A DTHSS-τ Preconditioner for the Discretized Linear Systems of Space-Fractional Diffusion Equations

doi:10.1007/s42967-025-00491-5

Abstract

Abstract: In this paper, the backward Euler method and the shifted Grünwald-Letnikov formulas are utilized to discretize the space-fractional diffusion equations. The discretized result is a system of linear equations with a coefficient matrix being the sum of a diagonal matrix and a non-Hermitian Toeplitz matrix. By utilizing the Hermitian and skew-Hermitian splitting of the Toeplitz matrix, we develop a two-parameter DT_HSS iteration method to solve the linear systems. The convergence is also discussed. A DT_HSS-τ(α,γ) preconditioner is proposed and the preconditioned GMRES method combined with the proposed preconditioner is applied to solve the linear systems. The spectral analysis of the T_HSS-τ(α,γ) preconditioned matrix is provided. Experimental results demonstrate the effectiveness of the proposed methods in solving the space-fractional diffusion equations.

Key words: Space-fractional diffusion equations, Matrix splitting iteration method, Convergence, Preconditioner, Spectral distribution

Shi-Ping Tang, Yu-Mei Huang. A DT_HSS-τ Preconditioner for the Discretized Linear Systems of Space-Fractional Diffusion Equations[J]. Communications on Applied Mathematics and Computation, 2025, 7(5): 2097-2119.

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