In this work, we numerically study the higher-ordered/extended Boussinesq system describing the propagation of water-waves over flat topography. A reformulation of the same order of precision that avoids the calculation of high order derivatives on the surface deformation is proposed. We show that this formulation enjoys an extended range of applicability while remaining stable. Moreover, a significant improvement in terms of linear dispersive properties in high frequency regime is made due to the suitable adjustment of a dispersion correction parameter. 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 under two main goals: validating the model and the numerical methods and assessing the potential need of such higher-order model. \red{The applicability of the proposed model and numerical method in practical problems is illustrated by a comparison with experimental data.}