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In this paper, a two-sided variable-coefficient space-fractional diffusionequation with fractional Neumann boundary condition is considered. To conquerthe weak singularity caused by the nonlocal space-fractional differentialoperators, by introducing an auxiliary fractional flux variable and usingpiecewise linear interpolations, a fractional block-centered finite difference(BCFD) method on general nonuniform grids is proposed. However, like othernumerical methods, the proposed method still produces linear algebraic systemswith unstructured dense coefficient matrices under the general nonuniformgrids.Consequently, traditional direct solvers such as Gaussian eliminationmethod shall require $\mathcal{O}(M^2)$ memory and $\mathcal{O}(M^3)$computational work per time level, where $M$ is the number of spatial unknownsin the numerical discretization. To address this issue, we combine thewell-known sum-of-exponentials (SOE) approximation technique with thefractional BCFD method to propose a fast version fractional BCFD algorithm.Based upon the Krylov subspace iterative methods, fast matrix-vectormultiplications of the resulting coefficient matrices with any vector aredeveloped, in which they can be implemented in only $\mathcal{O}(MN_{exp})$operations per iteration, where $N_{exp}\ll M$ is the number of exponentials inthe SOE approximation. Moreover, the coefficient matrices do not necessarilyneed to be generated explicitly, while they can be stored in$\mathcal{O}(MN_{exp})$ memory by only storing some coefficient vectors.Numerical experiments are provided to demonstrate the efficiency and accuracyof the method.

A fractional block-centered finite difference method for two-sided  space-fractional diffusion equations on general nonuniform grids and its fast  implementation