On a sinc-type MBE model

TL;DR

This paper introduces a new sinc-type MBE model derived from a cosine energy functional, with bounded derivatives, and provides a novel theoretical framework proving unconditional energy boundedness for semi-implicit methods.

Contribution

It presents a new sinc-type MBE model with bounded derivatives and establishes the first unconditional energy boundedness results for semi-implicit phase field models.

Findings

Explicit time step constraints for energy dissipation are quantified.

Numerical simulations confirm theoretical predictions.

First unconditional energy boundedness proof for semi-implicit methods without artificial stabilization.

Abstract

We introduce a new sinc-type molecular beam epitaxy model which is derived from a cosine-type energy functional. The landscape of the new functional is remarkably similar to the classical MBE model with double well potential but has the additional advantage that all its derivatives are uniformly bounded. We consider first order IMEX and second order BDF2 discretization schemes. For both cases we quantify explicit time step constraints for the energy dissipation which is in good accord with the practical numerical simulations. Furthermore we introduce a new theoretical framework and prove unconditional uniform energy boundedness with no size restrictions on the time step. This is the first unconditional (i.e. independent of the time step size) result for semi-implicit methods applied to the phase field models without introducing any artificial stabilization terms or fictitious variables.