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As is known that wave equations have physically very important propertieswhich should be respected by numerical schemes in order to predict correctlythe solution over a long time period. In this paper, the long-time behaviour ofmomentum and actions for energy-preserving methods is analysed for semilinearwave equations. A full discretisation of wave equations is derived and analysedby firstly using a spectral semi-discretisation in space and then by applyingthe adopted average vector field (AAVF) method in time. This numerical schemecan exactly preserve the energy of the semi-discrete system. The main theme ofthis paper is to analyse another important physical property of the scheme. Itis shown that this scheme yields near conservation of a modified momentum andmodified actions over long times. Both the results are rigorously proved basedon the technique of modulated Fourier expansions in two stages. First amulti-frequency modulated Fourier expansion of the AAVF method is constructedand then two almost-invariants of the modulation system are derived.

Long-time momentum and actions behaviour of energy-preserving methods for semilinear wave equations via spatial spectral semi-discretizations