An asymptotic preserving approach for nonlinear Schrodinger equation in the semiclassical limit

Abstract

We study numerically the semiclassical limit for the nonlinear Schroedinger equation thanks to a modification of the Madelung transform due to E.Grenier. This approach is naturally asymptotic preserving, and allows for the presence of vacuum. Even if the mesh size and the time step do not depend on the Planck constant, we recover the position and current densities in the semiclassical limit, with a numerical rate of convergence in accordance with the theoretical results, before shocks appear in the limiting Euler equation. By using simple projections, the mass and the momentum of the solution are well preserved by the numerical scheme, while the variation of the energy is not negligible numerically. Experiments suggest that beyond the critical time for the Euler equation, Grenier's approach yields smooth but highly oscillatory terms.

29 pages, 18 figures. More explanations, more references, and an extra experience past the breakup time

References in corpus (7)

• Stability of trapped Bose-Einstein condensates
• Geometric optics and instability for semi-classical Schrodinger equations
• Semi-classical limit of Schrodinger-Poisson equations in space dimension at least 3
• Supercritical geometric optics for nonlinear Schrodinger equations
• Geometric optics and boundary layers for Nonlinear Schrodinger equations
• WKB analysis for the Gross-Pitaevskii equation with non-trivial boundary conditions at infinity
• Numerical aspects of nonlinear Schrodinger equations in the presence of caustics

Cited by in corpus (3)

• Computational methods for the dynamics of the nonlinear Schrodinger/Gross-Pitaevskii equations
• On Fourier time-splitting methods for nonlinear Schrodinger equations in the semi-classical limit
• An asymptotic preserving scheme based on a new formulation for NLS in the semiclassical limit