High-order Time Stepping Schemes for Semilinear Subdiffusion Equations

TL;DR

This paper develops high-order time stepping schemes for semilinear subdiffusion equations using BDF convolution quadrature, achieving optimal convergence rates without extra regularity assumptions, and validates the approach with numerical examples.

Contribution

It extends high-order BDF schemes to nonlinear subdiffusion equations and provides rigorous convergence analysis with minimal regularity requirements.

Findings

Convergence order of the corrected BDF$k$ scheme is $O( au^{ ext{min}(k,1+2eta- ext{epsilon})})$.

Numerical experiments confirm theoretical convergence rates.

Method effectively handles nonlinear potential terms in subdiffusion equations.

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.