Abstract
We study the sine-transform-based splitting preconditioning technique for the linear systems arising in the numerical discretization of time-dependent one dimensional and two dimensional Riesz space fractional diffusion equations. Those linear systems are Toeplitz-like. By making use of diagonal-plus-Toeplitz splitting iteration technique, a sine-transform-based splitting preconditioner is proposed to accelerate the convergence rate efficiently when the Krylov subspace method is implemented. Theoretically, we prove that the spectrum of the preconditioned matrix of the proposed method is clustering around 1. In practical computations, by the fast sine transform the computational complexity at each time level can be done in \({{\mathcal {O}}}(n\log n)\) operations where n is the matrix size. Numerical examples are presented to illustrate the effectiveness of the proposed algorithm.
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We are very grateful to the anonymous referees for their invaluable comments and very detailed suggestions that have greatly improved the presentation of this paper.
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The research is funded in part by the National Natural Science Foundation of China under Grant 12001104, the Guangdong Basic and Applied Basic Research Foundation under Grant Nos. 2019A1515110279; 2019A1515110893; 2019A1515010789, and the Science and Technology Development Fund, Macau SAR (file no. 0118/2018/A3), MYRG2018-00015-FST from University of Macau.
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Lu, X., Fang, ZW. & Sun, HW. Splitting preconditioning based on sine transform for time-dependent Riesz space fractional diffusion equations. J. Appl. Math. Comput. 66, 673–700 (2021). https://doi.org/10.1007/s12190-020-01454-0
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DOI: https://doi.org/10.1007/s12190-020-01454-0
Keywords
- Riesz space fractional diffusion equations
- Shifted Grünwald discretization
- Symmetric positive definite Toeplitz matrix
- Sine-transform-based splitting preconditioner
- GMRES method