Analysis of the second-order BDF scheme with variable steps for the molecular beam epitaxial model without slope selection

In this work, we are concerned with the stability and convergence analysis of second order BDF (BDF2) scheme variable steps for molecular beam epitaxial model without slope selection. We first show that variable-step BDF2 is convex uniquely solvable under a weak time-step constraint. Then it preserves an energy dissipation law if adjacent ratios $r_k:=\tau_k/\tau_{k-1}<3.561.$ Moreover, novel discrete orthogonal convolution kernels argument some new convolutional inequalities, $L^2$ norm rigorous error estimates established, same step-ratios constraint ensuring stability., i.e., $0<r_k<3.561.$ This known to be best result in literature. finally adopt adaptive time-stepping strategy accelerate computations steady state solution confirm our theoretical findings by numerical examples.