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


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 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bambusi D.: Birkhoff normal form for some nonlinear PDEs. Comm. Math. Phys. 234, 253–285 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Benettin G., Giorgilli A.: On the Hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms. J. Stat. Phys. 74, 1117–1143 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bourgain J.: Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations. Geom. Funct. Anal. 6, 201–230 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cano B.: Conserved quantities of some Hamiltonian wave equations after full discretization. Numer. Math. 103, 197–223 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cohen D., Hairer E., Lubich C.: Numerical energy conservation for multi-frequency oscillatory differential equations. BIT 45, 287–305 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cohen D., Hairer E., Lubich C.: Long-time analysis of nonlinearly perturbed wave equations via modulated Fourier expansions. Arch. Ration. Mech. Anal. 187, 341–368 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Deuflhard P.: A study of extrapolation methods based on multistep schemes without parasitic solutions. Z. Angew. Math. Phys. 30, 177–189 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dujardin G., Faou E.: Normal form and long time analysis of splitting schemes for the linear Schrödinger equation with small potential. Numer. Math. 108, 223–262 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    García-Archilla B., Sanz-Serna J.M., Skeel R.D.: Long-time-step methods for oscillatory differential equations. SIAM J. Sci. Comput. 20, 930–963 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Grubmüller H., Heller H., Windemuth A., Tavan P.: Generalized Verlet algorithm for efficient molecular dynamics simulations with long-range interactions. Mol. Sim. 6, 121–142 (1991)CrossRefGoogle Scholar
  11. 11.
    Hairer, E., Lubich, C.: The life-span of backward error analysis for numerical integrators, Numer. Math 76, 441–462, (1997). Erratum: Google Scholar
  12. 12.
    Hairer E., Lubich C.: Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Numer. Anal. 38, 414–441 (2001)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Hairer E., Lubich C.: Spectral semi-discretisations of nonlinear wave equations over long times. Found. Comput. Math. 8, 319–334 (2008)CrossRefGoogle Scholar
  14. 14.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Structure-preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer Series in Computational Mathematics, vol. 31. Springer, Berlin (2006)Google Scholar
  15. 15.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I. Nonstiff Problems, 2nd edn. Springer Series in Computational Mathematics, vol. 8. Springer, Berlin (1993)Google Scholar
  16. 16.
    Reich S.: Backward error analysis for numerical integrators. SIAM J. Numer. Anal. 36, 1549–1570 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Tuckerman M., Berne B.J., Martyna G.J.: Reversible multiple time scale molecular dynamics. J. Chem. Phys. 97, 1990–2001 (1992)CrossRefGoogle Scholar

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

Personalised recommendations