  We extensively study the numerical accuracy of the well-known time splitting
Fourier spectral method for the approximation of singular solutions of the
Gross-Pitaevskii equation. In particular, we explore its capability of
preserving a steady-state vortex solution, whose density profile is
approximated by a very accurate diagonal Pad\'e expansion of order 8, here
explicitly derived for the first time. Although the Fourier spectral method
turns out to be only slightly more accurate than a time splitting finite
difference scheme, the former is reliable and efficient. Moreover, at a
post-processing stage, it allows an accurate evaluation of the solution outside
grid points, thus becoming particularly appealing when high resolution is
needed, such as in the study of quantum vortex interactions.
