Fast algorithms for high-order numerical methods for space-fractional diffusion equations
Fast algorithms for high-order numerical methods for space-fractional diffusion equations
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DOI：
10.1080/00207160.2016.1149579
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发表时间：
2017-05
影响因子：
1.8
通讯作者：
Siu-Long Lei;Yun-Chi Huang
中科院分区：
文献类型：
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作者：
Siu-Long Lei;Yun-Chi Huang
本文分两个阶段研究了求解空间分数阶扩散方程的快速数值方法。本文首先给出了Hao等人提出的一种隐式有限差分格式的快速直接解法。[A fourth order approximation of fractional derivatives with its applications，J. Comput. Phys. 281（2015），pp. 787-805]，该方法在空间上具有四阶精度，在时间上具有二阶精度。第二，Hao等[A fourth-order approximation of fractional derivatives with its applications，J. Comput. Phys. 281（2015），pp. 787-805]的方法，得到了在时间上具有高阶精度的数值解。特别地，可以实现在空间和时间上都具有四阶精度的方法。采用GMRES方法求解带有两个预条件子的离散线性方程组。基于Toeplitz矩阵求逆的CS表示，这两种预条件可以有效地应用，并证明了预条件GMRES方法的收敛速度快。最后给出了数值例子来支持理论分析。
ABSTRACT In this paper, fast numerical methods for solving space-fractional diffusion equations are studied in two stages. Firstly, a fast direct solver for an implicit finite difference scheme proposed by Hao et al. [A fourth-order approximation of fractional derivatives with its applications, J. Comput. Phys. 281 (2015), pp. 787–805], which is fourth-order accurate in space and second-order accurate in time, is developed based on a circulant-and-skew-circulant (CS) representation of Toeplitz matrix inversion. Secondly, boundary value method with spatial discretization of Hao et al. [A fourth-order approximation of fractional derivatives with its applications, J. Comput. Phys. 281 (2015), pp. 787–805] is adopted to produce a numerical solution with higher order accuracy in time. Particularly, a method with fourth-order accuracy in both space and time can be achieved. GMRES method is employed for solving the discretized linear system with two preconditioners. Based on the CS representation of Toeplitz matrix inversion, the two preconditioners can be applied efficiently, and the convergence rate of the preconditioned GMRES method is proven to be fast. Numerical examples are given to support the theoretical analysis.