A circulant preconditioner for the Riesz distributed-order space-fractional diffusion equations

ABSTRACT

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.

Keywords:

• Distributed-order
• space-fractional diffusion equations
• circulant preconditioner
• preconditioned conjugated gradient method

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research is supported by the Science and Technology Development Fund, Macau SAR (file no. 0118/2018/A3), and MYRG2018-00015-FST from University of Macau.