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

Sharp error analysis for averaging Crank-Nicolson schemes with corrections for subdiffusion with nonsmooth solutions