High-order Time Stepping Schemes for Semilinear Subdiffusion Equations
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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∈ (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 <cit.>, 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 BDFk 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.
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