Diagonal and Toeplitz splitting iteration methods for diagonal-plus-Toeplitz linear systems from spatial fractional diffusion equations

2017 ◽  
Vol 24 (4) ◽  
pp. e2093 ◽  
Author(s):  
Zhong-Zhi Bai ◽  
Kang-Ya Lu ◽  
Jian-Yu Pan
Keyword(s):  
Linear Systems ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Fractional Diffusion Equations ◽  
Iteration Methods

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

2015 ◽  
Vol 18 (2) ◽  
pp. 469-488 ◽  
Author(s):  
Xiao-Qing Jin ◽  
Fu-Rong Lin ◽  
Zhi Zhao
Keyword(s):  
Linear Systems ◽  
Iterative Methods ◽  
Dimensional Space ◽  
Fractional Diffusion ◽  
Second Order ◽  
Diffusion Equations ◽  
Two Dimensional ◽  
Fractional Diffusion Equations ◽  
Preconditioned Iterative ◽  
Preconditioned Iterative Methods

AbstractIn 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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