Thanks to the singularity of the solution of linear subdiffusion problems, most time-stepping methods on uniform meshes can result in $O(\tau)$ accuracy where $\tau$ denotes the time step. The present work aims to discover the reason why some type of Crank-Nicolson schemes (the averaging Crank-Nicolson scheme) for the subdiffusion can only yield $O(\tau^\alpha)$$(\alpha<1)$ accuracy, which is much lower than the desired. The existing well developed error analysis for the subdiffusion, which has been successfully applied to many time-stepping methods such as the fractional BDF-$p (1\leq p\leq 6)$, all requires singular points be out of the path of contour integrals involved. The averaging Crank-Nicolson scheme in this work is quite natural but fails to meet this requirement. By resorting to the residue theorem, some novel sharp error analysis is developed in this study, upon which correction methods are further designed to obtain the optimal $O(\tau^2)$ accuracy. All results are verified by numerical tests.

翻译：由于线性次扩散问题解的奇异性，大多数均匀网格上的时间步进方法只能达到$O(\tau)$精度，其中$\tau$表示时间步长。本文旨在揭示为何某类针对次扩散问题的Crank-Nicolson格式（平均Crank-Nicolson格式）仅能达到远低于预期的$O(\tau^\alpha)$$(\alpha<1)$精度。现有成熟的次扩散误差分析方法（已成功应用于分数阶BDF-$p (1\leq p\leq 6)$等多类时间步进方法）均要求奇点位于涉及的围道积分路径之外。本文研究的平均Crank-Nicolson格式虽然自然直观，但无法满足该要求。通过引入留数定理，本文发展了一种新颖的尖锐误差分析方法，并在此基础上进一步设计校正方法以获得最优$O(\tau^2)$精度。所有结果均通过数值实验验证。