Mathematics – Numerical Analysis
Scientific paper
2010-02-08
ESAIM: Mathematical Modelling and Numerical Analysis 45, 5 (2011) 981-1008
Mathematics
Numerical Analysis
29 pages, 18 figures. More explanations, more references, and an extra experience past the breakup time
Scientific paper
10.1051/m2an/2011005
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.
Carles Rémi
Mohammadi Bijan
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