  We present an advection-pressure flux-vector splitting method for the one and
two- dimensional shallow water equations following the approach first proposed
by Toro and V\'azquez for the compressible Euler equations. The resulting
first-order schemes turn out to be exceedingly simple, with accuracy and
robustness comparable to that of the sophisticated Godunov upwind method used
in conjunction with complete non- linear Riemann solvers. The technique splits
the full system into two subsystems, namely an advection system and a pressure
system. The sought numerical flux results from fluxes for each of the
subsystems. The basic methodology, extended on 2D unstructured meshes,
constitutes the building block for the construction of numerical schemes of
very high order of accuracy following the ADER approach. The presented
numerical schemes are systematically assessed on a carefully selected suite of
test problems with reference solutions, in one and two space dimensions.The
applicabil- ity of the schemes is illustrated through simulations of tsunami
wave propagation in the Pacific Ocean.
