论文标题
$ l^2 $ norm误差估计BDF方法最多五阶的相位场晶体模型
$L^2$ norm error estimates of BDF methods up to fifth-order for the phase field crystal model
论文作者
论文摘要
研究了第三次的众所周知的向后差公式（BDF），第四阶和第五阶以用于相位场晶体模型的时间整合。通过建立bdf-$ \ rmk $（$ \ rmk = 3,4,5 $）公式的新型离散梯度结构，我们在离散级别上建立了耗能耗散法，然后获得对相关数值方案的先验解决方案估计值（但是，我们无法为相应的BDF-6方案构建任何bdf-6 fore bdf-6-n ectient bdf-6 n ectient for bdf-6结构）。借助离散的正交卷积内核和Young-type卷积不平等，通过离散能源技术建立了一些简洁的$ l^2 $规范错误估计（相对于$ l^2 $规范的起始数据）。据我们所知，这是非A稳定BDF方案的这种类型$ l^2 $规范错误估计是针对非线性抛物线方程的。提出了数值示例以验证和支持理论分析。
The well-known backward difference formulas (BDF) of the third, the fourth and the fifth orders are investigated for time integration of the phase field crystal model. By building up novel discrete gradient structures of the BDF-$\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.
