Backward difference formula: The energy technique for subdiffusion equation
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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 (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 the higher-order BDF methods, due to the loss the A-stability. The core object of this paper is to fill in this gap.
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