ORIGINAL PAPERS

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

  • Shi-Ping Tang ,
  • Yu-Mei Huang
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  • 1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, Gansu, China;
    2. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, Gansu, China

Received date: 2023-12-01

  Revised date: 2024-07-06

  Accepted date: 2024-07-13

  Online published: 2025-06-17

Supported by

This work is supported by the National Natural Science Foundation of China (No. 11971215), the Science and Technology Project of Gansu Province of China (No. 22JR5RA391), the Center for Data Science of Lanzhou University, China, and the Key Laboratory of Applied Mathematics and Complex Systems of Lanzhou University, China.

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 DTHSS iteration method to solve the linear systems. The convergence is also discussed. A DTHSS-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 THSS-τ(α,γ) preconditioned matrix is provided. Experimental results demonstrate the effectiveness of the proposed methods in solving the space-fractional diffusion equations.

Cite this article

Shi-Ping Tang , Yu-Mei Huang . A DTHSS-τ Preconditioner for the Discretized Linear Systems of Space-Fractional Diffusion Equations[J]. Communications on Applied Mathematics and Computation, 2025 , 7(5) : 2097 -2119 . DOI: 10.1007/s42967-025-00491-5

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