Long-term analysis of symplectic or symmetric extended RKN methods for nonlinear wave equations

This paper analyses the long-time behaviour of one-stage symplectic or symmetric extended Runge--Kutta--Nystr\"{o}m (ERKN) methods when applied to nonlinear wave equations. It is shown that energy, momentum, and all harmonic actions are approximately preserved over a long time for one-stage explicit symplectic or symmetric ERKN methods when applied to nonlinear wave equations via spectral semi-discretisations. For the long-term analysis of symplectic or symmetric ERKN methods, we derive a multi-frequency modulated Fourier expansion of the ERKN method and show three almost-invariants of the modulation system. In the analysis of this paper, we neither assume symmetry for symplectic methods, nor assume symplecticity for symmetric methods. The results for symplectic and symmetric methods are obtained as a byproduct of the above analysis. We also give another proof by establishing a relationship between symplectic and symmetric ERKN methods and trigonometric integrators which have been researched for wave equations in the literature.