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

Published inFoundations of computational mathematics, vol. 8, no. 3, p. 319-334
Publication date2008
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.

Keywords
  • Nonlinear wave equation
  • Spectral semi-discretisation
  • Long-time behaviour
Citation (ISO format)
HAIRER, Ernst, LUBICH, Christian. Spectral semi-discretisations of weakly nonlinear wave equations over long times. In: Foundations of computational mathematics, 2008, vol. 8, n° 3, p. 319–334. doi: 10.1007/s10208-007-9014-9
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ISSN of the journal1615-3375
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