Skip to main content
SpringerLink
Log in

Accuracy of classical conservation laws for Hamiltonian PDEs under Runge–Kutta discretizations

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

We investigate conservative properties of Runge–Kutta methods for Hamiltonian partial differential equations. It is shown that multi-symplecitic Runge–Kutta methods preserve precisely the norm square conservation law. Based on the study of accuracy of Runge–Kutta methods applied to ordinary and partial differential equations, we present some results on the numerical accuracy of conservation laws of energy and momentum for Hamiltonian PDEs under Runge–Kutta discretizations.

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 T.J., Reich S.: Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve sysmplecticity. Phys. Lett. A 284, 184–193 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bridges T.J., Reich S.: Numerical methods for Hamiltonian PDEs. J. Phys. A Math. Gen. 39, 5287–5320 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cano B.: Conserved quantities of some Hamiltonian wave equations after full discretization. Numer. Math. 103, 197–223 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Hairer, E., Lubich, C., ROCHE, M.: The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods. Lecture Notes in Mathematics, vol. 1409. Springer, Berlin (1989)

  5. Hairer E., Lubich C., Wanner G.: Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, Berlin (2002)

    MATH  Google Scholar 

  6. Hong J., Li C.: Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations. J. Comput. Phys. 211, 448–472 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hong J., Liu H., Sun G.: The multi-symplecticity of partitioned Runge-Kutta methods for Hamiltonian PDEs. Math. Comput. 75, 167–181 (2006)

    MATH  MathSciNet  Google Scholar 

  8. Hong J., Liu Y., Munthe-Kaas H., Zanna A.: Global conservative properties and error estimation of a multi-symplectic scheme for Schrödinger equations with variable coefficients. Appl. Numer. Math. 56, 814–843 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hong J.: A survey of multi-symplectic Runge-Kutta type methods for Hamiltonian partial differential equations. In: Li, T., Zhang, P.(eds) Frontiers and Prospects of Contemporary Applied Mathematics. Series in Contemporary Applied Mathematics, vol CAM 6, pp. 71–113. World Scientific and Higher Education Press, Singapore (2005)

    Google Scholar 

  10. Hong, J., Li, C.: Some properties of multi-symplectic Runge-Kutta methods for Dirac equations. Research Report of ICMSEC (2004)

  11. Iserles A.: A First Course in the Numerical Analysis of Differential Equations. Cambridge University Press, Cambridge (1996)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Leimkuhler B., Reich S.: Simulating Hamiltonian Dynamics. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  15. Moore B., Reich S.: Multisymplectic integration methods for Hamiltonian PDEs. Future Gener. Comput. Syst. 19, 395–402 (2003)

    Article  Google Scholar 

  16. Moore B., Reich S.: Backward error analysis for multi-symplectic integration methods. Numer. Math. 95, 625–652 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Sanz-Serna J.M., Calvo M.P.: Numerical Hamiltonian Problems. Chapman & Hall, London (1994)

    MATH  Google Scholar 

Download references

Author information

Author notes

  1. Shanshan Jiang

    Present address: School of Mathematics Sciences, Peking University, 100871, Beijing, People’s Republic of China

Authors and Affiliations

  1. State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, P.O. Box 2719, 100080, Beijing, People’s Republic of China

    Jialin Hong, Shanshan Jiang & Chun Li

  2. Graduate School of the Chinese Academy of Sciences, 100080, Beijing, People’s Republic of China

    Shanshan Jiang & Chun Li

Authors
  1. Jialin Hong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shanshan Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chun Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jialin Hong.

Additional information

J. Hong, S. Jiang and C. Li are supported by the Director Innovation Foundation of ICMSEC and AMSS, the Foundation of CAS, the NNSFC (No. 19971089, No. 10371128, No. 60771054) and the Special Funds for Major State Basic Research Projects of China 2005CB321701.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hong, J., Jiang, S. & Li, C. Accuracy of classical conservation laws for Hamiltonian PDEs under Runge–Kutta discretizations. Numer. Math. 112, 1–23 (2009). https://doi.org/10.1007/s00211-008-0204-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-008-0204-4

Mathematics Subject Classification (2000)

Search

Navigation