Backward Difference Formulae: The Energy Technique for Subdiffusion Equation

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-Szego 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.