  The generalised continuum theory model of the dynamical evolution of surfaces
sputtered by ion-bombardment is a noisy Kuramoto-Sivashinsky type partial
differential equation. For some generic cases of sputtering parameters,
existing similar equations have shed a great deal of light and therefore
provided some understanding of the intricacies of evolving ion-sputtered
surfaces without a direct solution of the generalised model. However, recent
results have demonstrated a wider range of scaling regimes of the sputtering
conditions, a large number of which have no similar existing solved models in
other research fields for comparison, and whose characteristics are therefore
largely unknown. In this paper, a discretisation of the generalised continuum
model is performed for direct numerical simulations, the results of which are
applicable to all manner of simulations required for the different possible
scenarios in the dynamical evolution of sputtered surfaces. The approximation
errors and implementation of the results in any such simulation are also
discussed.
