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 - <p style="text-align: justify;">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.</p>