We propose a high-order numerical scheme for nonlinear time fractional reaction-diffusion equations with initial singularity, where L2-$1_{\sigma }$ scheme on graded mesh is used to approximate Caputo fractional derivative and Legendre spectral method is applied to discrete spatial variable. We give the priori estimate, existence and uniqueness of numerical solution. Then the unconditional stability and convergence are proved. The rate of convergence is $O(M^{-\min \{r\alpha ,2\}}+N^{-m})$, which is obtained without extra regularity assumption on the exact solution. Numerical results are given to confirm the sharpness of error analysis.

中文翻译：

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求解具有初始奇异性的非线性时间分数阶反应扩散方程的高阶数值格式

我们提出了具有初始奇异性的非线性时间分数阶反应扩散方程的高阶数值方案，其中L 2-$1_{\sigma }$梯度网格方案用于逼近 Caputo 分数阶导数，Legendre 谱方法用于离散空间变量。我们给出了数值解的先验估计、存在性和唯一性。然后证明了无条件稳定性和收敛性。收敛速度为$哦({米}^{-\mathrm{分钟}\{r\alpha ,2\}}+N^{-米})$，这是在没有对精确解进行额外正则假设的情况下获得的。给出了数值结果以确认误差分析的锐度。