Fast ADI Finite Difference Method for Two‐Dimensional Space‐Fractional Diffusion Equations on Nonuniform Spatial Grids

In this paper, a two‐dimensional space‐fractional diffusion equation (SFDE) with homogeneous Dirichlet boundary condition is studied. First, fast approximations for fractional integral operators on general nonuniform grids are developed through the integration of the sum‐of‐exponentials (SOE) and linear interpolation approximation techniques. Then, utilizing these fast approximations and combined with the alternating‐direction implicit (ADI) approach, an efficient ADI finite difference method on general nonuniform grids is proposed for the two‐dimensional SFDE model, which greatly improves the computational efficiency. The detailed implementation and computational cost, as well as the memory requirement of the method are thoroughly discussed. The fast ADI scheme is proven to be uniquely solvable and unconditionally stable, provided the SOE approximation is sufficiently accurate. Finally, numerical experiments demonstrate the effectiveness and accuracy of the proposed method, while appropriate nonuniform grids are shown to effectively address boundary singularities within the solutions. Meanwhile, comparisons with other finite difference methods without fast approximations are also presented.