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This article devotes to developing robust but simple correction techniquesand efficient algorithms for a class of second-order time stepping methods,namely the shifted fractional trapezoidal rule (SFTR), for subdiffusionproblems to resolve the initial singularity and nonlocality. The stabilityanalysis and sharp error estimates in terms of the smoothness of the initialdata and source term are presented. As a byproduct in numerical tests, we findamazingly that the Crank-Nicolson scheme ($\theta=\frac{1}{2}$) without initialcorrections can restore the optimal convergence rate for the subdiffusionproblem with smooth initial data and source terms. To deal with thenonlocality, fast algorithms are considered to reduce the computational costfrom $O(N^2)$ to $O(N \log N)$ and save the memory storage from $O(N)$ to$O(\log N)$, where $N$ denotes the number of time levels. Numerical tests areperformed to verify the sharpness of the theoretical results and confirm theefficiency and accuracy of initial corrections and the fast algorithms.

Efficient shifted fractional trapezoidal rule for subdiffusion problems with nonsmooth solutions on uniform meshes