The weighted and shifted seven-step BDF method for parabolic equations

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.