论文标题
半线性次要方程式的高阶时间阶程方案
High-order Time Stepping Schemes for Semilinear Subdiffusion Equations
论文作者
论文摘要
本文的目的是开发和分析用于求解半线性次扩散方程的高阶时间阶梯方案。我们将$ k $ - 步骤BDF卷积正交正交正交，以（0,1）$中的$α\订单$α\离散时间，并修改起始步骤以达到最佳收敛率。对于Jin，Li和Zhou \ cite {Jinlizhou：校正}的线性分数演化方程的方法已经进行了充分研究，而文献中仍缺少非线性问题的数值分析。通过将非线性电位项分为不规则的线性部分和更光滑的非线性部分，并使用生成功能技术，我们证明了校正后的BDF $ K $方案的收敛顺序为$ O（τ^{\ min（k，1+2α-am）} $，而无需对解决方案的常规性进行进一步的假设。提供数值示例以支持我们的理论结果。
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 $α\in (0,1)$, 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 $O(τ^{\min(k,1+2α-ε)})$, without imposing further assumption on the regularity of the solution. Numerical examples are provided to support our theoretical results.
