  We present a comparison between finite differences schemes and a
pseudospectral method applied to the numerical integration of stochastic
partial differential equations that model surface growth. We have studied, in
1+1 dimensions, the Kardar, Parisi and Zhang model (KPZ) and the Lai, Das Sarma
and Villain model (LDV). The pseudospectral method appears to be the most
stable for a given time step for both models. This means that the time up to
which we can follow the temporal evolution of a given system is larger for the
pseudospectral method. Moreover, for the KPZ model, a pseudospectral scheme
gives results closer to the predictions of the continuum model than those
obtained through finite difference methods. On the other hand, some numerical
instabilities appearing with finite difference methods for the LDV model are
absent when a pseudospectral integration is performed. These numerical
instabilities give rise to an approximate multiscaling observed in the
numerical simulations. With the pseudospectral approach no multiscaling is seen
in agreement with the continuum model.
