  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.
