Backward difference formulae: the energy technique for subdiffusion equation

Minghua Chen, Fan Yu, Zhi Zhou

Research output: Journal article publicationJournal articleAcademic researchpeer-review

3 Citations (Scopus)


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.

Original languageEnglish
Article number94
Pages (from-to)1-22
Number of pages22
JournalJournal of Scientific Computing
Issue number3
Publication statusPublished - Jun 2021


  • Backward difference formulae
  • Energy technique
  • Stability analysis
  • Subdiffusion equation

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • General Engineering
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics


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