A New Discrete Energy Technique for Multi-Step Backward Difference Formulas
Hong-Lin Liao , Tao Tang , Tao Zhou
CSIAM Trans. Appl. Math. ›› 2022, Vol. 3 ›› Issue (2) : 318 -334.
A New Discrete Energy Technique for Multi-Step Backward Difference Formulas
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.
Linear diffusion equations / backward differentiation formulas / discrete orthogonal convolution kernels / positive definiteness / stability and convergence
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