Circulant preconditioned iterations for fractional diffusion equations based on Hermitian and skew-Hermitian splittings

We employ the implicit finite difference scheme with the shifted Grünwald formula to discretize the fractional diffusion equations with constant coefficients. The coefficient matrix possesses the positive definite Toeplitz-like structure, so we can use the Hermitian and skew-Hermitian splitting method for solving the system. Krylov subspace methods with circulant preconditioners such as Strang’s and T. Chan’s preconditioners are proposed to solve each subsystem via using the fast Fourier transforms (FFTs). Moreover, we present convergence analysis and prove the spectrum of the preconditioned matrices to be clustered around 1. Superlinear convergence rates of the proposed algorithms are obtained. Numerical results illustrate the effectiveness and robustness of circulant preconditioners.