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

Abstract

Abstract. 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 ε 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 timereversible 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.

Numerical Methods for the Nonlinear Schrödinger Equation with Nonzero Far-field Conditions

“…When the TS-Chebyshev or TS-Cosine methods are applied for linear Schrödinger equation with constant/harmonic oscillator potential, e.g. Examples 1&2 in [2], we have observed the following meshing strategy: k-independent of ε and h = O(ε), which is the same as those for time-splitting Fourier-spectral or Sine-spectral methods [2,3]. We omitted the details here.…”

“…In theory, the strong convergence of the integral as ε goes to zero would imply the weak convergence of ρ ε = |u ε (x, t)| 2 with respect to x. This is motivated by the works [22,31] on the zero dispersion limit of the KDV equation and used for the numerical study of focusing nonlinear Schrödinger equation [3,9,27]. Figure 1 also displays the indefinite integral at t = 2.0 for four different values of ε: 0.64, 0.32, 0.16 and 0.08 for the three different types data.…”

“…Recently, Bao et al studied the time-splitting Fourierspectral (TS-Fourier) [2,3] and sine-spectral (TS-Sine) [5] methods for the NLS with zero far-field condition and applied them successfully to simulate Bose-Einstein condensation [4], in this case the periodic or homogeneous Dirichlet boundary condition can be applied. Because the TS-Fourier and TS-Sine methods are unconditionally stable, time reversible and time-transverse invariant, spectral accuracy in space and conserves the position density, they observed 'optimal' ε-resolution for obtaining 'correct' physical observables: time step k-independent of ε and mesh size h = O(ε) for linear case; k = O(ε) and h = O(ε) for defocusing nonlinear case, which is much more better than the finite difference methods [2,3]. Unfortunely, the TS-Fourier and TS-Sine methods can not be applied to approximate NLS (1.1) with nonzero far-field conditions (1.3), or (1.4), or (1.5).…”

Numerical solution of the Gross–Pitaevskii equation for Bose–Einstein condensation

“…[11], however they are included here since perturbation theory gives a systematic way to obtain rigorously (validate) these approximations and since they are used as important preparatory steps for the numerical simulations (approximate ground states usually serve as initial data). Then we use the time-splitting spectral method, which was studied in Bao et al [7,8] for the Schrödinger equation in the semiclassical regime, to discretize the time-dependent GPE. The merit of the numerical method is that it is explicit, unconditionally stable, time reversible, time-transverse invariant, and conserves the position density.…”

Parametric reduced models for the nonlinear Schrödinger equation

“…We assume that the observation time interval T obs = 0.02. The time series, consisting of 100,000 observations, is generated by using the Strang's splitting method in time, which has been implemented in [27] for the NLS equation. The data is generated from the solutions of the NLS in the Fourier domain (2.3), with K = 32.…”