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Numerische Mathematik

, Volume 110, Issue 2, pp 113–143 | Cite as

Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations

  • David Cohen
  • Ernst HairerEmail author
  • Christian Lubich
Article

Abstract

For classes of symplectic and symmetric time-stepping methods— trigonometric integrators and the Störmer–Verlet or leapfrog method—applied to spectral semi-discretizations of semilinear wave equations in a weakly non-linear setting, it is shown that energy, momentum, and all harmonic actions are approximately preserved over long times. For the case of interest where the CFL number is not a small parameter, such results are outside the reach of standard backward error analysis. Here, they are instead obtained via a modulated Fourier expansion in time.

Mathematics Subject Classification (2000)

35L70 65M70 65M15 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversity of BaselBaselSwitzerland
  2. 2.Dept. de MathématiquesUniversity of GenevaGeneva 4Switzerland
  3. 3.Mathematisches InstitutUniversity of TübingenTübingenGermany

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