Skip to main content
SpringerLink
Log in

An Energy-Preserving and Symmetric Scheme for Nonlinear Hamiltonian Wave Equations

  • Chapter
  • First Online:
Recent Developments in Structure-Preserving Algorithms for Oscillatory Differential Equations

  • 557 Accesses

Abstract

In this chapter, we derive and analyse an energy-preserving and symmetric scheme for nonlinear Hamiltonian wave equations, which can exactly preserve the energy of the underlying Hamiltonian wave equations. To this end, we first define and discuss the bounded operator-argument functions on the underlying domain. We then introduce an operator-variation-of-constants formula, based on which we present an energy-preserving scheme for nonlinear Hamiltonian wave equations. The scheme preserves the energy of the original continuous Hamiltonian system exactly. In comparison with the existing work on this topic, such as the well-known Average Vector Field (AVF) formula for Hamiltonian ordinary differential equations, the energy-preserving scheme avoids the semi-discretisation of spatial derivatives and exactly preserves the Hamiltonian of the original continuous Hamiltonian wave equation. This point is very significant in comparison with the AVF formula, since the AVF formula can preserve only the energy of Hamiltonian ordinary differential equations. Hence, the main theme of this chapter is to establish a scheme which can exactly preserve the energy of the nonlinear Hamiltonian wave equation. The chapter is also accompanied by some examples.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.00
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Berti, M.: Nonlinear Oscillations of Hamiltonian PDEs. Springer, Berlin (2007)

    MATH  Google Scholar 

  2. Biswas, A.: Soliton perturbation theory for phi-four model and nonlinear Klein–Gordon equations. Commun. Nonlinear Sci. Numer. Simul. 14, 3239–3249 (2009)

    Article  MathSciNet  Google Scholar 

  3. Celledoni, E., Grimm, V., McLachlan, R.I., McLaren, D.I., O’Neale, D., Owren, B., Quispel, G.R.W.: Preserving energy resp. dissipation in numerical PDEs using the "Average Vector Field" method. J. Comput. Phys. 231(20), 6770–6789 (2012)

    Article  MathSciNet  Google Scholar 

  4. Cohen, D., Jahnke, T., Lorenz, K., Lubich, C.: Numerical integrators for highly oscillatory Hamiltonian systems: a review. In: Mielke, A. (ed.) Analysis, Modeling and Simulation of Multiscale Problems, pp. 553–576. Springer, Berlin (2006)

    Google Scholar 

  5. Courant, R., Friedrichs, K., Lewy, H.: Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Annal. 100, 32–74 (1928); reprinted and translated. IBM J. Res. Dev. 11, 215–234 (1967)

    Google Scholar 

  6. Dehghan, M.: Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math. Comput. Simul. 71, 16–30 (2006)

    Article  MathSciNet  Google Scholar 

  7. Dehghan, M., Mirezaei, D.: Numerical solution to the unsteady two-dimensional Schrödinger equation using meshless local boundary integral equation method. Int. J. Numer. Methods Eng. 76, 501–520 (2008)

    Article  Google Scholar 

  8. Dehghan, M., Shokri, A.: Numerical solution of the nonlinear KleinCGordon equation using radial basis functions. J. Comput. Appl. Math. 230, 400–410 (2009)

    Article  MathSciNet  Google Scholar 

  9. Dodd, R.K., Eilbeck, I.C., Gibbon, J.D., Morris, H.C.: Solitons and Nonlinear Wave Equations. Academic, London (1982)

    MATH  Google Scholar 

  10. Eilbeck, J.C.: Numerical studies of solitons. In: Bishop, A.R., Schneider, T. (eds.) Solitons and Condensed Matter Physics, pp. 28–43. Springer, New York (1978)

    Chapter  Google Scholar 

  11. Fordy, A.P.: Soliton Theory: A Survey of Results. Manchester University Press (1990)

    Google Scholar 

  12. Franco, J.M.: New methods for oscillatory systems based on ARKN methods. Appl. Numer. Math. 56, 1040–1053 (2006)

    Article  MathSciNet  Google Scholar 

  13. 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 (1998)

    Article  MathSciNet  Google Scholar 

  14. Hairer, E., Lubich, C.: Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Numer. Anal. 38, 414–441 (2000)

    Article  MathSciNet  Google Scholar 

  15. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  16. Hochbruck, M., Lubich, C.: A Gautschi-type method for oscillatory second-order differential equations. Numer. Math. 83, 403-426 (1999)

    Article  MathSciNet  Google Scholar 

  17. Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010)

    Article  MathSciNet  Google Scholar 

  18. Infeld, E., Rowlands, G.: Nonlinear Waves, Solitons and Chaos. Cambridge University Press, New York (1990)

    MATH  Google Scholar 

  19. Liu, K., Wu, X.Y.: An extended discrete gradient formula for oscillatory Hamiltonian systems. J. Phys. A: Math. Theor. 46(165203), 19 (2013)

    Article  MathSciNet  Google Scholar 

  20. Liu, C., Wu, X.Y.: The boundness of the operator-valued functions for multidimensional nonlinear wave equations with applications. Appl. Math. Lett. 74, 60–67 (2017)

    Article  MathSciNet  Google Scholar 

  21. Matsuo, T.: New conservative schemes with discrete variational derivatives for nonlinear wave equations. J. Comput. Appl. Math. 203, 32–56 (2007)

    Article  MathSciNet  Google Scholar 

  22. Matsuo, T., Yamaguchi, H.: An energy-conserving Galerkin scheme for a class of nonlinear dispersive equations. J. Comput. Phys. 228, 4346–4358 (2009)

    Article  MathSciNet  Google Scholar 

  23. McLachlan, R.I., Quispel, G.R.W., Robidoux, N.: Geometric integration using discrete gradients. Philos. Trans. R. Soc. A 357, 1021–1045 (1999)

    Article  MathSciNet  Google Scholar 

  24. Quispel, G.R.W., McLaren, D.I.: A new class of energy-preserving numerical integration methods. J. Phys. A 41(045206), 7 (2008)

    Article  MathSciNet  Google Scholar 

  25. Ringler, T.D., Thuburn, J., Klemp, J.B., Skamarock, W.C.: A unified approach to energy conservation and potential vorticity dynamics for arbitrarily structured C-grids. J. Comput. Phys. 229, 3065–3090 (2010)

    Article  MathSciNet  Google Scholar 

  26. Schiesser, W.: The Numerical Methods Of Lines: Integration Of Partial Differential Equation. Academic Press, San Diego (1991)

    MATH  Google Scholar 

  27. Sevryuk, M. B., Lectures in Mathematics., 1211, Springer, Berlin (1986)

    Google Scholar 

  28. Shakeri, F., Dehghan, M.: Numerical solution of the Klein–Gordon equation via He’s variational iteration method. Nonlinear Dyn. 51, 89–97 (2008)

    Article  MathSciNet  Google Scholar 

  29. Shi, W., Wu, X.Y., Xia, J.: Explicit multi-symplectic extended leap-frog methods for Hamiltonian wave equations. J. Comput. Phys. 231, 7671–7694 (2012)

    Article  MathSciNet  Google Scholar 

  30. Taleei, A., Dehghan, M.: Time-splitting pseudo-spectral domain decomposition method for the soliton solutions of the one and multi-dimensional nonlinear Schrödinger equations. Comput. Phys. Commun. 185, 1515–1528 (2014)

    Article  MathSciNet  Google Scholar 

  31. Van de Vyver, H.: Scheifele two-step methods for perturbed oscillators. J. Comput. Appl. Math. 224, 415–432 (2009)

    Article  MathSciNet  Google Scholar 

  32. Wang, B., Liu, K., Wu, X.Y.: A Filon-type asymptotic approach to solving highly oscillatory second-order initial value problems. J. Comput. Phys. 243, 210–223 (2013)

    Article  MathSciNet  Google Scholar 

  33. Wang, B., Wu, X.Y.: A new high precision energy-preserving integrator for system of oscillatory second-order differential equations. Phys. Lett. A 376, 1185–1190 (2012)

    Article  MathSciNet  Google Scholar 

  34. Wang, B., Iserles, A., Wu, X.Y.: Arbitrary-order trigonometric Fourier collocation methods for multi-frequency oscillatory systems. Found. Comput. Math. 16, 151–181 (2016)

    Article  MathSciNet  Google Scholar 

  35. Wazwaz, A.M.: New travelling wave solutions to the Boussinesq and the Klein–Gordon equations. Commun. Nonlinear Sci. Numer. Simul. 13, 889–901 (2008)

    Article  MathSciNet  Google Scholar 

  36. Wu, X.Y., Liu, C.: An energy-preserving and symmetric scheme for nonlinear Hamiltonian wave equations. J. Math. Anal. Appl. 440, 167–182 (2016)

    Article  MathSciNet  Google Scholar 

  37. Wu, X.Y., Liu, C.: An integral formula adapted to different boundary conditions for arbitrarily high-dimensional nonlinear Klein–Gordon equations with its applications. J. Math. Phys. 57, 021504 (2016)

    Article  MathSciNet  Google Scholar 

  38. Wu, X.Y., Liu, C., Mei, L.J.: A new framework for solving partial differential equations using semi-analytical explicit RK(N)-type integrators. J. Comput. Appl. Math. 301, 74–90 (2016)

    Article  MathSciNet  Google Scholar 

  39. Wu, X.Y., Mei, L.J., Liu, C.: An analytical expression of solutions to nonlinear wave equations in higher dimensions with Robin boundary conditions. J. Math. Anal. Appl. 426, 1164–1173 (2015)

    Article  MathSciNet  Google Scholar 

  40. Wu, X.Y., Wang, B., Shi, W.: Efficient energy-preserving integrators for oscillatory Hamiltonian systems. J. Comput. Phys. 235, 587–605 (2013)

    Article  MathSciNet  Google Scholar 

  41. Wu, X.Y., You, X., Xia, J.: Order conditions for ARKN methods solving oscillatory systems. Comput. Phys. Commun. 180, 2250–2257 (2009)

    Article  MathSciNet  Google Scholar 

  42. Wu, X.Y., Wang, B., Xia, J.: Explicit symplectic multidimensional exponential fitting modified Runge–Kutta–Nystrom methods. BIT Numer. Math. 52, 773–795 (2012)

    Article  MathSciNet  Google Scholar 

  43. Wu, X.Y., You, X., Shi, W., Wang, B.: ERKN integrators for systems of oscillatory second-order differential equations. Comput. Phys. Commun. 181, 1873–1887 (2010)

    Article  MathSciNet  Google Scholar 

  44. Wu, X.Y., You, X., Wang, B.: Structure-Preserving Algorithms for Oscillatory Differential Equations. Springer, Berlin (2013)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, Jiangsu, China

    Xinyuan Wu

  2. School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, China

    Xinyuan Wu & Bin Wang

Authors
  1. Xinyuan Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xinyuan Wu .

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Singapore Pte Ltd. And Science Press

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Wu, X., Wang, B. (2018). An Energy-Preserving and Symmetric Scheme for Nonlinear Hamiltonian Wave Equations. In: Recent Developments in Structure-Preserving Algorithms for Oscillatory Differential Equations. Springer, Singapore. https://doi.org/10.1007/978-981-10-9004-2_10

Download citation

Publish with us

Policies and ethics

Search

Navigation