The BDF3/EP3 Scheme for MBE with No Slope Selection is Stable

Abstract

We consider the classical molecular beam epitaxy (MBE) model with logarithmic type potential known as no-slope-selection. We employ a third order backward differentiation (BDF3) in time with implicit treatment of the surface diffusion term. The nonlinear term is approximated by a third order explicit extrapolation (EP3) formula. We exhibit mild time step constraints under which the modified energy dissipation law holds. We break the second Dahlquist barrier and develop a new theoretical framework to prove unconditional uniform energy boundedness with no size restrictions on the time step. This is the first unconditional result for third order BDF methods applied to the MBE models without introducing any stabilization term or fictitious variable. The analysis can be generalized to a restrictive class of phase field models whose nonlinearity has bounded derivatives. A novel theoretical framework is also established for the error analysis of high order methods.