The energy technique for the six-step BDF method

Abstract

In combination with the Grenander-Szegö 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 BDF methods for parabolic equations with selfadjoint elliptic part. This is particularly useful for the six-step BDF method for which no Nevanlinna-Odeh multiplier exists. We introduce multipliers satisfying the positivity property for the six-step BDF method and establish stability of the method for parabolic equations.

BDF$6$ SAV schemes for time-fractional Allen-Cahn dissipative systems

“…We introduce the following BDF6 SAV schemes inspired by the six-step simple multiplier in [1] and the SAV approach introduced in [9,10]. The key for the SAV approach is to introduce a scalar auxiliary variable (SAV).…”

“…There are already a class of new multipliers for the six-step BDF method of the time-dependent PDEs [1,7]. For example, the multiplier µ 1 = 13/9, µ 2 = −25/36, µ 3 = 1/9, µ 4 = µ 5 = µ 6 = 0 was constructed in [1] for the parabolic equation. Here, based on the idea of [1,7], we develop the above multiplier to the time fractional problem.…”

“…For example, the multiplier µ 1 = 13/9, µ 2 = −25/36, µ 3 = 1/9, µ 4 = µ 5 = µ 6 = 0 was constructed in [1] for the parabolic equation. Here, based on the idea of [1,7], we develop the above multiplier to the time fractional problem.…”

“…China email: yuf20@lzu.edu.cn; chenmh@lzu.edu.cn key role. In contrast, it has been proved that the Nevanlinna-Odeh multipliers for the BDF6 method do not exist in [1]. Fortunately, a class of six-step simple multipliers are proposed in [1], which makes the energy technique applicable to the error analysis of BDF6 SAV schemes.…”

“…In comparison with the error analysis of BDFk (1 k 5) SAV schemes for classical Allen-Cahn equation in [10], the error analysis of BDF6 SAV schemes remains unavailable, which is the main motivation of the present work. In this paper, we apply the six-step simple multiplier in [1] and the SAV approach in [9,10] to construct the explicit-implicit BDF6 SAV schemes for time-fractional dissipative systems. We show that the proposed BDF6 SAV schemes are unconditional energy stable.…”