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.

翻译：反常扩散通常用子扩散方程建模，其中可能包含弱奇异源项。针对此类情形，许多主流时间步进方法（包括高阶BDF格式的校正方法 [Jin, Li, and Zhou, SIAM J. Sci. Comput., 39 (2017), A3129–A3152]）可能面临严重的阶数降低。为填补这一空白，我们提出一种时间步进格式的光滑化方法，通过采用$m$重积分-微分运算对奇异项进行正则化处理，并利用$k$步BDF卷积求积对方程进行离散——该方法称为ID$m$-BDF$k$方法。我们证明：即使源项为弱奇异且初始数据不兼容，仍可恢复预期的$k$阶收敛性。数值实验验证了理论结果。