In this paper we consider an initial-boundary value problem with a Caputo time derivative of order $\alpha\in(0,1)$. The solution typically exhibits a weak singularity near the initial time and this causes a reduction in the orders of convergence of standard schemes. To deal with this singularity, the solution is computed with a fitted difference scheme on a graded mesh. The convergence of this scheme is analysed using a discrete maximum principle and carefully chosen barrier functions. Sharp error estimates are proved, which show an enhancement in the convergence rate compared with the standard L1 approximation on uniform meshes, and also indicate an optimal choice for the mesh grading. This optimal mesh grading is less severe than the optimal grading for the standard L1 scheme. Furthermore, the dependence of the error on the final time forms part of our error estimate. Numerical experiments are presented which corroborate our theoretical results.

翻译：本文考虑具有阶数$\alpha\in(0,1)$的Caputo时间导数的初边值问题。解在初始时刻通常表现出弱奇异性，导致标准格式的收敛阶下降。针对该奇异性，本文采用基于分级网格的拟合差分格式进行求解。通过离散最大值原理和精心构造的障碍函数分析格式收敛性，证明了锐利误差估计，表明与均匀网格上的标准L1逼近相比，该格式的收敛速度得到提升，并给出了网格分级的最优选择。该最优网格分级比标准L1格式的最优分级更宽松。此外，误差对最终时刻的依赖性也纳入误差估计。数值实验验证了理论结果。