%0 Unpublished work
%T Uniformly accurate time-splitting methods for the semiclassical Schrödinger equationPart 1 : Construction of the schemes and simulations
%+ Invariant Preserving SOlvers (IPSO)
%+ Institut de Recherche Mathématique de Rennes (IRMAR)
%A Chartier, Philippe
%A Le Treust, Loïc
%A Méhats, Florian
%8 2015-04
%D 2015
%Z 1605.03446
%K nonlinear Schrödinger equation
%K semiclassical limit
%K Madelung transform
%K eikonal equation
%K splitting schemes
%K numerical simulation
%K uniformly accurate
%Z 35Q55, 35F21, 65M99, 76A02, 76Y05, 81Q20, 82D50
%Z Mathematics [math]/Analysis of PDEs [math.AP]
%Z Mathematics [math]/Numerical Analysis [math.NA]Preprints, Working Papers, ...
%X This article is devoted to the construction of new numerical methods for the semiclassical Schrödinger equation. A phase-amplitude reformulation of the equation is described where the Planck constant epsilon is not a singular parameter. This allows to build splitting schemes whose accuracy is spectral in space, of up to fourth order in time, and independent of epsilon before the caustics. The second-order method additionally preserves the L^2-norm of the solution just as the exact flow does. In this first part of the paper, we introduce the basic splitting scheme in the nonlinear case, reveal our strategy for constructing higher-order methods, and illustrate their properties with simulations. In the second part, we shall prove a uniform convergence result for the first-order splitting scheme applied to the linear Schrödinger equation with a potential.
%G English
%2 https://hal.science/hal-01140880v2/document
%2 https://hal.science/hal-01140880v2/file/UAmethodPart1.pdf
%L hal-01140880
%U https://hal.science/hal-01140880
%~ UNIV-RENNES1
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%~ CNRS
%~ INRIA
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%~ INRIA2016-PREPRINT
%~ INSA-GROUPE
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%~ INSTITUT-AGRO
%~ IRMAR-ANM
%~ IRMAR-ANUM