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

Published online by Cambridge University Press:  10 June 2011

Rémi Carles  and
Bijan Mohammadi
Rémi Carles
Affiliation:
CNRS and Univ. Montpellier 2, Mathématiques, CC 051, 34095 Montpellier, France. [email protected]
Bijan Mohammadi
Affiliation:
CNRS and Univ. Montpellier 2, Mathématiques, CC 051, 34095 Montpellier, France. [email protected]
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Abstract

We study numerically the semiclassical limit for the nonlinearSchrödinger equation thanks to a modification of the Madelungtransform due to Grenier. This approach allows for the presence ofvacuum. Even if the mesh size and the time step do not depend on the Planck constant, we recover the position and current densities in thesemiclassical limit, with a numerical rate of convergence inaccordance with the theoreticalresults, before shocks appear in the limiting Eulerequation. By using simple projections, the mass and the momentum ofthe solution are well preserved by the numerical scheme,while the variation of the energy is not negligiblenumerically. Experiments suggest that beyond the critical time for theEuler equation, Grenier's approach yields smooth but highlyoscillatory terms.

Keywords

Nonlinear Schrödinger equationsemiclassical limitcompressible Euler equationnumerical simulation
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 5 , September 2011 , pp. 981 - 1008
Copyright
© EDP Sciences, SMAI, 2011

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