$L^2$ norm error estimates of BDF methods up to fifth-order for the phase field crystal model

TL;DR

This work addresses stable, high-order time discretization for the phase field crystal (PFC) gradient-flow model by developing discrete gradient structures for BDF-$3$,$4$,$5$ and proving discrete energy dissipation laws. A discrete energy framework, aided by discrete orthogonal convolution (DOC) kernels, yields rigorous $L^2$-norm error estimates for non-A-stable BDF schemes in a nonlinear parabolic setting, with a notable demonstration that BDF-$6$ lacks such a structure. The authors provide explicit quadratic decompositions, energy functionals, and convolution inequalities, culminating in $O( au^{ m k})$ convergence in the $L^2$ norm under a mild time-step restriction, supported by numerical experiments verifying both convergence rates and physically relevant properties like energy dissipation and volume conservation. The results advance reliable long-time simulations of coarsening dynamics in the PFC model and offer a framework potentially extendable to fully discrete schemes preserving discrete energy laws.

Abstract

The well-known backward difference formulas (BDF) of the third, the fourth and the fifth orders are investigated for time integration of the phase field crystal model. By building up novel discrete gradient structures of the BDF-$\rmk$ ($\rmk=3,4,5$) formulas, we establish the energy dissipation laws at the discrete levels and then obtain the priori solution estimates for the associated numerical schemes (however, we can not build any discrete energy dissipation law for the corresponding BDF-6 scheme because the BDF-6 formula itself does not have any discrete gradient structures). With the help of the discrete orthogonal convolution kernels and Young-type convolution inequalities, some concise $L^2$ norm error estimates (with respect to the starting data in the $L^2$ norm) are established via the discrete energy technique. To the best of our knowledge, this is the first time such type $L^2$ norm error estimates of non-A-stable BDF schemes are obtained for nonlinear parabolic equations. Numerical examples are presented to verify and support the theoretical analysis.

Figures (2)

• Figure 1: The crystal growth process obtained at $t=1$, $200$, $300$, $400$, $500$ and $1000$ by the BDF-5 scheme (the BDF-3 and BDF-4 schemes generate similar profiles).
• Figure 2: Evolutions of original energy (left) and volume difference (right)

Theorems & Definitions (29)

• Lemma 2.1
• Theorem 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• Remark 1
• Theorem 2.2
• proof
• Lemma 2.4
• proof