High-order BDF convolution quadrature for subdiffusion models with a singular source term
Abstract
Anomalous diffusion is often modelled in terms of the subdiffusion equation, which can involve a weakly singular source term. For this case, many predominant time stepping methods, including the correction of high-order BDF schemes [{\sc Jin, Li, and Zhou}, SIAM J. Sci. Comput., 39 (2017), A3129--A3152], may suffer from a severe order reduction. To fill in this gap, we propose a smoothing method for time stepping schemes, where the singular term is regularized by using a -fold integral-differential calculus and the equation is discretized by the -step BDF convolution quadrature, called ID-BDF method. We prove that the desired th-order convergence can be recovered even if the source term is a weakly singular and the initial data is not compatible. Numerical experiments illustrate the theoretical results.
Cite
@article{arxiv.2305.03384,
title = {High-order BDF convolution quadrature for subdiffusion models with a singular source term},
author = {Jiankang Shi and Minghua Chen},
journal= {arXiv preprint arXiv:2305.03384},
year = {2023}
}