We investigate a second-order accurate time-stepping scheme for solving a time-fractional diffusion equation with a Caputo derivative of order~$\alpha \in (0,1)$. The basic idea of our scheme is based on local integration followed by linear interpolation. It reduces to the standard Crank--Nicolson scheme in the classical diffusion case, that is, as $\alpha\to 1$. Using a novel approach, we show that the proposed scheme is $\alpha$-robust and second-order accurate in the $L^2(L^2)$-norm, assuming a suitable time-graded mesh. For completeness, we use the Galerkin finite element method for the spatial discretization and discuss the error analysis under reasonable regularity assumptions on the given data. Some numerical results are presented at the end.

翻译：本文研究求解含有Caputo导数的时间分数阶扩散方程（阶数$\alpha \in (0,1)$）的二阶精确时间步进格式。该格式的基本思想基于局部积分与线性插值，在经典扩散情形（即$\alpha\to 1$）下退化为标准的Crank-Nicolson格式。通过一种新方法，我们证明在适当的时间分级网格下，该格式在$L^2(L^2)$范数下具有$\alpha$-稳健性和二阶精确性。为完整性考虑，我们采用Galerkin有限元方法进行空间离散，并在给定数据的合理正则性假设下进行误差分析。最后给出若干数值结果。