High-order BDF convolution quadrature for subdiffusion models with a singular source term
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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 [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 IDm-BDFk method. We prove that the desired kth-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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