# Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. Part I: Finite dimensional discretization.

Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. Part I: Finite dimensional discretization. - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.

1 IPSO - Invariant Preserving SOlvers IRMAR - Institut de Recherche Mathématique de Rennes, Inria Rennes – Bretagne Atlantique 2 IRMAR - Institut de Recherche Mathématique de Rennes 3 LMJL - Laboratoire de Mathématiques Jean Leray

Abstract : We consider {\em discretized} Hamiltonian PDEs associated with a Hamiltonian function that can be split into a linear unbounded operator and a regular nonlinear part. We consider splitting methods associated with this decomposition. Using a finite dimensional Birkhoff normal form result, we show the almost preservation of the {\em actions} of the numerical solution associated with the splitting method over arbitrary long time, provided the Sobolev norms of the initial data is small enough, and for asymptotically large level of space approximation. This result holds under {\em generic} non resonance conditions on the frequencies of the linear operator and on the step size. We apply this results to nonlinear Schrödinger equations as well as the nonlinear wave equation. }

Autor: Erwan Faou - Benoît Grébert - Eric Paturel -

Fuente: https://hal.archives-ouvertes.fr/

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