The BDF3/EP3 scheme for MBE with no slope selection is stable

TL;DR

This paper introduces a stable third-order BDF3/EP3 numerical scheme for the no-slope-selection MBE model, proving unconditional energy boundedness and establishing a new error analysis framework without stability restrictions.

Contribution

It develops the first unconditional stability proof for third-order BDF methods applied to MBE models without stabilization or fictitious variables.

Findings

Proves unconditional energy boundedness of the scheme.

Breaks the second Dahlquist barrier for this application.

Establishes a new error analysis framework for high-order methods.

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 terms or fictitious variables. A novel theoretical framework is also established for the error analysis of high order methods.