Propagation of Gevrey regularity over long times for the fully discrete Lie Trotter splitting scheme applied to the linear Schrödinger equation

François Castella; Guillaume Dujardin

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

  • Volume: 43, Issue: 4, page 651-676
  • ISSN: 0764-583X

Abstract

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In this paper, we study the linear Schrödinger equation over the d-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.

How to cite

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Castella, François, and Dujardin, Guillaume. "Propagation of Gevrey regularity over long times for the fully discrete Lie Trotter splitting scheme applied to the linear Schrödinger equation." ESAIM: Mathematical Modelling and Numerical Analysis 43.4 (2009): 651-676. <http://eudml.org/doc/250580>.

@article{Castella2009,
abstract = { In this paper, we study the linear Schrödinger equation over the d-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. },
author = {Castella, François, Dujardin, Guillaume},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Splitting; KAM theory; resonance; normal forms; Gevrey regularity; Schrödinger equation.; Lie-Trotter splitting method; symplectic method; Schrödinger equation; fast Fourier transform},
language = {eng},
month = {7},
number = {4},
pages = {651-676},
publisher = {EDP Sciences},
title = {Propagation of Gevrey regularity over long times for the fully discrete Lie Trotter splitting scheme applied to the linear Schrödinger equation},
url = {http://eudml.org/doc/250580},
volume = {43},
year = {2009},
}

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: Mathematical Modelling and Numerical Analysis
DA - 2009/7//
PB - EDP Sciences
VL - 43
IS - 4
SP - 651
EP - 676
AB - In this paper, we study the linear Schrödinger equation over the d-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.
LA - eng
KW - Splitting; KAM theory; resonance; normal forms; Gevrey regularity; Schrödinger equation.; Lie-Trotter splitting method; symplectic method; Schrödinger equation; fast Fourier transform
UR - http://eudml.org/doc/250580
ER -

References

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