Asymptotically compatible energy law of the Crank–Nicolson type schemes for time-fractional MBE models

Abstract

This paper is concerned with the numerical simulations for the dynamics of the time-fractional molecular beam epitaxy models. The variable-step Crank–Nicolson-type schemes are proposed and analyzed for model with and without slope selection respectively. By using the discrete gradient structures of discrete fractional derivative, the numerical schemes preserve discrete variational energy dissipation law on general meshes unconditionally. These are generalizations of classical energy dissipation laws and are asymptotically compatible with the classical MBE model. Numerical examples with an adaptive time-stepping strategy are provided to show the effectiveness of our schemes.