Numerical Study of Time-Splitting Spectral Discretizations of Nonlinear Schrödinger Equations in the Semiclassical Regimes

In this paper we study the performance of time-splitting spectral approximations for general nonlinear Schrödinger equations (NLS) in the semiclassical regimes, where the Planck constant $\varepsilon$ is small. The time-splitting spectral approximation under study is explicit, unconditionally stable and conserves the position density in L^1. Moreover it is time-transverse invariant and time-reversible when the corresponding NLS is. Extensive numerical tests are presented for weak/strong focusing/defocusing nonlinearities, for the Gross--Pitaevskii equation, and for current-relaxed quantum hydrodynamics. The tests are geared towards the understanding of admissible meshing strategies for obtaining "correct" physical observables in the semiclassical regimes. Furthermore, comparisons between the solutions of the NLS and its hydrodynamic semiclassical limit are presented.