  Starting from the 3D Gross-Pitaevskii equation we revisit the dimensional
reduction to an effective one-dimensional wave-equation that describes the
longitudinal dynamics of a Bose condensate in an axially-symmetric external
potential. Using a variational approach, Salasnich et al. introduced the
non-polynomial Schr\"odinger equation (NPSE) which takes into account radial
broadening of the wave function due to particle interactions. A closed form was
derived by neglecting slow variation of the radial wave function along the
longitudinal direction. We show that the full equation can efficiently be
implemented using a time splitting spectral scheme coupled to a constraint
equation. We confirm the validity of the approximation for experimentally
relevant parameters. The corrections are found to be localized to regions of
strong density gradients for which we highlight the differences through a
number of numerical examples comparing the different 1D models.
