# 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

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topCastella, 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

top- D. Bambusi and B. Grebert, Birkhoff normal form for PDEs with tame modulus. Duke Math. J.135 (2006) 507–567. Zbl1110.37057
- C. Besse, B. Bidégaray and S. Descombes, Order estimates in time of splitting methods for the nonlinear Schrödinger equation. SIAM J. Numer. Anal.40 (2002) 26–40. Zbl1026.65073
- B. Cano, Conserved quantities of some Hamiltonian wave equations after full discretization. Numer. Math.103 (2006) 197–223. Zbl1096.65125
- D. Cohen, E. Hairer and C. Lubich, Long-time analysis of nonlinearly perturbed wave equations via modulated Fourier expansions. Arch. Ration. Mech. Anal.187 (2008) 341–368. Zbl1145.35087
- G. Dujardin, Analyse de méthodes d'intégration en temps des équation de Schrödinger. Ph.D. Thesis, University Rennes 1, France (2008).
- G. Dujardin and E. Faou, Normal form and long time analysis of splitting schemes for the linear Schrödinger equation with small potential. Numer. Math.108 (2007) 223–262. Zbl1137.65062
- G. Dujardin and E. Faou, Long time behavior of splitting methods applied to the linear Schrödinger equation. C. R. Math. Acad. Sci. Paris344 (2007) 89–92. Zbl1110.65089
- H.L. Eliasson and S.B. Kuksin, KAM for non-linear Schrödinger equation. Preprint (2006). Zbl1140.37021
- E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems, Springer Series in Computational Mathematics8. Second Edition, Springer, Berlin (1993). Zbl0789.65048
- E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration – Structure-preserving Algorithms for Ordinary Differential Equations. Springer, Berlin (2002). Zbl0994.65135
- T. Jahnke and C. Lubich, Error bounds for exponential operator splittings. BIT40 (2000) 735–744. Zbl0972.65061
- B. Leimkuhler and S. Reich, Simulating Hamiltonian dynamics, Cambridge Monographs on Applied and Computational Mathematics14. Cambridge University Press, Cambridge (2004). Zbl1069.65139
- C. Lubich, On splitting methods for the Schödinger-Poisson and cubic nonlinear Schrödinger equations. Math. Comp.77 (2008) 2141–2153. Zbl1198.65186
- M. Oliver, M. West and C. Wulff, Approximate momentum conservation for spatial semidiscretizations of semilinear wave equations. Numer. Math.97 (2004) 493–535. Zbl1060.65106
- Z. Shang, Resonant and Diophantine step sizes in computing invariant tori of Hamiltonian systems. Nonlinearity13 (2000) 299–308. Zbl1008.65092