A New Discrete Energy Technique for Multi-Step Backward Difference Formulas

Hong-Lin Liao, Tao Tang, Tao Zhou*

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

7 Citations (Scopus)

Abstract

The backward differentiation formula (BDF) is a popular family of implicit methods for the numerical integration of stiff differential equations. It is well noticed that the stability and convergence of the A-stable BDF1 and BDF2 schemes for parabolic equations can be directly established by using the standard discrete energy analysis. However, such classical analysis seems not directly applicable to the BDF-k with 3 ≤ k ≤ 5. To overcome the difficulty, a powerful analysis tool based on the Nevanlinna-Odeh multiplier technique [Numer. Funct. Anal. Optim., 3:377-423, 1981] was developed by Lubich et al. [IMA J. Numer. Anal., 33:1365-1385, 2013]. In this work, by using the so-called discrete orthogonal convolution kernel technique, we recover the classical energy analysis so that the stability and convergence of the BDF-k with 3 ≤ k ≤ 5 can be established.

Original languageEnglish
Pages (from-to)318-334
Number of pages17
JournalCSIAM Transactions on Applied Mathematics
Volume3
Issue number2
DOIs
Publication statusPublished - Jun 2022

Scopus Subject Areas

  • Applied Mathematics

User-Defined Keywords

  • backward differentiation formulas
  • stability and convergence
  • discrete orthogonal convolution kernels
  • Linear diffusion equations
  • positive definiteness

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