ABSTRACT

In this paper, we study a fast algorithm for the numerical solution of the 1D distributed-order space-fractional diffusion equation. After discretization by the finite difference method, the resulting system is the symmetric positive definite Toeplitz matrix. The preconditioned conjugate gradient method with a circulant preconditioner is employed to solve the linear system. Theoretically, the spectrum of the preconditioned matrix is proved to be clustered around 1, which can guarantee the superlinear convergence rate of the proposed method. Numerical experiments are carried out to demonstrate the effectiveness of our proposed method.

中文翻译：

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Riesz 分布阶空间分数扩散方程的循环预条件子

摘要

在本文中，我们研究了一维分布阶空间分数扩散方程数值解的快速算法。通过有限差分法离散化后，得到的系统为对称正定 Toeplitz 矩阵。采用循环预条件子的预条件共轭梯度法求解线性系统。理论上，证明了预处理矩阵的谱在1附近聚类，可以保证所提方法的超线性收敛速度。进行了数值实验来证明我们提出的方法的有效性。