Energy dissipation law of the variable time-step fractional BDF2 scheme for the time fractional molecular beam epitaxial model

Abstract

In this work, we take a consideration of the time fractional molecular beam epitaxial(MBE) models. The variable time-step BDF2 methods are proposed for time fractional MBE models in order to obtain the high-order accuracy in time since the low regularity occurring in the initial state. By virtue of discrete gradient structures, energy dissipation laws of the schemes for time fractional MBE model with and without slope selection are proved, respectively. Besides, the unique solvabilities of two schemes are also guaranteed. The discrete modified energy and the corresponding energy dissipation law are asymptotically compatible with the associated discrete energy and the energy dissipation law of the variable time-step BDF2 method for the classical MBE model. Numerical experiments with the time adaptive strategies are provided to verify the effectiveness of the schemes.

Keywords:

• Time-fractional molecular beam epitaxial
• fractional BDF2 scheme
• energy dissipation law
• adaptive time-stepping method
• unique solvability

2010 AMS Subject Classifications:

• 65M06
• 65M12
• 35R11

Disclosure statement

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

Additional information

Funding

The authors would like to acknowledge support by the National Natural Science Foundation of China (No. U22B2046, 61833005,11701081), Jiangsu Province Natural Science Foundation (No. BK20222023).