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A matrix splitting preconditioning method for solving the discretized tempered fractional diffusion equations

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Abstract

The initial boundary value problem of the tempered fractional diffusion equations is a kind of important equations arising in many application fields. In this paper, the Crank-Nicolson scheme is applied in the discretization of the tempered fractional diffusion equations. We then get the discretized system of linear equations with the coefficient matrix having the structure of the sum of an identity matrix and the product of a diagonal and a symmetric positive-definite Toeplitz matrix. A scaled diagonal and Toeplitz-approximate splitting (SDTAS) preconditioner is developed, and the GMRES method combined with this preconditioner is applied to solve the linear system. The spectral distribution of the preconditioned matrix is analyzed and some theoretical results are given. Numerical results demonstrate that the proposed preconditioner is efficient in accelerating the convergence rate of the GMRES method.

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Funding

This work is supported by the National Natural Science Foundation of China No. 11971215 and the Key Laboratory of Applied Mathematics and Complex Systems of Lanzhou University.

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  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Shi-Ping Tang & Yu-Mei Huang

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  1. Shi-Ping Tang
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  2. Yu-Mei Huang
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Correspondence to Yu-Mei Huang.

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Tang, SP., Huang, YM. A matrix splitting preconditioning method for solving the discretized tempered fractional diffusion equations. Numer Algor 92, 1311–1333 (2023). https://doi.org/10.1007/s11075-022-01341-8

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  • DOI: https://doi.org/10.1007/s11075-022-01341-8

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Mathematics Subject Classification (2010)