  In this work, we numerically study the higher-ordered/extended Boussinesq
system describing the propagation of water-waves over flat topography. An
equivalent suitable reformulation is proposed, making the model more
appropriate for the numerical implementation and significantly improved in
terms of linear dispersive properties in high frequency regimes due to the
suitable adjustment of a dispersion correction parameter. Moreover, we show
that a significant interest is behind the derivation of a new formulation of
the higher-ordered/extended Boussinesq system that avoids the calculation of
high order derivatives existing in the model. We show that this formulation
enjoys an extended range of applicability while remaining stable. We develop a
second order splitting scheme where the hyperbolic part of the system is
treated with a high-order finite volume scheme and the dispersive part is
treated with a finite difference approach. Numerical simulations are then
performed to validate the model and the numerical methods.
