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.

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.

DOI : 10.1051/m2an/2009028
Classification : 65P10, 37M15, 37K55
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 = {https://www.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  - https://www.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 https://www.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. https://www.numdam.org/articles/10.1051/m2an/2009028/

[1] D. Bambusi and B. Grebert, Birkhoff normal form for PDEs with tame modulus. Duke Math. J. 135 (2006) 507-567. | MR | Zbl

[2] 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. | MR | Zbl

[3] B. Cano, Conserved quantities of some Hamiltonian wave equations after full discretization. Numer. Math. 103 (2006) 197-223. | MR | Zbl

[4] 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. | MR | Zbl

[5] G. Dujardin, Analyse de méthodes d'intégration en temps des équation de Schrödinger. Ph.D. Thesis, University Rennes 1, France (2008).

[6] 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. | MR | Zbl

[7] G. Dujardin and E. Faou, Long time behavior of splitting methods applied to the linear Schrödinger equation. C. R. Math. Acad. Sci. Paris 344 (2007) 89-92. | MR | Zbl

[8] H.L. Eliasson and S.B. Kuksin, KAM for non-linear Schrödinger equation. Preprint (2006).

[9] E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems, Springer Series in Computational Mathematics 8. Second Edition, Springer, Berlin (1993). | MR | Zbl

[10] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration - Structure-preserving Algorithms for Ordinary Differential Equations. Springer, Berlin (2002). | MR | Zbl

[11] T. Jahnke and C. Lubich, Error bounds for exponential operator splittings. BIT 40 (2000) 735-744. | MR | Zbl

[12] B. Leimkuhler and S. Reich, Simulating Hamiltonian dynamics, Cambridge Monographs on Applied and Computational Mathematics 14. Cambridge University Press, Cambridge (2004). | MR | Zbl

[13] C. Lubich, On splitting methods for the Schödinger-Poisson and cubic nonlinear Schrödinger equations. Math. Comp. 77 (2008) 2141-2153. | MR

[14] M. Oliver, M. West and C. Wulff, Approximate momentum conservation for spatial semidiscretizations of semilinear wave equations. Numer. Math. 97 (2004) 493-535. | MR | Zbl

[15] Z. Shang, Resonant and Diophantine step sizes in computing invariant tori of Hamiltonian systems. Nonlinearity 13 (2000) 299-308. | MR | Zbl

Cité par Sources :