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The aim of this paper is to study the time stepping scheme for approximatelysolving the subdiffusion equation with a weakly singular source term. In thiscase, many popular time stepping schemes, including the correction ofhigh-order BDF methods, may lose their high-order accuracy. To fill in thisgap, in this paper, we develop a novel time stepping scheme, where the sourceterm is regularized by using a $k$-fold integral-derivative and the equation isdiscretized by using a modified BDF2 convolution quadrature. We prove that theproposed time stepping scheme is second-order, even if the source term isnonsmooth in time and incompatible with the initial data. Numerical results arepresented to support the theoretical results.

Modified BDF2 schemes for subdiffusion models with a singular source term