  We report systematic numerical simulations of the collision of a bright
matter-wave soliton made of Bose-condensed alkali-metal atoms through a narrow
potential barrier by using the three-dimensional Gross-Pitaevskii equation. In
this way, we determine how the transmission coefficient depends on the soliton
impact velocity and the barrier height. Quite remarkably, we also obtain the
regions of parameters where there is the collapse of the bright soliton induced
by the collision. We compare these three-dimensional results with the ones
obtained by three different one-dimensional nonlinear Schr\"odinger equations.
We find that a specifically modified nonpolynomial Schr\"odinger equation is
able to accurately assess the transmission coefficient even in a region in
which the usual nonpolynomial Schr\"odinger equation does collapse. In
particular, this simplified but very effective one-dimensional model takes into
account the transverse width dynamics of the soliton with an ordinary
differential equation coupled to the partial differential equation of the axial
wave function of the Bose-Einstein condensate.
