A DT

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

_{\text{HS}}^{}

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S iteration method to solve the linear systems. The convergence is also discussed. A DT

_{\text{HS}}^{}

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S-

\tau (\alpha ,\gamma )

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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 DT

_{\text{HS}}^{}

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S-

\tau (\alpha ,\gamma )

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preconditioned matrix is provided. Experimental results demonstrate the effectiveness of the proposed methods in solving the space-fractional diffusion equations.

Keywords

Space-fractional diffusion equations / Matrix splitting iteration method / Convergence / Preconditioner / Spectral distribution

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Shanghai University