Volume 18, Issue 2
Preconditioned Iterative Methods for Two-Dimensional Space-Fractional Diffusion Equations

Xiao-Qing Jin, Fu-Rong Lin & Zhi Zhao

Commun. Comput. Phys., 18 (2015), pp. 469-488.

Published online: 2018-04

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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.

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@Article{CiCP-18-469, author = {}, title = {Preconditioned Iterative Methods for Two-Dimensional Space-Fractional Diffusion Equations}, journal = {Communications in Computational Physics}, year = {2018}, volume = {18}, number = {2}, pages = {469--488}, 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.120314.230115a}, url = {http://global-sci.org/intro/article_detail/cicp/11036.html} }
TY - JOUR T1 - Preconditioned Iterative Methods for Two-Dimensional Space-Fractional Diffusion Equations JO - Communications in Computational Physics VL - 2 SP - 469 EP - 488 PY - 2018 DA - 2018/04 SN - 18 DO - http://doi.org/10.4208/cicp.120314.230115a UR - https://global-sci.org/intro/article_detail/cicp/11036.html KW - AB -

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.

Xiao-Qing Jin, Fu-Rong Lin & Zhi Zhao. (2020). Preconditioned Iterative Methods for Two-Dimensional Space-Fractional Diffusion Equations. Communications in Computational Physics. 18 (2). 469-488. doi:10.4208/cicp.120314.230115a
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