Skip to main content
SpringerLink
Log in

Conserved quantities of some Hamiltonian wave equations after full discretization

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

Hamiltonian PDEs have some invariant quantities, which would be good to conserve with the numerical integration. In this paper, we concentrate on the nonlinear wave and Schrödinger equations. Under hypotheses of regularity and periodicity, we study how a symmetric space discretization makes that the space discretized system also has some invariants or `nearly' invariants which well approximate the continuous ones. We conjecture some facts which would explain the good numerical approximation of them after time integration when using symplectic Runge-Kutta methods or symmetric linear multistep methods for second-order systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bridges, Th.J.: Multi-symplectic structures and wave propagation. Math. Proc. Camb. Phi. Soc. 121, 141–190 (1997)

    Article  MathSciNet  Google Scholar 

  2. Calvo, M.P., Hairer, E.: Accurate long-term integration of dynamical systems. Applied Numerical Mathematics 18, 95–105 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Calvo, M.P., Murua, A., Sanz-Serna, J.M.: Modified equations for ODEs, Chaotic numerics. In: P. E. Kloeden, K. J. Palmer (eds.) American Mathematical Society, Providence, Contemporary Mathematics, Vol. 172, 1994, pp. 63–74

  4. Cano, B.: Integración numérica de órbitas periódicas mediante métodos multipaso, Doctoral thesis, Universidad de Valladolid, 1996

  5. Cano, B.: Conserved quantities of some Hamiltonian wave equations after full discretization, Applied Mathematics Report, Report 2005/5, Universidad de Valladolid

  6. Cano, B., Durán, A.: Analysis of variable-stepsize linear multistep methods with special emphasis on symmetric ones. Math. Comp. 72, 1769–1801 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cano, B., Durán, A.: A technique to construct symmetric variable-stepsize linear multistep methods for second-order systems. Math. Comp. 72, 1803–1816 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cano, B., Sanz-Serna, J.M.: Error growth in the numerical integration of periodic orbits, with application to Hamiltonian and reversible systems. SIAM J. Numer. Anal., 34, 1391–1417 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cano, B., Sanz-Serna, J.M.: Error growth in the numerical integration of periodic orbits by multistep methods, with application to reversible systems. IMA J. Numer. Anal., 18, 57–75 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dahlquist, G., Bjorck, A.: Numerical Methods, Prentice-Hall, 1974

  11. Durán, A., Sanz-Serna, J.M.: The numerical integration of relative equilibrium solutions. The nonlinear Schrödinger equation. IMA J. Numer. Anal. 20, 235–261 (2000)

    MATH  Google Scholar 

  12. Frutos, de J., Sanz-Serna, J.M.: Accuracy and conservation properties in numerical integration: the case of the Korteweg-de Vries equation. Numer. Math. 75, 421–445 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hairer, E.: Backward error analysis for multistep methods. Numer. Math. 84, 199–232 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hairer, E., Lubich, C.: The lifespan of backward error analysis for numerical integrators. Numer. Math. 76, 441–462 (1997). 414–441 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hairer, E., Lubich, C.: Symmetric multistep methods over long times. Numer. Math. 97, 699–723 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations. Springer-Verlag, Berlin Heidelberg New York, 2002

  17. Islas, A.L., Karpeev, D.A., Schober, C.M.: Geometric integrators for the Nonlinear Schrödinger equation. J. Comp. Phys. 173, 116–148 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  18. Islas, A.L., Schober, C.M.: On the preservation of phase space structure under multisymplectic discretization. J. Comput. Phys. 197(2), 585–609 (2004)

    Article  MathSciNet  Google Scholar 

  19. Kuksin, S.B.: Nearly integrable infinite-dimensional Hamiltonian systems. Lecture Notes in Math. 1556, Springer-Verlag, Berlin, 1993

  20. Mclachlan, R., Robidoux, N.: Antisymmetry, pseudospectral methods, and conservative PDEs. Fiedler, B. et al., (ed.) International conference on differential equations. Proceedings of the conference, Equadiff '99, Berlin, Germany, August 1–7, 1999. Vol. 2. Singapore: World Scientific. (2000), pp. 994–999.

  21. Moore, B., Reich, S.: Backward error analysis for multisymplectic integration methods. Num. Math. 95, 625–652 (2003)

    Article  MATH  Google Scholar 

  22. Oliver, M., West, M., Wulff, C.: Approximate momentum conservation for spatial semidiscretization of semilinear wave equations. Numer. Math. 97, 493–535 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Reich, S.: Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations. J. Comput. Phys. 157, 473–499 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  24. Sanz-Serna, J.M., Calvo, M.P.: Numerical Hamiltonian problems. Chapman and Hall, London, 1994

  25. Tadmor, E.: The exponential accuracy of Fourier and Chebyshev differencing methods. SIAM J. Num. Anal. 23, 1–10 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  26. Weinstein.: Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16, 472–491 (1985)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática Aplicada, Universidad de Valladolid, C/Doctor Mergelina s/n, 47011, Valladolid, Spain

    B. Cano

Authors
  1. B. Cano
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. Cano.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cano, B. Conserved quantities of some Hamiltonian wave equations after full discretization. Numer. Math. 103, 197–223 (2006). https://doi.org/10.1007/s00211-006-0680-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-006-0680-3

Mathematics Subject Classification

Search

Navigation