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Preconditioned Iterative Methods for Two-Dimensional Space-Fractional Diffusion Equations

Published online by Cambridge University Press:  30 July 2015

Xiao-Qing Jin ,
Fu-Rong Lin  and
Zhi Zhao
Xiao-Qing Jin
Affiliation:
Department of Mathematics, University of Macau, Macao 999078, China
Fu-Rong Lin
Affiliation:
Department of Mathematics, Shantou University, Shantou 515063, China
Zhi Zhao*
Affiliation:
Department of Mathematics, University of Macau, Macao 999078, China
*
*Corresponding author. Email addresses: xqjin@umac.mo (X.-Q. Jin), frlin@stu.edu.cn (F.-R. Lin), zhaozhi231@163.com (Z. Zhao)
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Abstract

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 Grünwald 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.

Keywords

65F1065L0665U05Fractional diffusion equationCN-WSGD schemepreconditioned GMRES methodpreconditioned CGNR methodToeplitz matrixfast Fourier transform

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Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 2 , August 2015 , pp. 469 - 488
Copyright
Copyright © Global-Science Press 2015 

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