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arXiv · 2003.03607

High-order Time Stepping Schemes for Semilinear Subdiffusion Equations

Kai Wang·Zhi Zhou
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Abstract

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.

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Kai Wang, Zhi Zhou. 2020-03-07. High-order Time Stepping Schemes for Semilinear Subdiffusion Equations. https://arxiv.org/abs/2003.03607

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