Quasi-Toeplitz splitting iteration methods for unsteady space-fractional diffusion equations

We construct a class of quasi-Toeplitz splitting iteration methods to solve the two-sided unsteady space-fractional diffusion equations with variable coefficients. By making full use of the structural characteristics of the coefficient matrix, the method only requires computational costs of O(n log n) with n denoting the number of degrees of freedom. We develop an appropriate circulant matrix to replace the Toeplitz matrix as a preconditioner. We discuss the spectral properties of the quasi-circulant splitting preconditioned matrix. Numerical comparisons with existing approaches show that the present method is both effective and efficient when being used as matrix splitting preconditioners for Krylov subspace iteration methods.