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

Abstract  We consider 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 actions of the numerical solution associated with the splitting method over arbitrary long time and for asymptotically large level
of space approximation, provided the Sobolev norm of the initial data is small enough. This result holds under generic non-resonance conditions on the frequencies of the linear operator and on the step size. We apply these results to nonlinear
Schrödinger equations as well as the nonlinear wave equation.

• Content Type Journal Article
• DOI 10.1007/s00211-009-0258-y
• Authors
  
  
  • Erwan Faou, INRIA and Ecole Normale Supérieure de Cachan, Bretagne Avenue Robert Schumann 35170 Bruz France
  • Benoît Grébert, Université de Nantes Laboratoire de Mathématiques Jean Leray 2 rue de la Houssinière 44322 Nantes France
  • Eric Paturel, Université de Nantes Laboratoire de Mathématiques Jean Leray 2 rue de la Houssinière 44322 Nantes France
  
  

• Journal Numerische Mathematik
• Online ISSN 0945-3245
• Print ISSN 0029-599X

Published by

Original source : http://www.springerlink.com/content/k11508g52p710w...