无倾斜选择的分子束外延模型变步长BDF2格式的最优误差估计

SHARP ERROR ESTIMATE OF BDF2 SCHEME WITH VARIABLE TIME STEPS FOR MOLECULAR BEAM EPITAXIAL MODELS WITHOUT SLOP SELECTION

对于没有斜率选择的分子束外延模型,具有可变时间步长的两步向后微分公式(BDF2)的稳定性和收敛性仍未被完全解决。在本文中,我们首先证明了该BDF2格式在新的相邻时间步长比条件下保持修正的能量耗散定律:r_(k)=τ_(k)/τ_(k-1)≤4.8645-δ,其中δ>0是给定的任意小常数。然后,我们介绍了最近发展的离散正交卷积(DOC)和离散互补卷积(DCC)核技巧,并在新的比率条件r_(k)≤4.8645-δ下给出了BDF2格式的鲁棒且最优的二阶收敛性。鲁棒性意味着,除了r_(k)≤4.8645-δ以外,收敛性不需要其他时间步长上的约束条件。此外,我们的分析表明,使用一阶BDF1格式计算第一步数值解足以确保全局最优收敛阶。也就是说,选择BDF1格式计算起始步的数值解不会导致全局二阶收敛的损失。数值算例验证了我们的理论分析。

The stability and convergence of two-step backward differentiation formula(BDF2)with variable time steps still remain incomplete for solving the molecular beam epitaxial model without slope selection.In this paper,we first prove the proposed BDF2 scheme to preserve a modified energy dissipation law under a new adjacent time-step ratio condition:r_(k)=τ_(k)/τ_(k-1)≤4.8645-δ,whereδ>0 is a given arbitrarily small constant.After that,we introduce the recently developed techniques of the discrete orthogonal convolution(DOC)and discrete complementary convolution(DCC)kernels,and present the robust and sharp second-order convergence of the BDF2 scheme with the new ratio condition:r_(k)≤4.8645-δ.The robustness means the convergence does not need other constrained condition on the time steps except for r_(k)≤4.8645-δ.In addition,our analysis shows that the first-order BDF1 scheme for the start step is enough to ensure the globally optimal convergence order.This is,the choice of BDF1 scheme for the start step does not bring the loss of global second-order convergence.Numerical examples are provided to demonstrate the theoretical analysis.