论文标题
多维空间分数allen-cahn方程的快速两级strang分裂方法，具有离散的最大原理
A fast two-level Strang splitting method for multi-dimensional spatial fractional Allen-Cahn equations with discrete maximum principle
论文作者
论文摘要
在本文中，我们研究了多维空间分数Allen-CAHN方程的数值解。在空间分数riesz衍生物的半差异后，获得了具有Toeplitz结构的非线性普通微分方程系统。为了降低计算复杂性，提出了一种两级的strang分裂方法，其中系统中的toeplitz矩阵分为循环矩阵和偏斜循环矩阵的总和。因此，提出的方法可以通过快速的傅立叶变换快速实现，替换以计算昂贵的toeplitz矩阵指数。从理论上讲，我们方法的离散最大原理是无条件保留的。此外，在时间和空间中都进行了二阶精度的无限规范中的误差分析。最后，进行数值测试以证实我们的理论结论和所提出方法的效率。
In this paper, we study the numerical solutions of the multi-dimensional spatial fractional Allen-Cahn equations. After semi-discretization for the spatial fractional Riesz derivative, a system of nonlinear ordinary differential equations with Toeplitz structure is obtained. For the sake of reducing the computational complexity, a two-level Strang splitting method is proposed, where the Toeplitz matrix in the system is split into the sum of a circulant matrix and a skew-circulant matrix. Therefore, the proposed method can be quickly implemented by the fast Fourier transform, substituting to calculate the expensive Toeplitz matrix exponential. Theoretically, the discrete maximum principle of our method is unconditionally preserved. Moreover, the analysis of error in the infinite norm with second-order accuracy is conducted in both time and space. Finally, numerical tests are given to corroborate our theoretical conclusions and the efficiency of the proposed method.
