  In this paper we derive and prove the wellposedness of a deep water model
that generalizes the Saut-Xu system for nonflat bottoms. Then, we present a new
numerical method based on a splitting approach for studying this system. The
advantage of this method is that it does not require any low pass filter to
avoid spurious oscillations. We prove a local error estimate and we show that
our scheme represents a good approximation of order one in time. Then, we
perform some numerical experiments which confirm our theoretical result and we
study three physical phenomena : the evolution of water waves over a rough
bottom; the evolution of a KdV soliton when the shallowness parameter
increases; the homogenization effect of rapidly varying topographies on water
waves.
