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A circulant preconditioner for the Riesz distributed-order space-fractional diffusion equations. (English) Zbl 1500.65041

Summary: In this paper, we study a fast algorithm for the numerical solution of the 1D distributed-order space-fractional diffusion equation. After discretization by the finite difference method, the resulting system is the symmetric positive definite Toeplitz matrix. The preconditioned conjugate gradient method with a circulant preconditioner is employed to solve the linear system. Theoretically, the spectrum of the preconditioned matrix is proved to be clustered around 1, which can guarantee the superlinear convergence rate of the proposed method. Numerical experiments are carried out to demonstrate the effectiveness of our proposed method.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65F08 Preconditioners for iterative methods
65F10 Iterative numerical methods for linear systems
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
15B05 Toeplitz, Cauchy, and related matrices
15A18 Eigenvalues, singular values, and eigenvectors
26A33 Fractional derivatives and integrals
35R11 Fractional partial differential equations
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