Reliability of the time splitting Fourier method for singular solutions in quantum fluids
This work addresses the need for accurate numerical methods for quantum fluid simulations, particularly for studying vortex interactions, but the improvement is incremental.
The paper evaluates the time splitting Fourier spectral method for solving the Gross-Pitaevskii equation with singular vortex solutions, finding it reliable and efficient, with accuracy slightly better than finite difference schemes. It also provides a new diagonal Padé expansion of order 8 for the vortex density profile.
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 Padé 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.