Dominant Hermitian Splitting Iteration Method For Discrete Space-Fractional Diffusion Equations

The discretizations of the left and the right fractional derivatives based on the shifted finite-difference formulas of the Grunwald-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 beta 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. (c) 2020 IMACS. Published by Elsevier B.V. All rights reserved.