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 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 [Formula: see text]-fold integral-differential calculus and the equation is discretized by the [Formula: see text]-step BDF convolution quadrature, called the ID[Formula: see text]-BDF[Formula: see text] method. We prove that the desired [Formula: see text]th-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.