In this paper, we study the linear Schrödinger equation over the -dimensional torus, with small values of the perturbing potential. We consider numerical approximations of the associated solutions obtained by a symplectic splitting method (to discretize the time variable) in combination with the Fast Fourier Transform algorithm (to discretize the space variable). In this fully discrete setting, we prove that the regularity of the initial datum is preserved over long times, i.e. times that are exponentially long with the time discretization parameter. We here refer to Gevrey regularity, and our estimates turn out to be uniform in the space discretization parameter. This paper extends [G. Dujardin and E. Faou, Numer. Math. 97 (2004) 493-535], where a similar result has been obtained in the semi-discrete situation, i.e. when the mere time variable is discretized and space is kept a continuous variable.
Mots-clés : splitting, KAM theory, resonance, normal forms, Gevrey regularity, Schrödinger equation
@article{M2AN_2009__43_4_651_0,
author = {Castella, Fran\c{c}ois and Dujardin, Guillaume},
title = {Propagation of {Gevrey} regularity over long times for the fully discrete {Lie} {Trotter} splitting scheme applied to the linear {Schr\"odinger} equation},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {651--676},
publisher = {EDP-Sciences},
volume = {43},
number = {4},
year = {2009},
doi = {10.1051/m2an/2009028},
mrnumber = {2542870},
zbl = {1171.65089},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2009028/}
}
TY - JOUR AU - Castella, François AU - Dujardin, Guillaume TI - Propagation of Gevrey regularity over long times for the fully discrete Lie Trotter splitting scheme applied to the linear Schrödinger equation JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2009 SP - 651 EP - 676 VL - 43 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2009028/ DO - 10.1051/m2an/2009028 LA - en ID - M2AN_2009__43_4_651_0 ER -
%0 Journal Article %A Castella, François %A Dujardin, Guillaume %T Propagation of Gevrey regularity over long times for the fully discrete Lie Trotter splitting scheme applied to the linear Schrödinger equation %J ESAIM: Modélisation mathématique et analyse numérique %D 2009 %P 651-676 %V 43 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2009028/ %R 10.1051/m2an/2009028 %G en %F M2AN_2009__43_4_651_0
Castella, François; Dujardin, Guillaume. Propagation of Gevrey regularity over long times for the fully discrete Lie Trotter splitting scheme applied to the linear Schrödinger equation. ESAIM: Modélisation mathématique et analyse numérique, Special issue on Numerical ODEs today, Tome 43 (2009) no. 4, pp. 651-676. doi : 10.1051/m2an/2009028. http://archive.numdam.org/articles/10.1051/m2an/2009028/
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