  A general conversion strategy by involving a shifted parameter $\theta$ is
proposed to construct high-order accuracy difference formulas for fractional
calculus operators. By converting the second-order backward difference formula
with such strategy, a novel $\theta$-scheme with correction terms is developed
for the subdiffusion problem with nonsmooth data, which is robust even for very
small $\alpha$ and can resolve the initial singularity.The optimal error
estimates are carried out with essential arguments and are verified by
numerical tests.
