Numerical aspects of the nonlinear Schrödinger equation in the semiclassical limit in a supercritical regime

Rémi Carles; Bijan Mohammadi

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

  • Volume: 45, Issue: 5, page 981-1008
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Carles, Rémi, and Mohammadi, Bijan. "Numerical aspects of the nonlinear Schrödinger equation in the semiclassical limit in a supercritical regime." ESAIM: Mathematical Modelling and Numerical Analysis 45.5 (2011): 981-1008. <http://eudml.org/doc/276351>.

@article{Carles2011,
abstract = { 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. },
author = {Carles, Rémi, Mohammadi, Bijan},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Nonlinear Schrödinger equation; semiclassical limit; compressible Euler equation; numerical simulation; nonlinear Schrödinger equation; compressible Euler equation; numerical examples; spectral method; Madelung transform; convergence; shocks},
language = {eng},
month = {6},
number = {5},
pages = {981-1008},
publisher = {EDP Sciences},
title = {Numerical aspects of the nonlinear Schrödinger equation in the semiclassical limit in a supercritical regime},
url = {http://eudml.org/doc/276351},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Carles, Rémi
AU - Mohammadi, Bijan
TI - Numerical aspects of the nonlinear Schrödinger equation in the semiclassical limit in a supercritical regime
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2011/6//
PB - EDP Sciences
VL - 45
IS - 5
SP - 981
EP - 1008
AB - 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.
LA - eng
KW - Nonlinear Schrödinger equation; semiclassical limit; compressible Euler equation; numerical simulation; nonlinear Schrödinger equation; compressible Euler equation; numerical examples; spectral method; Madelung transform; convergence; shocks
UR - http://eudml.org/doc/276351
ER -

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