Stability of the BDF methods of order up to five for parabolic equations can be established by the energy technique via Nevanlinna--Odeh multipliers. The nonexistence of Nevanlinna--Odeh multipliers makes the six-step BDF method special; however, the energy technique was recently extended by the authors in [Akrivis et al., SIAM J. Numer. Anal. \textbf{59} (2021) 2449--2472] and covers all six stable BDF methods. The seven-step BDF method is unstable for parabolic equations, since it is not even zero-stable. In this work, we construct and analyze a stable linear combination of two non zero-stable schemes, the seven-step BDF method and its shifted counterpart, referred to as WSBDF7 method. The stability regions of the WSBDF$q, q\leqslant 7$, with a weight $\vartheta\geqslant1$, increase as $\vartheta$ increases, are larger than the stability regions of the classical BDF$q,$ corresponding to $\vartheta=1$. We determine novel and suitable multipliers for the WSBDF7 method and establish stability for parabolic equations by the energy technique. The proposed approach is applicable for mean curvature flow, gradient flows, fractional equations and nonlinear equations.

翻译：对于抛物方程，基于Nevanlinna--Odeh乘子的能量技术可证明最高达五阶BDF方法的稳定性。由于Nevanlinna--Odeh乘子的缺失，六步BDF方法具有特殊性；然而，作者近期在[Akrivis等, SIAM J. Numer. Anal. \textbf{59} (2021) 2449--2472]中扩展了该能量技术，使其覆盖所有六种稳定的BDF方法。七步BDF方法因不具备零稳定性而无法用于抛物方程。本文构造并分析了一种稳定的线性组合方法——七步BDF方法及其移位形式的组合（记为WSBDF7方法），其中后者具有非零稳定性。对于权重$\vartheta\geqslant1$的WSBDF$q$方法（$q\leqslant7$），其稳定区域随$\vartheta$增大而扩大，且大于经典BDF$q$方法（对应$\vartheta=1$）的稳定区域。我们为WSBDF7方法确定了新型的适用乘子，并通过能量技术建立了抛物方程的稳定性。该方法可应用于平均曲率流、梯度流、分数阶方程及非线性方程。