High-order time-splitting Hermite and Fourier spectral methods for the Gross-Pitaevskii equation ?

Abstract

In this paper, we are concerned with the numerical solution of the time-dependent Gross‐Pitaevskii Equation (GPE) involving a quasi-harmonic potential. Primarily, we consider discretisations that are based on spectral methods in space and higherorder exponential operator splitting methods in time. The resulting methods are favourable in view of accuracy and eciency; moreover, geometric properties of the equation such as particle number and energy conservation are well captured. Regarding the spatial discretisation of the GPE, we consider two approaches. In the unbounded domain, we employ a spectral decomposition of the solution into Hermite basis functions; on the other hand, restricting the equation to a suciently large bounded domain, Fourier techniques are applicable. For the time integration of the GPE, we study various exponential operator splitting methods of convergence orders two, four, and six. Our main objective is to provide accuracy and eciency comparisons of exponential operator splitting Fourier and Hermite pseudospectral methods for the time evolution of the GPE. Furthermore, we illustrate the eectiveness of higher-order time-splitting methods compared to standard integrators in a long-term integration.