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Preconditioned iterative methods for twodimensional space-fractional diffusion equations
Xiao-Qing Jin1; Fu-Rong Lin2; Zhi Zhao1
Source PublicationCommunications in Computational Physics

In this paper, preconditioned iterative methods for solving two-dimensional space-fractional diffusion equations are considered. The fractional diffusion equation is discretized by a second-order finite difference scheme, namely, the Crank-Nicolson weighted and shifted Gr ¨unwald difference (CN-WSGD) scheme proposed in [W. Tian, H. Zhou and W. Deng, A class of second order difference approximation for solving space fractional diffusion equations, Math. Comp., 84 (2015) 1703-1727]. For the discretized linear systems, we first propose preconditioned iterative methods to solve them. Then we apply the D’Yakonov ADI scheme to split the linear systems and solve the obtained splitting systems by iterative methods. Two preconditioned iterative methods, the preconditioned generalized minimal residual (preconditioned GMRES) method and the preconditioned conjugate gradient normal residual (preconditioned CGNR) method, are proposed to solve relevant linear systems. By fully exploiting the structure of the coefficient matrix, we design two special kinds of preconditioners, which are easily constructed and are able to accelerate convergence of iterative solvers. Numerical results show the efficiency of our preconditioners.

KeywordFractional Diffusion Equation Toeplitz Matrix Fast Fourier Transform Cn-wsgd Scheme Preconditioned Gmres Method Preconditioned Cgnr Method
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Document TypeJournal article
CollectionFaculty of Science and Technology
Corresponding AuthorZhi Zhao
Affiliation1.Department of Mathematics, University of Macau, Macao 999078, China
2.Department of Mathematics, Shantou University, Shantou 515063, China
First Author AffilicationUniversity of Macau
Corresponding Author AffilicationUniversity of Macau
Recommended Citation
GB/T 7714
Xiao-Qing Jin,Fu-Rong Lin,Zhi Zhao. Preconditioned iterative methods for twodimensional space-fractional diffusion equations[J]. Communications in Computational Physics,2015,18(2):469-488.
APA Xiao-Qing Jin,Fu-Rong Lin,&Zhi Zhao.(2015).Preconditioned iterative methods for twodimensional space-fractional diffusion equations.Communications in Computational Physics,18(2),469-488.
MLA Xiao-Qing Jin,et al."Preconditioned iterative methods for twodimensional space-fractional diffusion equations".Communications in Computational Physics 18.2(2015):469-488.
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