Almost conservation of the harmonic actions for fully discretized nonlinear Klein--Gordon equations at low regularity

TL;DR

This paper demonstrates that symplectic mollified impulse methods nearly conserve harmonic actions in fully discretized nonlinear Klein-Gordon equations at low regularity, extending previous results to non-smooth solutions.

Contribution

It proves near conservation of harmonic actions for discretized Klein-Gordon equations at low regularity using symplectic mollified impulse methods.

Findings

Harmonic actions are nearly conserved at low regularity.

The methods work with a CFL number of size 1.

Extension of previous results to non-smooth solutions.

Abstract

Close to the origin, the nonlinear Klein--Gordon equations on the circle are nearly integrable Hamiltonian systems which have infinitely many almost conserved quantities called harmonic actions or super-actions. We prove that, at low regularity and with a CFL number of size 1, this property is preserved if we discretize the nonlinear Klein--Gordon equations with the symplectic mollified impulse methods. This extends previous results of D. Cohen, E. Hairer and C. Lubich to non-smooth solutions.