论文标题
向后差公式：次扩散方程的能量技术
Backward difference formula: The energy technique for subdiffusion equation
论文作者
论文摘要
基于A稳定性和G稳定性的等效性，在[Akrivis，Chen，Yu，Zhou，Math中讨论了六步BDF方法的热量BDF方法的能量技术。 comp。，修订]。不幸的是，这种理论很难扩展时间分数PDE。在这项工作中，我们考虑了三种类型的次扩散模型，即单期，多项和分布式阶分数扩散方程。我们为$ k $ - 步骤向后差公式（bdf $ k $）生成的时间阶梯方案进行了新颖而简洁的稳定分析，以大致求解次扩散方程。分析主要依赖于能量技术，通过应用Grenander-Szegö定理。这种参数已被广泛用于确认各种$ a $稳定方案的稳定性（例如$ k = 1,2 $）。但是，由于$ a $稳定性损失，对于高阶BDF方法而言，这不是一件容易的事。本文的核心对象是填补此空白。
Based on the equivalence of A-stability and G-stability, the energy technique of the six-step BDF method for the heat equation has been discussed in [Akrivis, Chen, Yu, Zhou, Math. Comp., Revised]. Unfortunately, this theory is hard to extend 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 time stepping schemes generated by $k$-step backward difference formula (BDF$k$), 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 the higher-order BDF methods, due to the loss the $A$-stability. The core object of this paper is to fill in this gap.
