Based on the equivalence of A-stability and G-stability, the energy technique of the six-step BDF method for heat equation has been discussed in [Akrivis, Chen, Yu, Zhou, SIAM J. Numer. Anal., Minor Revised]. Unfortunately, this theory is hard to apply in the time-fractional PDEs. In this work, we consider three types of subdiffusion models, namely single-term, multi-term and distributed order fractional diffusion equations. We present a novel and concise stability analysis of the time stepping schemes generated by k-step backward difference formulae (BDFk), for approximately solving the subdiffusion equation. The analysis mainly relies on the energy technique by applying Grenander-Szegö theorem. This kind of argument has been widely used to confirm the stability of various A-stable schemes (e.g., \(k=1,2\)). However, it is not an easy task for higher order BDF methods, due to lack of the A-stability. The core object of this paper is to fill in this gap.

基于A稳定性和G稳定性的等价性，在[Akrivis，Chen，Yu，Zhou，SIAM J. Numer。Chem。，2004，6（1）]中讨论了用于热方程的六步BDF方法的能量技术。肛门，未成年人修订]。不幸的是，这种理论很难应用于时间分数PDE中。在这项工作中，我们考虑了三种类型的子扩散模型，即单项，多项和分布式阶分数扩散方程。我们提出了一种由k阶后向差分公式（BDF k）生成的时间步进方案的新颖简洁的稳定性分析，用于近似求解子扩散方程。该分析主要通过应用Grenander-Szegö定理来依赖能量技术。这种说法已被广泛用于确认各种-稳定方案（例如\（k = 1,2 \））。但是，由于缺乏A稳定性，对于高阶BDF方法而言，这并不是一件容易的事。本文的核心目标是填补这一空白。