High-order Time Stepping Schemes for Semilinear Subdiffusion Equations
Kai Wang, Zhi Zhou
Abstract
The aim of this paper is to develop and analyze high-order time stepping schemes for approximately solving semilinear subdiffusion equations. We apply the convolution quadrature generated by $k$-step backward differentiation formula (BDF$k$) 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 [SIAM J. Sci. Comput., 39 (2017), pp. A3129--A3152], 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.