In this paper, we study the linear Schrödinger equation over the
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://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 - http://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 http://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. http://www.numdam.org/articles/10.1051/m2an/2009028/
[1] Birkhoff normal form for PDEs with tame modulus. Duke Math. J. 135 (2006) 507-567. | MR | Zbl
and ,[2] Order estimates in time of splitting methods for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 40 (2002) 26-40. | MR | Zbl
, and ,[3] Conserved quantities of some Hamiltonian wave equations after full discretization. Numer. Math. 103 (2006) 197-223. | MR | Zbl
,[4] Long-time analysis of nonlinearly perturbed wave equations via modulated Fourier expansions. Arch. Ration. Mech. Anal. 187 (2008) 341-368. | MR | Zbl
, and ,[5] Analyse de méthodes d'intégration en temps des équation de Schrödinger. Ph.D. Thesis, University Rennes 1, France (2008).
,[6] 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
and ,[7] Long time behavior of splitting methods applied to the linear Schrödinger equation. C. R. Math. Acad. Sci. Paris 344 (2007) 89-92. | MR | Zbl
and ,[8] KAM for non-linear Schrödinger equation. Preprint (2006).
and ,[9] Solving Ordinary Differential Equations I. Nonstiff Problems, Springer Series in Computational Mathematics 8. Second Edition, Springer, Berlin (1993). | MR | Zbl
, and ,[10] Geometric Numerical Integration - Structure-preserving Algorithms for Ordinary Differential Equations. Springer, Berlin (2002). | MR | Zbl
, and ,[11] Error bounds for exponential operator splittings. BIT 40 (2000) 735-744. | MR | Zbl
and ,[12] Simulating Hamiltonian dynamics, Cambridge Monographs on Applied and Computational Mathematics 14. Cambridge University Press, Cambridge (2004). | MR | Zbl
and ,[13] On splitting methods for the Schödinger-Poisson and cubic nonlinear Schrödinger equations. Math. Comp. 77 (2008) 2141-2153. | MR
,[14] Approximate momentum conservation for spatial semidiscretizations of semilinear wave equations. Numer. Math. 97 (2004) 493-535. | MR | Zbl
, and ,[15] Resonant and Diophantine step sizes in computing invariant tori of Hamiltonian systems. Nonlinearity 13 (2000) 299-308. | MR | Zbl
,- One-stage exponential integrators for nonlinear Schrödinger equations over long times, BIT, Volume 52 (2012) no. 4, pp. 877-903 | DOI:10.1007/s10543-012-0385-1 | Zbl:1257.65055
- Splitting integrators for nonlinear Schrödinger equations over long times, Foundations of Computational Mathematics, Volume 10 (2010) no. 3, pp. 275-302 | DOI:10.1007/s10208-010-9063-3 | Zbl:1189.65301
- KAM Theorem of Symplectic Algorithms, Symplectic Geometric Algorithms for Hamiltonian Systems (2010), p. 549 | DOI:10.1007/978-3-642-01777-3_14
- Modified Energy for Split-Step Methods Applied to the Linear Schrödinger Equation, SIAM Journal on Numerical Analysis, Volume 47 (2009) no. 5, p. 3705 | DOI:10.1137/080744578
Cité par 4 documents. Sources : Crossref, zbMATH