BIT Numerical Mathematics

, Volume 55, Issue 3, pp 705–732 | Cite as

Long-term analysis of numerical integrators for oscillatory Hamiltonian systems under minimal non-resonance conditions

  • David Cohen
  • Ludwig GaucklerEmail author
  • Ernst Hairer
  • Christian Lubich


For trigonometric and modified trigonometric integrators applied to oscillatory Hamiltonian differential equations with one or several constant high frequencies, near-conservation of the total and oscillatory energies are shown over time scales that cover arbitrary negative powers of the step size. This requires non-resonance conditions between the step size and the frequencies, but in contrast to previous results the results do not require any non-resonance conditions among the frequencies. The proof uses modulated Fourier expansions with appropriately modified frequencies.


Oscillatory Hamiltonian systems Modulated Fourier expansions Trigonometric integrators Störmer–Verlet scheme IMEX scheme Long-time energy conservation Numerical resonances Non-resonance condition 

Mathematics Subject Classification

65P10 65L05 34E13 


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • David Cohen
    • 1
  • Ludwig Gauckler
    • 2
    Email author
  • Ernst Hairer
    • 3
  • Christian Lubich
    • 4
  1. 1.Matematik och matematisk statistikUmeå universitetUmeåSweden
  2. 2.Institut für MathematikTU BerlinBerlinGermany
  3. 3.Section de mathématiquesUniversité de GenèveGeneva 4Switzerland
  4. 4.Mathematisches InstitutUniversität TübingenTübingenGermany

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