2008/11/27 by Erwan Faou, Faou, Erwan, Benoît Grébert +3 · 1 citation
Engineering · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.0811.4538
openalex publication_date 2008/11/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.