High-order IMEX-spectral schemes for computing the dynamics of systems of nonlinear Schrödinger /Gross-Pitaevskii equations

Xavier Antoine (1, 2) , Christophe Besse (3) , Vittorio Rispoli (3)

1 SPHINX - Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization
2 IECL - Institut Élie Cartan de Lorraine
3 IMT - Institut de Mathématiques de Toulouse UMR5219

The aim of this paper is to build and validate some explicit high-order schemes, both in space and time, for simulating the dynamics of systems of nonlinear Schrödinger /Gross-Pitaevskii equations. The method is based on the combination of high-order IMplicit-EXplicit (IMEX) schemes in time and Fourier pseudo-spectral approximations in space. The resulting IMEXSP schemes are highly accurate, efficient and easy to implement. They are also robust when used in conjunction with an adaptive time stepping strategy and appear as an interesting alternative to time-splitting pseudo-spectral (TSSP) schemes. Finally, a complete numerical study is developed to investigate the properties of the IMEXSP schemes, in comparison with TSSP schemes, for one-and two-components systems of Gross-Pitaevskii equations.

Mots clés: IMplicit-EXplicit schemes; pseudo-spectral schemes; high-order discretization; dynamics; Bose-Einstein Condensates; spin-orbit; Nonlinear Schrödinger equation; Gross-Pitaevskii equation; adaptive time stepping; time-splitting

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Xavier Antoine, Christophe Besse, Vittorio Rispoli. High-order IMEX-spectral schemes for computing the dynamics of systems of nonlinear Schrödinger /Gross-Pitaevskii equations. Journal of Computational Physics, 2016, 327, pp.252-269. ⟨10.1016/j.jcp.2016.09.020⟩. ⟨hal-01271463⟩