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PCG method with Strang’s circulant preconditioner for Hermitian positive definite linear system in Riesz space fractional advection–dispersion equations

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Abstract

In this paper, preconditioned conjugate gradient (PCG) method with Strang’s circulant preconditioner is investigated to solve the Hermitian positive definite linear systems, which is result from the Crank–Nicolson (C-N) finite difference scheme with the weighted and shifted Grünwald difference (WSGD) operators to discretize the Riesz space fractional advection–dispersion equation (RSFADE). We show that the spectrum of the preconditioned matrix is clustered around 1, and the singular values of the preconditioned matrix are uniformly bounded away from zero under a certain condition, respectively; hence the PCG method, when applied to solving the preconditioned system, converges superlinearly. Moreover, the complexity in each iteration of the PCG method is \(O(N\log N)\) via using the fast Fourier transforms, where N is the matrix size. Numerical experiments are included to demonstrate the effectiveness of our approach.

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Acknowledgements

The authors would like to convey their thanks to the anonymous referees and the associate editor for their very helpful comments and suggestions, which greatly improved this paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shaoguan University, Shaoguan, 512005, China

    Wei Qu

  2. State Key Laboratory of Quality Research in Chinese Medicines and Faculty of Information Technology, Macau University of Science and Technology, Avenida Wai Long, Taipa, 999078, Macau, China

    Wei Qu, Hai-Wei Shen & Yong Liang

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  1. Wei Qu
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  2. Hai-Wei Shen
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  3. Yong Liang
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Correspondence to Wei Qu.

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Communicated by José Tenreiro Machado.

Wei Qu is supported by the National Natural Science Foundation of China (Grant No. 11601340). Yong Liang is supported by the National Natural Science Foundation of China (Grant No. 61661166011), and Macau Science and Technology Development Fund (FDCT) 003/2016/AFJ.

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Qu, W., Shen, HW. & Liang, Y. PCG method with Strang’s circulant preconditioner for Hermitian positive definite linear system in Riesz space fractional advection–dispersion equations. Comp. Appl. Math. 37, 4554–4569 (2018). https://doi.org/10.1007/s40314-018-0586-6

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  • DOI: https://doi.org/10.1007/s40314-018-0586-6

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