We derive unconditionally stable and convergent variable-step BDF2 scheme for solving the MBE model with slope selection. The discrete orthogonal convolution kernels of the variable-step BDF2 method is commonly utilized recently for solving the phase field models. In this paper, we further prove some new inequalities, concerning the vector forms, for the kernels especially dealing with the nonlinear terms in the slope selection model. The convergence rate of the fully discrete scheme is proved to be two both in time and space in $L^2$ norm under the setting of the variable time steps. Energy dissipation law is proved rigorously with a modified energy by adding a small term to the discrete version of the original free energy functional. Two numerical examples including an adaptive time-stepping strategy are given to verify the convergence rate and the energy dissipation law.

翻译：本文针对含斜率选择的分子束外延模型，提出了无条件稳定且收敛的变步长BDF2格式。目前，变步长BDF2方法的离散正交卷积核常被用于求解相场模型。本文进一步证明了该类核的若干新不等式（涉及向量形式），特别针对斜率选择模型中的非线性项。在全离散格式框架下，证明了在变时间步长设置下时间与空间$L^2$范数的收敛阶均为二阶。通过在原自由能泛函离散形式中添加小修正项，严格证明了修正能量的耗散律。给出了包含自适应时间步进策略的两个数值算例，验证了收敛阶和能量耗散律。