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

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

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

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topCarles, 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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