The energy technique for the six-step BDF method

Abstract

In combination with the Grenander-Szeg\H o theorem, we observe that a relaxed positivity condition on multipliers, milder than the basic requirement of the Nevanlinna-Odeh multipliers that the sum of the absolute values of their components is strictly less than 1, makes the energy technique applicable to the stability analysis of backward difference formula (BDF) methods for parabolic equations with self-adjoint elliptic part. This is particularly useful for the six-step BDF method, for which we show that no Nevanlinna-Odeh multipliers exist. We introduce multipliers satisfying the positivity property for the six-step BDF method and establish stability of the method for parabolic equations.