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:摘要:本文研究了第三、第四和第五阶反向差分公式(BDF)在晶相场模型的时间积分中的应用。通过构建BDF-$k$($k=3,4,5$)公式的新型离散梯度结构,建立了离散级别上的能量耗散定律,并通过离散能量技术获得了相关数值方案的应先估计(然而,由于BDF-6公式本身没有任何离散梯度结构,因此我们无法为相应的BDF-6方案建立任何离散能量耗散定律)。借助于离散正交卷积核和Young型卷积不等式,通过离散能量技术建立了$L^2$范数误差估计(相对于起始数据在$L^2$范数下)的简明表达。据我们所知,这是第一次为非A-稳定BDF方案在非线性抛物线方程中获得此类$L^2$范数误差估计。最后,通过数值实验验证和支持了理论分析。