**Equation-Free Accelerated Simulations of the Morphological Relaxation
of Crystal Surfaces**

G. J. Wagner, X. Zhou, S. J. Plimpton, Int J for Multiscale Computational Engineering, 8, 423-439 (2010).

A method for accelerating kinetic Monte Carlo simulations of solid surface morphology evolution, based on the equation-free 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.

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