Finite Difference Method on Non-Uniform Meshes for Time Fractional Diffusion Problem

In this paper, we consider the time fractional diffusion equation with Caputo fractional derivative. Due to the singularity of the solution at the initial moment, it is difficult to achieve an ideal convergence rate when the time discretization is performed on uniform meshes. Therefore, in order to improve the convergence order, the Caputo time fractional derivative term is discretized by the L2 - 1 sigma format on non-uniform meshes, with sigma = 1 - alpha/2, while the spatial derivative term is approximated by the classical central difference scheme on uniform meshes. According to the summation formula of positive integer k power, and considering k = 3, 4, 5, we propose three non-uniform meshes for time discretization. Through theoretical analysis, different time convergence orders O(N-(min{k alpha,2})) can be obtained, where N denotes the number of time splits. Finally, the theoretical analysis is verified by several numerical examples.