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

Erwan Faou (1, 2) , Benoît Grébert (3) , Eric Paturel (3)

1 IPSO - Invariant Preserving SOlvers
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 LMJL - Laboratoire de Mathématiques Jean Leray

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. }

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]; Systèmes dynamiques [math.DS]

Erwan Faou, Benoît Grébert, Eric Paturel. Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. Part I: Finite dimensional discretization.. Numerische Mathematik, 2010, 114 (3), pp.429-458. ⟨10.1007/s00211-009-0258-y⟩. ⟨hal-00341241⟩