We study numerically the semiclassical limit for the nonlinear Schrödinger equation thanks to a modification of the Madelung transform due to Grenier. This approach 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.
@article{M2AN_2011__45_5_981_0,
author = {Carles, R\'emi and Mohammadi, Bijan},
title = {Numerical aspects of the nonlinear Schr\"odinger equation in the semiclassical limit in a supercritical regime},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {45},
year = {2011},
pages = {981-1008},
doi = {10.1051/m2an/2011005},
mrnumber = {2817553},
zbl = {1269.65104},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2011__45_5_981_0}
}
Carles, Rémi; Mohammadi, Bijan. Numerical aspects of the nonlinear Schrödinger equation in the semiclassical limit in a supercritical regime. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 45 (2011) pp. 981-1008. doi : 10.1051/m2an/2011005. http://gdmltest.u-ga.fr/item/M2AN_2011__45_5_981_0/
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