A circulant preconditioner for fractional diffusion equations

► The coefficient matrix of the fractional diffusion equation is Toeplitz-like. ► The PCGRN method with a circulant preconditioner is proposed to solve the resulting Toeplitz-like system. ► The superlinear convergence rate for the proposed method has been theoretically proven under some conditions....

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Bibliographic Details
Published in:Journal of computational physics 2013-06, Vol.242, p.715-725
Main Authors: Lei, Siu-Long, Sun, Hai-Wei
Format: Article
Language:English
Subjects:
CGNR method
Circulant preconditioner
Computational efficiency
Convergence
Diffusion
Diffusion coefficient
Fast Fourier transform
Finite difference method
Fractional diffusion equations
Linear systems
Mathematical analysis
Mathematical models
Shifted Grünwald discretization
Toeplitz
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Summary:► The coefficient matrix of the fractional diffusion equation is Toeplitz-like. ► The PCGRN method with a circulant preconditioner is proposed to solve the resulting Toeplitz-like system. ► The superlinear convergence rate for the proposed method has been theoretically proven under some conditions. ► Numerical results show the robustness and efficiency of our circulant preconditioner. The implicit finite difference scheme with the shifted Grünwald formula, which is unconditionally stable, is employed to discretize fractional diffusion equations. The resulting systems are Toeplitz-like and then the fast Fourier transform can be used to reduce the computational cost of the matrix–vector multiplication. The preconditioned conjugate gradient normal residual method with a circulant preconditioner is proposed to solve the discretized linear systems. The spectrum of the preconditioned matrix is proven to be clustered around 1 if diffusion coefficients are constant; hence the convergence rate of the proposed iterative algorithm is superlinear. Numerical experiments are carried out to demonstrate that our circulant preconditioner works very well, even though for cases of variable diffusion coefficients.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2013.02.025