A circulant preconditioner for the Riesz distributed-order space-fractional diffusion equations. Issue 16 (2nd November 2022)

• Record Type: Journal Article
• Authors: Huang, Xin; Fang, Zhi-Wei; Sun, Hai-Wei; Zhang, Chun-Hua
• Is Part Of: Linear & multilinear algebra. Volume 70:Issue 16(2022)
• Page Start: 3081
• Page End: 3096
• Publication Date: 2022-11-02
• Subjects: Distributed-order -- space-fractional diffusion equations -- circulant preconditioner -- preconditioned conjugated gradient method; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2020.1823309
• Languages: English
• ISSNs: 0308-1087

Abstract

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.