ENATE scheme (Enhanced Numerical Approximation of a Transport Equation) was proposed by the first author and further developed by the second one in his Ph.D. Thesis. The improvements to be presented refer to the splitting of the convective flux within a discretization interval. This flux is divided into a mean plus a deviation that accounts for its variation inside the interval. This variation is then shifted to the RHS and considered as a pseudo-source. This treatment of the convection term allows to use the formulae developed in ENATE for the source discretization with Hermite polynomials. This approach gives machine accuracy even with very few discretization intervals if the solution can be approximated by polynomials of up to fifth degree, although this can be improved by changing the Hermite polynomials employed. There are no restrictions on the Reynolds/Pèclet values; machine accuracy can be reached for the whole range. The detailed derivation of ENATE for a 2D scalar transport equation will be described and its application to the Navier-Stokes equations highlighted. Problems inherent to the Navier-Stokes equations such as the pressure-velocity coupling will commented upon. Several computational cases will be shown that evidence the good accuracy of the approach.