  The aim of this work is to construct efficient finite volume schemes for the
numerical study of sediment transport in shallow water, in the framework of the
Exner model.In most cases, the velocity related to the sediment is much lower
that the fluid velocity, which, in turn, may be much lower that the
free-surface wave speed. Explicit methods that resolve all waves require small
time steps due to the CFL stability restriction because of fast surface waves.
Furthermore, if Rusanov flux is adopted, slow sediment waves may be affected by
the large numerical diffusion. The objective of the present work is to
drastically improve the efficiency in the computation of the evolution of the
sediment by treating water waves implicitly, thus allowing much larger time
steps than the one required by fully explicit schemes. The goal is reached by
suitably semi-implicit schemes obtained by the use of implicit-explicit
Runge-Kutta methods.
