In this paper, a two-sided variable-coefficient space-fractional diffusion equation with fractional Neumann boundary condition is considered. To conquer the weak singularity caused by the nonlocal space-fractional differential operators, by introducing an auxiliary fractional flux variable and using piecewise linear interpolations, a fractional block-centered finite difference (BCFD) method on general nonuniform grids is proposed. However, like other numerical methods, the proposed method still produces linear algebraic systems with unstructured dense coefficient matrices under the general nonuniform grids.Consequently, traditional direct solvers such as Gaussian elimination method shall require $\mathcal{O}(M^2)$ memory and $\mathcal{O}(M^3)$ computational work per time level, where $M$ is the number of spatial unknowns in the numerical discretization. To address this issue, we combine the well-known sum-of-exponentials (SOE) approximation technique with the fractional BCFD method to propose a fast version fractional BCFD algorithm. Based upon the Krylov subspace iterative methods, fast matrix-vector multiplications of the resulting coefficient matrices with any vector are developed, 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 in the SOE approximation. Moreover, the coefficient matrices do not necessarily need 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 accuracy of the method.

翻译：本文考虑带有分数阶Neumann边界条件的双侧变系数空间分数阶扩散方程。为克服非局部空间分数阶微分算子引起的弱奇异性，通过引入辅助分数阶通量变量并利用分段线性插值，提出了一种在一般非均匀网格上的分数阶块中心有限差分（BCFD）方法。然而，与其他数值方法类似，该方法在一般非均匀网格下仍会产生系数矩阵非结构化稠密的线性代数系统。因此，传统直接求解器（如高斯消元法）在每个时间层需要$\mathcal{O}(M^2)$内存和$\mathcal{O}(M^3)$计算量，其中$M$为数值离散中的空间未知量个数。为解决该问题，我们将著名的指数和（SOE）近似技术与分数阶BCFD方法相结合，提出了一种快速版本的分数阶BCFD算法。基于Krylov子空间迭代方法，发展了所得系数矩阵与任意向量的快速矩阵-向量乘法，每次迭代仅需$\mathcal{O}(MN_{exp})$次运算，其中$N_{exp}\ll M$为SOE近似中的指数个数。此外，系数矩阵无需显式生成，仅需存储$\mathcal{O}(MN_{exp})$内存的若干系数向量。数值实验验证了该方法的效率与精度。