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A Dimensional Splitting Exponential Time Differencing Scheme for Multidimensional Fractional Allen-Cahn Equations
Journal of Scientific Computing ( IF 2.5 ) Pub Date : 2021-03-06 , DOI: 10.1007/s10915-021-01431-0
Hao Chen , Hai-Wei Sun

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.



中文翻译:

多维分数阶Allen-Cahn方程的维分裂指数时差方案

本文涉及解决具有空间分数Riesz导数的多维Allen-Cahn方程的数值方法。针对空间有限差分离散化的时间积分,提出了一种使用维数分裂指数时间微分近似的全离散数值格式。从理论上讲,我们证明了所提出的数值方案可以无条件地保留离散最大值原理。在完全离散的意义上也建立了所提出的方案的最大范数中的误差估计。在实际计算中,可以通过计算线性系统和仅与具有Toeplitz结构的一维离散矩阵相关的矩阵指数来执行所提出的算法。同时,利用反转Toeplitz矩阵和计算Toeplitz指数乘以向量的快速方法来降低复杂度。给出了在两个和三个空间维度上的数值示例,以说明所提出方案的有效性和效率。

更新日期:2021-03-07
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