Backward difference formula: The energy technique for subdiffusion equation

TL;DR

This paper develops a novel energy technique-based stability analysis for higher-order backward difference formulas applied to subdiffusion equations, extending stability results beyond classical A-stability limitations.

Contribution

It introduces a new stability analysis method for k-step BDF schemes solving subdiffusion equations, overcoming challenges posed by loss of A-stability in higher orders.

Findings

Stability of BDFk schemes for subdiffusion models is established.

The energy technique effectively handles multi-term and distributed order fractional equations.

The approach extends stability analysis to higher-order methods beyond classical A-stability constraints.

Abstract

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-Szeg\"{o} 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.