A high-order energy stable method for the MBE models with slope selection by using Lagrange multiplier approach

In this work, we develop high-order convex splitting implicit–explicit Runge–Kutta methods for Molecular Beam Epitaxy (MBE) model with slope selection, which plays key roles in materials science and physics for describing various phenomena, such as phase transitions, interactions and interfacial dynamics. Since the epitaxy surface height evolution equation is viewed as a dynamical form of a $L^2$-gradient flow, MBE has the highly similar growing processes as the growing facets in phase-ordering process in magnetic systems. Within this context, we focus our attention on the systems with multiple components possess greater physical significance than their classical (single-component) counterparts. We rigorously prove that the proposed schemes both preserve the energy dissipation and mass conservation. Finally, the accuracy and efficiency of proposed schemes are demonstrated by some numerical experiments.