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arXiv · 2305.03384

High-order BDF convolution quadrature for subdiffusion models with a singular source term

Jiankang Shi·Minghua Chen
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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 $m$-fold integral-differential calculus and the equation is discretized by the $k$-step BDF convolution quadrature, called ID$m$-BDF$k$ method. We prove that the desired $k$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.

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Jiankang Shi, Minghua Chen. 2023-05-05. High-order BDF convolution quadrature for subdiffusion models with a singular source term. https://arxiv.org/abs/2305.03384

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