Abstract
By exploiting Toeplitz-like structure and non-Hermitian dense property of the discrete coefficient matrix, a new double-layer iterative method called SHSS-PCG method is employed to solve the linear systems originating from the implicit finite difference discretization of fractional diffusion equations (FDEs). The method is a combination of the single-step Hermitian and skew-Hermitian splitting (SHSS) method with the preconditioned conjugate gradient (PCG) method. Further, the new circulant preconditioners are proposed to improve the efficiency of SHSS-PCG method, and the computation cost is further reduced via using the fast Fourier transform (FFT). Theoretical analysis shows that the SHSS-PCG iterative method with circulant preconditioners is convergent. Numerical experiments are given to show that our SHSS-PCG method with circulant preconditioners preforms very well, and the proposed circulant preconditioners are very efficient in accelerating the convergence rate.
Highlights
IntroductionMethod to solve the discretized system of fractional diffusion equations (FDEs) with Meerschaet–Tadjeran’s method, and numerically showed that its convergence is fast with smaller diffusion coefficients (in that case the discretized system is well-conditioned)
Fractional calculus has a long history and its origin goes back to describing half order (α = 1/2) derivative by Leibnitz in 1695
The main reason is that it is more challenging or sometimes even impossible to obtain the analytical solution of fractional partial differential equations (FPDEs), or that the obtained analytical solution is less valuable
Summary
Method to solve the discretized system of FDEs with Meerschaet–Tadjeran’s method, and numerically showed that its convergence is fast with smaller diffusion coefficients (in that case the discretized system is well-conditioned). Gu [39] proposed the Hermitian and skew-Hermitian splitting methods and the circulant preconditioner to accelerate the convergence rate for solving the fractional diffusion equations with constant coefficients; and numerical results show that the methods and circulant preconditioners are efficient. The novelty of this paper is to present a new double-layer SHSS-PCG iterative method to further utilize matrix structure to improve the computational efficiency for solving the full non-Hermitian linear systems originating in the discretization of FDEs. The new method includes two aspects. 3, the SHSS-PCG iterative method is proposed for solving the discretized Toeplitz-like linear systems.
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