CSIAM Trans. Appl. Math., 3 (2022), pp. 318-334.
Published online: 2022-05
Cited by
- BibTex
- RIS
- TXT
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.
}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.SO-2021-0032}, url = {http://global-sci.org/intro/article_detail/csiam-am/20540.html} }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.