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hal-00341226, version 1

Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. Part II: Abstract splitting.

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

Numerische Mathematik 114, 3 (2010) 459-490

Abstract: We consider Hamiltonian PDEs that can be split into a linear unbounded operator and a regular non linear part. We consider abstract splitting methods associated with this decomposition where no discretization in space is made. We prove a normal form result for the corresponding discrete flow under generic non resonance conditions on the frequencies of the linear operator and on the step size. This result implies the conservation of the regularity of the numerical solution associated with the splitting method over arbitrary long time, provided the initial data is small enough. This result holds for numerical schemes controlling the round-off error at each step to avoid possible high frequency energy drift. We apply this results to nonlinear Schrödinger equations as well as the nonlinear wave equation. }

  • 1:  Institut de Recherche Mathématique de Rennes (IRMAR)
  • CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne
  • 2:  IPSO (INRIA - IRMAR)
  • CNRS : UMR6074 – INRIA – Université de Rennes 1
  • 3:  Laboratoire de Mathématiques Jean Leray (LMJL)
  • CNRS : UMR6629 – Université de Nantes – École Centrale de Nantes
  • Domain : Mathematics/Numerical Analysis
    Mathematics/Dynamical Systems
    Mathematics/Analysis of PDEs
  • Keywords : Birkhoff normal form – splitting methods – long time analysis
 
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  • From: Benoit Grebert <>
  • Submitted on: Monday, 24 November 2008 17:00:51
  • Updated on: Monday, 9 July 2012 10:35:15