Reliability of the time splitting Fourier method for singular solutions   in quantum fluids
  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. 
Marco Caliari and Simone Zuccher
math.NA