Volume 3, Issue 2
A New Discrete Energy Technique for Multi-Step Backward Difference Formulas

CSIAM Trans. Appl. Math., 3 (2022), pp. 318-334.

Published online: 2022-05

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• 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.

• Keywords

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

65M06, 65M12

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@Article{CSIAM-AM-3-318, author = {Hong-Lin and Liao and and 23339 and and Hong-Lin Liao and Tao and Tang and and 23340 and and Tao Tang and Tao and Zhou and and 23341 and and Tao Zhou}, title = {A New Discrete Energy Technique for Multi-Step Backward Difference Formulas}, journal = {CSIAM Transactions on Applied Mathematics}, year = {2022}, volume = {3}, number = {2}, pages = {318--334}, 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.

}, 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} }
TY - JOUR T1 - A New Discrete Energy Technique for Multi-Step Backward Difference Formulas AU - Liao , Hong-Lin AU - Tang , Tao AU - Zhou , Tao JO - CSIAM Transactions on Applied Mathematics VL - 2 SP - 318 EP - 334 PY - 2022 DA - 2022/05 SN - 3 DO - http://doi.org/10.4208/csiam-am.SO-2021-0032 UR - https://global-sci.org/intro/article_detail/csiam-am/20540.html KW - Linear diffusion equations, backward differentiation formulas, discrete orthogonal convolution kernels, positive definiteness, stability and convergence. AB -

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.

Hong-Lin Liao, Tao Tang & Tao Zhou. (2022). A New Discrete Energy Technique for Multi-Step Backward Difference Formulas. CSIAM Transactions on Applied Mathematics. 3 (2). 318-334. doi:10.4208/csiam-am.SO-2021-0032
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