A family of novel time-stepping methods for the fractional calculus operators is presented with a shifted parameter. The truncation error with second-order accuracy is proved under the framework of the shifted convolution quadrature. To improve the efficiency, two aspects are considered, that i) a fast algorithm is developed to reduce the computation complexity from $O(N_t^2)$ to $O(N_t\log N_t)$ and the memory requirement from $O(N_t)$ to $O(\log N_t)$, where $N_t$ denotes the number of successive time steps, and ii) correction terms are added to deal with the initial singularity of the solution. The stability analysis and error estimates are provided in detail where in temporal direction the novel time-stepping methods are applied and the spatial variable is discretized by the finite element method. Numerical results for d-dimensional examples ($d=1,2,3$) confirm our theoretical conclusions and the efficiency of the fast algorithm.

提出了带有移位参数的分数阶微积分算子的一系列新颖的时间步长方法。在移位卷积正交框架下证明了具有二阶精度的截断误差。为了提高效率，考虑了两个方面：i）开发了一种快速算法来减少$Ø（{ñ}_{Ť}^2）$ 到 $Ø（{ñ}_{Ť}\mathrm{日志}{ñ}_{Ť}）$ 以及来自的内存需求 $Ø（{ñ}_{Ť}）$ 到 $Ø（\mathrm{日志}{ñ}_{Ť}）$， 在哪里 ${ñ}_{Ť}$表示连续时间步长的数量，并且ii）添加了校正项以应对解的初始奇异性。详细介绍了稳定性分析和误差估计，其中在时间方向上应用了新颖的时间步长方法，并通过有限元方法离散化了空间变量。d维示例的数值结果（$d=\mathrm{1个}，2，3$）证实了我们的理论结论和快速算法的效率。