The aim of this paper is to develop and analyze high-order time stepping schemes for solving semilinear subdiffusion equations. We apply the $k$-step BDF convolution quadrature to discretize the time-fractional derivative with order $\alpha\in (0,1)$, and modify the starting steps in order to achieve optimal convergence rate. This method has already been well-studied for the linear fractional evolution equations in Jin, Li and Zhou \cite{JinLiZhou:correction}, while the numerical analysis for the nonlinear problem is still missing in the literature. By splitting the nonlinear potential term into an irregular linear part and a smoother nonlinear part, and using the generating function technique, we prove that the convergence order of the corrected BDF$k$ scheme is $O(\tau^{\min(k,1+2\alpha-\epsilon)})$, without imposing further assumption on the regularity of the solution. Numerical examples are provided to support our theoretical results.

中文翻译：

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半线性子扩散方程的高阶时间步进方案

本文的目的是开发和分析求解半线性子扩散方程的高阶时间步进方案。我们应用 $k$-step BDF 卷积求积以 $\alpha\in (0,1)$ 阶离散时间分数阶导数，并修改起始步骤以实现最佳收敛速度。该方法已经在 J​​in、Li 和 Zhou\cite{JinLiZhou:correction} 中的线性分数阶演化方程中得到了很好的研究，而对于非线性问题的数值分析在文献中仍然缺失。通过将非线性势项拆分为不规则的线性部分和更平滑的非线性部分，并使用生成函数技术，我们证明了修正后的 BDF$k$ 方案的收敛阶数为 $O(\tau^{\min(k ,1+2\alpha-\epsilon)})$, 不对解的规律性作进一步的假设。提供了数值例子来支持我们的理论结果。