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.