Error Estimates of Operator Splitting Spectral Method for Semiclassical Sub-Diffusive Gross-Pitaevskii Equation

Abstract

Nonlinear Gross-Pitaevskii-type models are frequently seen in the fields of Bose-Einstein condensation and quantum mechanics. We derive error estimates for the Lie-Trotter operator splitting spectral method for semiclassical sub-diffusive Gross-Pitaevskii equation in the unbounded domain or with the periodic boundary condition. After establishing a priori estimates for the analytic solution in fractional Sobolev space, the local error estimates for the Lie-Trotter splitting operator method are derived. The related estimates for the Lie commutator of nonlocal linear operator and nonlinear operator play key roles in deriving the local error estimates. We then obtain the global error bounds for the fully discrete scheme based on the space approximation with mapped Chebyshev spectral-Galerkin methods in the case of the unbounded domain and with Fourier spectral methods in the case of the periodic boundary condition. Especially, their convergence orders with respect to the small (scaled) Planck constant $ε$ are obtained for the first time under the framework of Wentzel-Kramers-Brillouin analysis. Numerical experiments verify and complement our theoretical results.