Birkhoff normal form and splitting methods for semi linear Hamiltonian PDEs. Part I: Finite dimensional discretization

TL;DR

This paper demonstrates that splitting methods for discretized Hamiltonian PDEs nearly preserve the actions over long times, under certain conditions, by employing a finite-dimensional Birkhoff normal form approach.

Contribution

It introduces a finite-dimensional Birkhoff normal form technique to analyze long-term action preservation for splitting methods applied to discretized Hamiltonian PDEs.

Findings

Actions are almost preserved over long times with small initial data.

Results hold under generic non-resonance conditions.

Applicable to nonlinear Schrödinger and wave equations.

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 Schr\"odinger equations as well as the nonlinear wave equation.}