Discrete Energy Analysis of the Third-Order Variable-Step BDF Time-Stepping for Diffusion Equations

Hong Lin Liao*, Tao Tang, Tao Zhou

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

4 Citations (Scopus)


This is one of our series works on discrete energy analysis of the variable-step BDF schemes. In this part, we present stability and convergence analysis of the third-order BDF (BDF3) schemes with variable steps for linear diffusion equations, see, e.g., [SIAM J. Numer. Anal., 58:2294-2314] and [Math. Comp., 90: 1207-1226] for our previous works on the BDF2 scheme. To this aim, we first build up a discrete gradient structure of the variable-step BDF3 formula under the condition that the adjacent step ratios are less than 1.4877, by which we can establish a discrete energy dissipation law. Mesh-robust stability and convergence analysis in the L2norm are then obtained. Here the mesh robustness means that the solution errors are well controlled by the maximum time-step size but independent of the adjacent time-step ratios. We also present numerical tests to support our theoretical results.

Original languageEnglish
Pages (from-to)325-344
Number of pages20
JournalJournal of Computational Mathematics
Issue number2
Early online dateFeb 2023
Publication statusPublished - Mar 2023

Scopus Subject Areas

  • Computational Mathematics

User-Defined Keywords

  • Diffusion equations
  • Discrete gradient structure
  • Discrete orthogonal convolution kernels
  • Stability and convergence
  • Variable-step third-order BDF scheme


Dive into the research topics of 'Discrete Energy Analysis of the Third-Order Variable-Step BDF Time-Stepping for Diffusion Equations'. Together they form a unique fingerprint.

Cite this