Spectral semi-discretisations of weakly non-linear wave equations over long times
The long-time behaviour of spectral semi-discretisations of weakly non-linear wave equations is analysed. It is shown that the harmonic actions are approximately conserved for the semi-discretised system as well. This permits to prove that the energy of the wave equation along the interpolated semi-discrete solution remains well conserved over long times and close to the Hamiltonian of the semi-discrete equation. Although the momentum is no longer an exact invariant of the semi-discretisation, it is shown to be approximately conserved. All these results are obtained with the technique of modulated Fourier expansions.
non-linear wave equation long-time behaviour modulated fourier expansion spectral semi-discretization regularity of solutions near conservation of harmonic actions energy and momentum