Reliability of the time splitting Fourier method for singular solutions in quantum fluids

TL;DR

This paper evaluates the accuracy and reliability of the time splitting Fourier spectral method for simulating singular solutions in quantum fluids, demonstrating its effectiveness in preserving vortex solutions and enabling high-resolution analysis.

Contribution

The study introduces a highly accurate diagonal Padé expansion for vortex density profiles and assesses the Fourier method's performance in quantum fluid simulations.

Findings

Fourier spectral method is slightly more accurate than finite difference schemes.

The method reliably preserves steady-state vortex solutions.

It allows precise evaluation outside grid points for high-resolution needs.

Abstract

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.