HIGH-ORDER BDF CONVOLUTION QUADRATURE FOR SUBDIFFUSION MODELS WITH A SINGULAR SOURCE TERM\ast

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 backward differentiation formula (BDF) schemes [B. Jin, B. Y. Li, and Z. Zhou, SIAM J. Sci. Comput., 39 (2017), pp. 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 an m-fold integral-differential calculus and the equation is discretized by the k-step BDF convolution quadrature, called the IDm-BDFk method. We prove that the desired kth-order convergence can be recovered even if the source term is weakly singular and the initial data is not compatible. Numerical experiments illustrate the theoretical results.