Skip to main content
SpringerLink
Log in

Structure-Preserving Numerical Methods for Infinite-Dimensional Birkhoffian Systems

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

A universal infinite-dimensional Birkhoffian formalism for system of partial differential equations (PDEs) including non-conservative cases is introduced as a generalization of infinite-dimensional Hamiltonian system. A class of generalized Hamilton–Jacobi equation for functionals is investigated to construct a kind of generating functional which is connected to solution of PDEs in Birkhoffian formalism. It is a new extension for generalizing the generating function method for finite-dimensional systems to generating functional method for infinite-dimensional systems. Based on the theory of generating functional, a way to construct structure-preserving schemes for Birkhoffian systems is explained. Numerical experiments are carried out on both acoustic wave and transverse-electric wave propagations with dissipations. The numerical results show that the proposed structure-preserving schemes developed by generating functionals have a clear superiority over symplectic or/and multi-symplectic schemes for long term computation. Moreover, the proposed S4 scheme preserves globally the Poynting’s energy on the electromagnetic fields in the perfectly matched layer medium.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry. Springer, New York (1999)

    Book  MATH  Google Scholar 

  2. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1978)

    Book  MATH  Google Scholar 

  3. Feng, K., Wang, D.L.: Symplectic difference schemes for Hamiltonian systems in general symplectic structure. J. Comput. Math. 9, 86–96 (1991)

    MathSciNet  MATH  Google Scholar 

  4. Feng, K., Qin, M.: Symplectic Geometric Algorithms for Hamiltonian Systems. Springer, Heiddelberg (2009)

    Google Scholar 

  5. Shang, Z.: Generating functions for volume-preserving mappings and Hamilton–Jacobi equations for source-free systems. Sci. China (Ser. A) 37, 1172–1188 (1994)

    MathSciNet  MATH  Google Scholar 

  6. Marsden, J.E., Patrick, G.W., Shkoller, S.: Multisymplectic geometry, variational integrators, and nonlinear PDEs. Commun. Math. Phys. 199, 351–395 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Marsden, J.E., Shkoller, S.: Multisymplectic geometry, covariant Hamiltonians and water waves. Math. Proc. Camb. Philos. Soc. 125, 553–575 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Bridges, T.J.: Multi-symplectic structures and wave propagation. Math. Proc. Camb. Philos. Soc. 121, 147–190 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hong, J., Sun, Y.: Generating functions of multi-symplectic RK methods via DW Hamilton–Jacobi equations. Numer. Math. 110, 491–519 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Tang, Y.F.: The necessary condition for a Runge–Kutta scheme to be symplectic for Hamiltonian systems. Comp. Math. Appl. 26, 13–20 (1993)

    Article  MATH  Google Scholar 

  12. McLachlan, R.: Symplectic integration of Hamiltonian wave equations. Numer. Math. 66, 465–492 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Saitoh, I., Suzuki, Y., Takahashi, N.: Stability of symplectic finite diffrence time domain method. IEEE Trans. Magn. 38(6), 665–668 (2002)

    Article  Google Scholar 

  14. Reich, S.: Multi-symplectic Runge–Kutta collocation methods for Hamiltonian wave equations. J. Comput. Phys. 157, 473–499 (2002)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Santilli, R.M.: Foundations of Theoretical Mechanics II. Springer, New York (1983)

    Book  MATH  Google Scholar 

  18. Su, H.-L., Qin, M.-Z.: Symplectic schemes for Birkhoffian system. Commun. Theor. Phys. (Beijing, China) 41, 329–334 (2004)

  19. Su, H., Qin, M., Wang, Y., Scherer, R.: Multi-symplectic Birkhoffian structure for PDEs with dissipation terms. Phys. Lett. A 374, 2410–2416 (2010)

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

    Book  Google Scholar 

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

    MATH  Google Scholar 

  22. Feng, K., Wang, D.L.: A note on conservation laws of symplectic difference schemes for Hamiltonian systems. JCM 9, 229–237 (1991)

    MATH  Google Scholar 

  23. Feng, K., Qin, M.Z.: The symplectic methods for the computation of Hamilton’s equations. In: Guo Ben-Yu, Y.L. (ed.) Proceedings of Conference on Numerical Methods for PDEs, Lecture Notes in Math, 1297, pp. 1–37. Springer, Berlin (1987)

  24. Berenger, J.P.: A perfectly matched layer for the absorption of electromagnetic waves. JCP 114, 185–200 (1994)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Renmin University of China, Beijing, 100872, China

    Hongling Su

  2. Theoretical Division, Los Alamos National Lab, Los Alamos, NM, 87544, USA

    Shengtai Li

Authors
  1. Hongling Su
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shengtai Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hongling Su.

Additional information

Project 10701081 and 11071251 supported by the NNSFC.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Su, H., Li, S. Structure-Preserving Numerical Methods for Infinite-Dimensional Birkhoffian Systems. J Sci Comput 65, 196–223 (2015). https://doi.org/10.1007/s10915-014-9958-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-014-9958-2

Keywords

Mathematics Subject Classification

Search

Navigation