Dominant Hermitian splitting iteration method for discrete space-fractional diffusion equations

Abstract

The discretizations of the left and the right fractional derivatives based on the shifted finite-difference formulas of the Grünwald-Letnikov type can result in Toeplitz matrices T and T⁎. Combining with the generating function of Toeplitz matrix T, we analyse the dominant property of the Hermitian part of T relative to its skew-Hermitian part. Then we construct a dominant Hermitian splitting iteration method for solving the discrete linear system of the considered space-fractional diffusion equations, and design a more practical dominant Hermitian-circulant splitting preconditioner to accelerate the convergence rates of the Krylov subspace iteration methods. Theoretical analyses demonstrate that all eigenvalues of the corresponding preconditioned matrix are clustered in a complex disk centered at 1 with the radius much less than 1, especially when the order β of the fractional derivative is close to 2. In addition, the numerical results show that the constructed preconditioner can effectively solve the discrete linear systems of one-dimensional and two-dimensional space-fractional diffusion equations.