  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.
