Adaptive time-stepping algorithms for molecular beam epitaxy: Based on energy or roughness

Abstract Due to the small parameter, high order derivatives and nonlinear term, it takes a long time to reach the steady state in the simulation of the molecular beam epitaxy (MBE) model. In this work, based on the numerical scheme in Qiao et al. (2011) and the good properties it holds, we introduce two types of adaptive time-stepping methods, in which the physical quantities like energy or roughness are involved to produce time steps. Similarly to the spatial adaptive method, we take equidistribution of the physical quantities in time direction to control the simulation error. The numerical experiment shows that the computational time is significantly saved by these methods.