Skip to main content
SpringerLink
Log in

Exponential Integrators Based on Discrete Gradients for Linearly Damped/Driven Poisson Systems

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

Abstract

Exponential integrators based on discrete gradient methods are applied to non-canonical Hamiltonian systems with added linear forcing/damping terms, which may be time-dependent. Changes in the dynamics, such as conservation of energy or Casimirs, which result from inclusion of the linear forcing/damping terms, are not exactly preserved by standard discrete gradient methods. However, those changes are shown to be exactly preserved by the exponential integrators in special circumstances. The methods are also symmetric, second order, and linearly stable. To demonstrate advantages in both accuracy and efficiency over other standard methods, the exponential integrators are applied to a three dimensional Lotka-Volterra system and a damped/driven Ablowitz-Ladik system.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Discrete dissipative solitons: Abdullaev, FKh. Lect. Notes Phys. 661, 327–341 (2005)

    Article  Google Scholar 

  2. Ascher, U.M., McLachlan, R.I.: Multi-symplectic box schemes and the Korteweg-de Vries equation. Appl. Numer. Math. 48, 255–269 (2004)

    Article  MathSciNet  Google Scholar 

  3. Bhatt, A., Floyd, D., Moore, B.E.: Second order conformal symplectic schemes for damped Hamiltonian systems. J. Sci Comput. 66, 1234–1259 (2016)

    Article  MathSciNet  Google Scholar 

  4. Bhatt, A., Moore, B.E.: Structure-preserving exponential Runge-Kutta methods. SIAM J. Sci. Comput. 39(2), A593–A612 (2017)

    Article  MathSciNet  Google Scholar 

  5. Bhatt, A., Moore, B.E.: Exponential integrators preserving local conservation laws of PDEs with time-dependent damping/driving forces. J. Comput. Appl. Math. 352, 341–351 (2019)

    Article  MathSciNet  Google Scholar 

  6. Cai, D., Bishop, A.R., Grønbech-Jensen, N., Malomed, B.A.: Moving solitons in the damped Ablowitz-Ladik model driven by a standing wave. Phys. Rev. E 50(2), R694–R697 (1994)

    Article  Google Scholar 

  7. Cai, W., Zhang, H., Wang, Y.: Modelling damped acoustic waves by a dissipation-preserving conformal symplectic method. Proc. R. Soc. A 473, 20160798 (2017)

    Article  MathSciNet  Google Scholar 

  8. 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. Comp. Phys. 231, 6770-6789 (2012)

  9. Cohen, D., Hairer, E.: Linear energy-preserving integrators for Poisson systems. BIT Numer. Math. 51, 91–101 (2011)

    Article  MathSciNet  Google Scholar 

  10. Dressler, U.: Symmetry property of the Lyapunov spectra of a class of dissipative dynamical systems with viscous damping. Phys. Rev. A 38(4), 2103–2109 (1988)

    Article  Google Scholar 

  11. Fu, H., Zhou, W.-E., Qian, X., Song, S.-H., Zhang, L.-Y.: Conformal structure-preserving method for damped nonlinear Schrödinger equation. Chin. Phys. B 25(11), 405 (2016)

    Article  Google Scholar 

  12. Gonzalez, O.: Time integration of discrete Hamiltonian systems. J. Nonlinear Sci. 6, 449–467 (1996)

    Article  MathSciNet  Google Scholar 

  13. Hairer, E.: Energy-preserving variant of collocation methods. J. Numer. Anal. Ind. Appl. Math. 5, 73–84 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Hairer, E., Lubich, Ch.: Energy-diminishing integration of gradient systems. IMA J. Numer. Anal. 34, 452–461 (2014)

    Article  MathSciNet  Google Scholar 

  15. Hairer, E., Lubich, Ch., Wanner, G.: Geometric numerical integration: structure preserving algorithms for ordinary differential equations. Springer-Verlag, Berlin (2002)

    Book  Google Scholar 

  16. Harten, A., Lax, P.D., van Leer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SAIM Rev. 25, 35–61 (1983)

    MathSciNet  MATH  Google Scholar 

  17. Itoh, T., Abe, K.: Hamiltonian preserving discrete canonical equations based on variational difference quotients. J. Comp. Phys. 76(1), 85–102 (1988)

    Article  Google Scholar 

  18. Kong, X., Wu, H., Mei, F.: Structure-preserving algorithms for Birkhoffian systems. J. Geom. Phys. 62, 1157–1166 (2012)

    Article  MathSciNet  Google Scholar 

  19. Lawson, J.D.: Generalized Runge-Kutta processes for stable systems with large Lipschitz constants. SIAM J. Numer. Anal. 4(3), 372–380 (1967)

    Article  MathSciNet  Google Scholar 

  20. Leimkuhler, B., Reich, S.: Simulating Hamiltonian Dynamics, Cambridge (2004)

  21. Li, Y.W., Wu, X.: Exponential integrators preserving first integrals or Lyapunov functions for conservative or dissipative systems. SIAM J. Sci. Comput. 38(3), A1876–A1895 (2016)

    Article  MathSciNet  Google Scholar 

  22. Mei, L., Huang, L., Huang, S.: Exponential integrators with quadratic energy preservation for linear Poisson systems. J. Comp. Phys. 387, 446–454 (2019)

    Article  MathSciNet  Google Scholar 

  23. McLachlan, R.I., Perlmutter, M.: Conformal Hamiltonian systems. J. Geom. Phys. 39(4), 276–300 (2001)

    Article  MathSciNet  Google Scholar 

  24. McLachlan, R.I., Quispel, G.R.W.: What kinds of dynamics are there? Lie pseudogroups, dynamical systems and geometric integration, Nonlinearity 14(6), 1689–1705 (2001)

    MATH  Google Scholar 

  25. McLachlan, R.I., Quispel, G.R.W.: Splitting methods. Acta Numer. 11, 341–434 (2002)

    Article  MathSciNet  Google Scholar 

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

  27. Modin, K., Söderlind, G.: Geometric integration of Hamiltonian systems perturbed by Rayleigh damping. BIT Numer. Math. 51, 977–1007 (2011)

    Article  MathSciNet  Google Scholar 

  28. Moore, B.E.: Conformal multi-symplectic integration methods for forced-damped semi-linear wave equations. Math. Comput. Simulat. 80, 20–28 (2009)

    Article  MathSciNet  Google Scholar 

  29. Moore, B.E.: Multi-conformal-symplectic PDEs and discretizations. J. Comput. Appl. Math. 323, 1–15 (2017)

    Article  MathSciNet  Google Scholar 

  30. Moore, B.E., Noreña, L., Schober, C.M.: Conformal conservation laws and geometric integration for damped Hamiltonian PDEs. J. Comp. Phys. 232, 214–233 (2013)

    Article  MathSciNet  Google Scholar 

  31. Quispel, G.R.W., McLaren, D.I.: A new class of energy-preserving numerical integration methods. J. Phys. A-Math. Theor. 41(4), 36 (2008)

    Article  MathSciNet  Google Scholar 

  32. Ruan, J., Wang, L.: Exponential discrete gradient schemes for stochastic differential equations, arXiv:1711.02522 [math.NA] (2017)

  33. Sanz-Serna, J.M., Calvo, M.P.: Numerical hamiltonian problems. Chapman & Hall, London (1994)

    Book  Google Scholar 

  34. Shang, X., Ottinger, H.C.: Structure-preserving integrators for dissipative systems based on reversible-irreversible splitting. Proc. R. Soc. A 476, 2019446 (2020)

    Article  MathSciNet  Google Scholar 

  35. Shchesnovich, V.S., Barashenkov, I.V.: Soliton-radiation in the parametrically driven, damped nonlinear Schrödinger equation. Physica D 164, 83–109 (2002)

    Article  MathSciNet  Google Scholar 

  36. Shen, X., Leok, M.: Geometric exponential integrators. J. Comp. Phys. 382, 27–42 (2019)

    Article  MathSciNet  Google Scholar 

  37. Stuart, A.M., Humphries, A.R.: Dynamical systems and numerical analysis. Cambridge Press, Cambridge (1998)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  39. Sun, Y., Shang, Z.: Structure-preserving algorithms for Birkhoffian systems. Phys. Lett. A 336, 358–369 (2005)

    Article  Google Scholar 

  40. Wang, B.: Exponential average-vector-field integrator for conservative or ,dissipative systems Recent Developments in Structure-Preserving algorithms for oscillatory differential equations. Springer, Singapore (2018)

    Google Scholar 

  41. Wang, B., Wu, X.: Exponential collocation methods for conservative or dissipative systems. J. Comput. Appl. Math. 360, 99–116 (2019)

    Article  MathSciNet  Google Scholar 

  42. Zemlyanaya, E.V., Barashenkov, I.V.: Traveling solitons in the damped-driven nonlinear Schrödinger equation. SIAM J. Appl. Math. 64, 800–818 (2004)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author thanks Elena Celledoni, Brynjulf Owren, and Reinout Quispel for helpful conversations as the ideas in this article were being developed.

Author information

Authors and Affiliations

  1. University of Central Florida, Orlando, USA

    Brian E. Moore

Authors
  1. Brian E. Moore
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Brian E. Moore.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Moore, B.E. Exponential Integrators Based on Discrete Gradients for Linearly Damped/Driven Poisson Systems. J Sci Comput 87, 56 (2021). https://doi.org/10.1007/s10915-021-01468-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-021-01468-1

Keywords

Mathematics Subject Classification

Search

Navigation