无斜率选择的分子束外延模型的二阶变步长BDF格式分析
Analysis of the second order BDF scheme with variable steps for the molecular beam epitaxial model without slope selection
摘要：
本文研究了无斜率选择的分子束外延模型的二阶变步长BDF（BDF 2）格式的稳定性和收敛性。我们首先证明了变步长BDF2格式在弱时间步长约束下是凸的且唯一可解。然后证明了当相邻时间步长比为$r_k:=τ_k/τ_{k-1}<3.561.$时，它保持一个能量耗散律。此外，在保证能量稳定的相同步长比约束下，利用一种新的离散正交卷积核参数和一些新的离散卷积不等式，建立了$L^2$范数稳定性和严格的误差估计。也就是说，$0<r_k<3.561.$已知这是文献中的最佳结果。最后，我们采用自适应的时间步长策略来加速稳态解的计算，并通过数值例子证实了我们的理论结果。
英文摘要：
In this work, we are concerned with the stability and convergence analysis of the second order BDF (BDF2) scheme with variable steps for the molecular beam epitaxial model without slope selection. We first show that the variable-step BDF2 scheme is convex and uniquely solvable under a weak time-step constraint. Then we show that it preserves an energy dissipation law if the adjacent time-step ratios $r_k:=τ_k/τ_{k-1}<3.561.$ Moreover, with a novel discrete orthogonal convolution kernels argument and some new discrete convolutional inequalities, the $L^2$ norm stability and rigorous error estimates are established, under the same step-ratios constraint that ensuring the energy stability., i.e., $0<r_k<3.561.$ This is known to be the best result in literature. We finally adopt an adaptive time-stepping strategy to accelerate the computations of the steady state solution and confirm our theoretical findings by numerical examples.
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