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.

翻译：针对相场晶体模型的时间积分问题，研究了著名的三阶、四阶和五阶向后差分公式（BDF）。通过构建BDF-$\rmk$（$\rmk=3,4,5$）公式的新型离散梯度结构，我们在离散层次上建立了能量耗散定律，并由此获得了相关数值格式的解的先验估计（然而，由于BDF-6公式本身不具备任何离散梯度结构，我们无法为相应的BDF-6格式建立任何离散能量耗散定律）。借助离散正交卷积核与杨型卷积不等式，通过离散能量技术建立了一些简洁的$L^2$范数误差估计（关于$L^2$范数中的起始数据）。据我们所知，这是首次针对非线性抛物型方程获得非A-稳定的BDF格式的此类$L^2$范数误差估计。数值算例验证并支持了理论分析。