  We present a splitting method for the one-dimensional Saint-Venant-Exner
equations used for describing the bed evolution in shallow water systems. We
adapt the flux vector splitting approach of Toro and Vazquez-Cend\`on and
identify one subsystem of conservative equations (advection system) and one of
non-conservative equations (pressure system), both having a very simple
eigenstructure compared to the full system. The final numerical scheme is
constructed using a Godunov-type path-conservative scheme for the pressure
system and a simple conservative Godunov method for the advection system and
solved following a coupled solution strategy. The resulting first-order
accurate method is extended to second order of accuracy in space and time via
the ADER approach together with an AENO reconstruction technique. Accuracy,
robustness and well-balanced properties of the resulting scheme are assessed
through a carefully selected suite of testcases. The scheme is exceedingly
simple, accurate and robust as the sophisticated Godunov methods. A distinctive
feature of the novel scheme is its flexibility in the choice of the sediment
transport closure formula, which makes it particularly attractive for
scientific and engineering applications.
