Energy stable and convergent BDF3-5 schemes for the molecular beam epitaxial model with slope selection

Abstract

The backward differential formulas of order 3-5 (BDF3-5) are applied to simulate the molecular beam epitaxial (MBE) model with slop selection. The fully implicit BDF3-5 schemes are uniquely solvable under an estimated time-step requirement. We prove that the schemes conserve the modified discrete energy dissipation laws by utilizing the discrete gradient structures of BDF3-5 formulas. Furthermore, we present a new convolution inequality to handle the nonlinear term for the error estimate. More importantly, the norm convergence analysis of the BDF3-5 schemes are proved rigorously without the assumption of Lipschitz boundedness. Numerical examples are shown to verify the efficiency and the accuracy of the developed schemes.