Resonances in long time integration of semi linear Hamiltonian PDEs

TL;DR

This paper investigates the long-term behavior of symplectic numerical methods applied to semi-linear Hamiltonian PDEs, identifying resonance issues and proposing truncation techniques to preserve actions over extended times.

Contribution

It demonstrates how standard methods can encounter resonance problems and introduces a truncation approach with bounds to ensure long-time action preservation.

Findings

Resonance issues can cause pathological behavior in standard integrators.

Truncation combined with splitting methods can avoid resonance problems.

Long-time action preservation is achieved under certain frequency conditions.

Abstract

We consider a class of Hamiltonian PDEs that can be split into a linear unbounded operator and a regular non linear part, and we analyze their numerical discretizations by symplectic methods when the initial value is small in Sobolev norms. The goal of this work is twofold: First we show how standard approximation methods cannot in general avoid resonances issues, and we give numerical examples of pathological behavior for the midpoint rule and implicit-explicit integrators. Such phenomena can be avoided by suitable truncations of the linear unbounded operator combined with classical splitting methods. We then give a sharp bound for the cut-off depending on the time step. Using a new normal form result, we show the long time preservation of the actions for such schemes for all values of the time step, provided the initial continuous system does not exhibit resonant frequencies.