An initial-boundary value problem with a Caputo time derivative of fractional order $\alpha\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. For this problem, we give a simple and general numerical-stability analysis using barrier functions, which yields sharp pointwise-in-time error bounds on quasi-graded temporal meshes with arbitrary degree of grading. L1-type and Alikhanov-type discretization in time are considered. In particular, those results imply that milder (compared to the optimal) grading yields optimal convergence rates in positive time. Semi-discretizations in time and full discretizations are addressed. The theoretical findings are illustrated by numerical experiments.

翻译：考虑一个具有分数阶$\alpha\in(0,1)$的Caputo时间导数的初边值问题，这类问题的解通常在初始时刻表现出奇异行为。针对该问题，我们采用屏障函数给出了一种简单且通用的数值稳定性分析，该方法在具有任意分级程度的拟分级时间网格上得到了尖锐的逐时点误差界。研究考虑了L1型离散格式和Alikhanov型时间离散格式。特别地，这些结果表明，相较于最优分级，采用较温和的分级方式能在正时间区域获得最优收敛速率。本文还讨论了时间半离散化和全离散化情形。理论发现通过数值实验进行了验证。