We propose a fast algorithm for the variable-order (VO) space-fractional advection-diffusion equations with nonlinear source terms on a finite domain. Due to the impact of the space-dependent the VO, the resulting coefficient matrices arising from the finite difference discretization of the fractional advection-diffusion equation are dense without Toeplitz-like structure. By the properties of the elements of coefficient matrices, we show that the off-diagonal blocks can be approximated by low-rank matrices. Then we present a fast algorithm based on the polynomial interpolation to approximate the coefficient matrices. The approximation can be constructed in \({\mathcal {O}}(kN)\) operations and requires \({\mathcal {O}}(kN)\) storage with N and k being the number of unknowns and the approximants, respectively. Moreover, the matrix-vector multiplication can be implemented in \({\mathcal {O}} (kN\log N)\) complexity, which leads to a fast iterative solver for the resulting linear systems. The stability and convergence of the new scheme are also studied. Numerical tests are carried out to exemplify the accuracy and efficiency of the proposed method.

我们提出了一种在有限域上具有非线性源项的可变阶（VO）空间分数对流扩散方程的快速算法。由于空间依赖的VO的影响，分数对流扩散方程的有限差分离散化所产生的系数矩阵是密集的，没有Toeplitz样的结构。通过系数矩阵元素的性质，我们表明非对角线块可以由低秩矩阵近似。然后，我们提出了一种基于多项式插值的快速算法来近似系数矩阵。可以通过\（{\ mathcal {O}}（kN）\）运算来构造近似值，并且需要使用N和k的\（{\ mathcal {O}}（kN）\）存储分别是未知数和近似值。此外，矩阵向量乘法可以以\（{\ mathcal {O}}（kN \ log N）\）复杂度来实现，这导致了针对所得线性系统的快速迭代求解器。还研究了新方案的稳定性和收敛性。数值测试表明了该方法的准确性和有效性。