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In this paper, fast numerical methods for solving space-fractional diffusion equations are studied in two stages. Firstly, a fast direct solver for an implicit finite difference scheme proposed by Haoet al.[A fourth-order approximation of fractional derivatives with its applications, J. Comput. Phys. 281 (2015), pp. 787–805], which is fourth-order accurate in space and second-order accurate in time, is developed based on a circulant-and-skew-circulant (CS) representation of Toeplitz matrix inversion. Secondly, boundary value method with spatial discretization of Haoet al.[A fourth-order approximation of fractional derivatives with its applications, J. Comput. Phys. 281 (2015), pp. 787–805] is adopted to produce a numerical solution with higher order accuracy in time. Particularly, a method with fourth-order accuracy in both space and time can be achieved. GMRES method is employed for solving the discretized linear system with two preconditioners. Based on the CS representation of Toeplitz matrix inversion, the two preconditioners can be applied efficiently, and the convergence rate of the preconditioned GMRES method is proven to be fast. Numerical examples are given to support the theoretical analysis. [ABSTRACT FROM PUBLISHER]

In this paper, finite difference schemes for differential equations with both temporal and spatial fractional derivatives are studied. When the order of the time fractional derivative is in, an alternating direction implicit (ADI) scheme with second-order accuracy in both space and time is constructed. For equations with time fractional derivatives of order lying in, a scheme is derived and solved by the generalized minimal residual method. We also propose a preconditioner to improve the efficiency for the implementation of the scheme in this situation. [ABSTRACT FROM PUBLISHER]