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 $\sigma =1-\frac{\alpha }{2}$, 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 $O\left(N^{-\min \{k\alpha ,2\}}\right)$ can be obtained, where N denotes the number of time splits. Finally, the theoretical analysis is verified by several numerical examples.

中文翻译：

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时间分数阶扩散问题的非均匀网格有限差分法

在本文中，我们考虑了具有Caputo分数导数的时间分数扩散方程。由于初始时刻解的奇异性，在均匀网格上进行时间离散化时，很难达到理想的收敛速度。因此，为了改善收敛阶数，Caputo时间分数导数项通过$\mathrm{大号}2--{\mathrm{1个}}_{\sigma }$ 在非均匀网格上设置格式 $\sigma =\mathrm{1个}--\frac{\alpha }{2}$，而空间导数项是通过经典的中心差分方案对均匀网格进行近似的。根据正整数k次幂的求和公式，并考虑$ķ=3，4，5$，我们提出了三个非均匀网格用于时间离散化。通过理论分析，得出不同的时间收敛阶数$Ø（{ñ}^{--分\{ķ\alpha ，2\}}）$可以获得，其中N表示时间分割数。最后，通过几个数值例子验证了理论分析。