  We propose a novel numerical method for the solution of the shallow water
equations in different regimes of the Froude number making use of general
polygonal meshes. The fluxes of the governing equations are split such that
advection and acoustic-gravity sub-systems are derived, hence separating slow
and fast phenomena. This splitting allows the nonlinear convective fluxes to be
discretized explicitly in time, while retaining an implicit time marching for
the acoustic-gravity terms. Consequently, the novel schemes are particularly
well suited in the low Froude limit of the model, since no numerical viscosity
is added in the implicit solver. Besides, stability follows from a milder CFL
condition which is based only on the advection speed and not on the celerity.
High order time accuracy is achieved using the family of semi-implicit IMEX
Runge-Kutta schemes, while high order in space is granted relying on two
discretizations: (i) a cell-centered finite volume (FV) scheme for the
nonlinear convective contribution on the polygonal cells; (ii) a staggered
discontinuous Galerkin (DG) scheme for the solution of the linear system
associated to the implicit discretization of the pressure sub-system.
Therefore, three different meshes are used, namely a polygonal Voronoi mesh, a
triangular subgrid and a staggered quadrilateral subgrid. The novel schemes are
proved to be Asymptotic Preserving (AP), hence a consistent discretization of
the limit model is retrieved for vanishing Froude numbers, which is the given
by the so-called "lake at rest" equations. Furthermore, the novel methods are
well-balanced by construction, and this property is also demonstrated. Accuracy
and robustness are then validated against a set of benchmark test cases with
Froude numbers ranging in the interval $\Fr \approx [10^{-6};5]$, hence showing
that multiple time scales can be handled by the novel methods.
