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

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