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

TL;DR

This paper introduces a stable linear combination of the seven-step BDF method and its shifted version, called WSBDF7, extending stability analysis to higher-order methods for parabolic equations.

Contribution

The authors construct and analyze a new stable scheme, WSBDF7, by combining non-zero-stable schemes and establish its stability for parabolic equations using energy techniques.

Findings

WSBDF7 has larger stability regions than classical BDF7.

The stability of WSBDF7 is proven for various types of equations.

The approach extends to nonlinear and fractional equations.

Abstract

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.