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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.

Keywords

Preconditioned conjugate gradient method Riesz space fractional advection–dispersion equation Crank–Nicolson finite difference scheme Strang’s circulant preconditioner Fast Fourier transforms 

Mathematics Subject Classification

26A33 34A08 65L12 65F08 65F10 65T50 

Notes

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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Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsShaoguan UniversityShaoguanChina
  2. 2.State Key Laboratory of Quality Research in Chinese Medicines and Faculty of Information TechnologyMacau University of Science and TechnologyTaipaChina

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