Spectral collocation methods using sine functions for a rotating Bose–Einstein condensation in optical lattices

We study spectral-Galerkin methods (SGM) and spectral collocation methods (SCM) for parameter-dependent problems, where the Fourier sine functions are used as the basis functions. When the SGM and the SCM are incorporated in the context of a Taylor predictor–inexact Newton corrector continuation algorithm for tracing solution curves of the Gross–Pitaevskii equation (GPE), they can efficiently provide accurate numerical solutions for the GPE. We show how the inexact Newton method outperforms the classical Newton method in the continuation algorithm. In our numerical experiments, the centered difference method (CDM), the SGM and SCM are exploited to compute energy levels and wave functions of a rotating Bose–Einstein condensation (BEC) and a rotating BEC in optical lattices in 2D. Sample numerical results are reported.