Circulant preconditioners for a kind of spatial fractional diffusion equations. (English) Zbl 1437.65013
Summary: In this paper, circulant preconditioners are studied for discretized matrices arising from finite difference schemes for a kind of spatial fractional diffusion equations. The fractional differential operator is comprised of left-sided and right-sided derivatives with order in \((\frac{1}{2},1)\). The resulting discretized matrices preserve Toeplitz-like structure and hence their matrix-vector multiplications can be computed efficiently by the fast Fourier transform. Theoretically, the spectra of the circulant preconditioned matrices are shown to be clustered around 1 under some conditions. Numerical experiments are presented to demonstrate that the preconditioning technique is very efficient.
MSC:
| 65F08 | Preconditioners for iterative methods |
| 35R11 | Fractional partial differential equations |
| 65F10 | Iterative numerical methods for linear systems |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
fractional diffusion equation; Toeplitz matrix; circulant preconditioner; fast Fourier transform; Krylov subspace methodsReferences:
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