The two-step backward differential formula (BDF2) implicit method with unequal time-steps is investigated for the Cahn-Hilliard model by focusing on the numerical influences of time-step variations. The suggested method is proved to preserve a modified energy dissipation law at the discrete levels if the adjoint time-step ratios fulfill a new step-ratio restriction $0<r_k:=\tau_k/\tau_{k-1}\le r_{\mathrm{user}}$ ($r_{\mathrm{user}}$ can be chosen by the user such that $r_{\mathrm{user}}<4.864$), such that it is mesh-robustly stable in an energy norm. We view the BDF2 formula as a convolution approximation of the first time derivative and perform the error analysis by using the recent suggested discrete orthogonal convolution kernels. By developing some novel convolution embedding inequalities with respect to the orthogonal convolution kernels, an $L^2$ norm error estimate is established at the first time under the updated step-ratio restriction $0<r_k\le r_{\mathrm{user}}$.The time-stepping scheme is mesh-robustly convergent in the sense that the convergence constant (prefactor) in the error estimate is independent of the adjoint time-step ratios. On the basis of ample tests on random time meshes, a useful adaptive time-stepping strategy is applied to efficiently capture the multi-scale behaviors and to accelerate the long-time simulation approaching the steady state.

翻译：用于 Cahn- Hilliard 模型的双步后退偏差公式( BDF2 2) 隐含法且时间步步不均的隐性公式( BDF2 2) 被调查为 Cahn- Hilliard 模型, 重点是时间步变的数值影响。 推荐的双步后退比率( BDF2 ) 符合新的一步步差限制 $0 <r_k:\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\