There has been a significant research interest in the dynamics of the molecular beam epitaxy (MBE) growth lately. The MBE technique is among the most refined methods for the growth of thin solid films and it is of great importance for applied studies. Numerical methods for solving the continuum model of the MBE require very large time simulation, and therefore large time steps become necessary. The main purpose of this talk is to present and analyze highly stable time discretizations which allow much larger time step than that for a standard implicit-explicit approach. To this end, an extra term, which is consistent with the order of the time discretization, is added to stabilize the numerical schemes. Then the stability properties of the resulting schemes are established rigorously. Numerical experiments are carried out to support the theoretical claims. The proposed methods are also applied to simulate the MBE models with large solution times. The power laws for the coarsening process are obtained and are compared with previously published results.

The generalization of the proposed method to the Cahn-Hilliard equation is also discussed.