Equation-free accelerated simulations of the morphological relaxation of crystal surfaces

A method for accelerating kinetic Monte Carlo simulations of solid surface morphology evolution, based on the equationfree projective integration (EFPI) technique, is developed and investigated. This method is demonstrated through application to the 1+1 dimensional solid-on-solid model for surface evolution. EFPI exploits the multiscale nature of a physics problem, using fine-scale simulations at short times to evolve coarse length scales over long times. The method requires identification of a set of coarse variables that parameterize the system, and it is found that the most obvious coarse variables for this problem, those related to the ensemble-averaged surface position, are inadequate for capturing the dynamics of the system. This is remedied by including among the coarse variables a statistical description of the fine scales in the problem, which in this case can be captured by a two-point correlation function. Projective integration allows speedup of the simulations, but if speed-up of more than a factor of around 3 is attempted the solution can become oscillatory or unstable. This is shown to be caused by the presence of both fast and slow components of the two-point correlation function, leading to the equivalent of a stiff system of equations that is hard to integrate. By fixing the fast components of the solution over each projection step, we are able to achieve speedups of a factor of 20 without oscillations, while maintaining accuracy.