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Spectral semi-discretisations of weakly nonlinear wave equations over long times

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E. Hairer and Ch. Lubich

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Abstract. The long-time behaviour of spectral semi-discretisations of weakly nonlinear wave equations is analysed. It is shown that the harmonic actions are approximately conserved also for the semi-discretised system. 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.

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Key Words. Nonlinear wave equation, spectral semi-discretisation, long-time behaviour.

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