization in space. Theoretically, we prove that the proposed numerical scheme can unconditionally preserve the discrete maximum principle. The error estimate in maximum-norm of the proposed scheme is also established in the fully discrete sense. In practical computation, the proposed algorithm can be carried out by computing linear systems and the matrix exponential associated with only one dimensional discretized matrices that possess Toeplitz structure. Meanwhile, fast methods for inverting the Toeplitz matrix and computing the Toeplitz exponential multiplying a vector are exploited to reduce the complexity. Numerical examples in two and three spatial dimensions are given to illustrate the effectiveness and efficiency of the proposed scheme.
引用
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页数:25
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