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 $L^2$ 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.

Keywords:

• The molecular beam epitaxial model
• slope selection
• BDF3-5 schemes
• energy dissipation law
• error estimate

MSC Classifications (2020):

• 74A50
• 35Q99
• 65M12

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

We would like to acknowledge support by the National Natural Science Foundation of China (No. 11701229, 11701081, 11861060), Key Project of Natural Science Foundation of China (No. 61833005) and ZhiShan Youth Scholar Program of SEU, China Postdoctoral Science Foundation (No. 2019M651634), High-level Scientific Research foundation for the introduction of talent of Nanjing Institute of Technology (No. YKL201856).