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Sixth-order symplectic and symmetric explicit ERKN schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations

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Abstract

This paper is devoted to the analysis of the sixth-order symplectic and symmetric explicit extended Runge–Kutta–Nyström (ERKN) schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations. Fourteen practical sixth-order symplectic and symmetric explicit ERKN schemes are constructed, and their phase properties are investigated. The paper is accompanied by five numerical experiments, including a nonlinear two-dimensional wave equation. The numerical results in comparison with the sixth-order symplectic and symmetric Runge–Kutta–Nyström methods and a Gautschi-type method demonstrate the efficiency and robustness of the new explicit schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations.

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References

  1. Alolyan, I., Anastassi, Z.A., Simos, T.E.: A new family of symmetric linear four-step methods for the efficient integration of the Schrödinger equation and related oscillatory problems. Appl. Math. Comput. 218, 5370–5382 (2012)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Cohen, D., Hairer, E., Lubich, C.: Numerical Energy Conservation for Multi-Frequency Oscillatory Differential Equations. BIT 45, 287–305 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Franco, J.M.: Runge-Kutta-Nyström methods adapted to the numerical integration of perturbed oscillators. Comput. Phys. Comm. 147, 770–787 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. García, A., Martín, P., González, A.B.: New methods for oscillatory problems based on classical codes. Appl. Numer. Math. 42, 141–157 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. González, A.B., Martín, P., Farto, J.M.: A new family of Runge-Kutta type methods for the numerical integration of perturbed oscillators. Numer. Math. 82, 635–646 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Okunbor, D., Skeel, R.D.: Canonical Runge-Kutta-Nyström methods of order 5 and 6. J. Comput. Appl. Math. 51, 375–382 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  12. Panopoulos, G. A., Simos, T. E.: An eight-step semi-embedded predictor-corrector method for orbital problems and related IVPs with oscillatory solutions for which the frequency is unknown. J. Comput. Appl. Math. 1–15 (2015)

  13. Stavroyiannis, S., Simos, T.E.: Optimization as a function of the phase-lag order of two-step P-stable method for linear periodic IVPs. App. Numer. Math. 59, 2467–2474 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Stiefel, E.L., Scheifele, G.: Linear and regular celestial mechanics. Springer-Verlag, New York (1971)

    Book  MATH  Google Scholar 

  15. Tocino, A., Vigo-Aguiar, J.: Symplectic conditions for exponential fitting Runge-Kutta-Nyström methods. Math. Comput. Modell. 42, 873–876 (2005)

    Article  MATH  Google Scholar 

  16. Van de Vyver, H.: Stability and phase-lag analysis of explicit Runge-Kutta methods with variable coefficients for oscillatory problems. Comput. Phys. Comm. 173, 115–130 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Vigo-Aguiar, J., Simos, T.E., Ferrándiz, J.M.: Controlling the error growth in long-term numerical integration of perturbed oscillations in one or more frequencies. Proc. Roy. Soc. London Ser. A 460, 561–567 (2004)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang, B., Li, G.: Bounds on asymptotic-numerical solvers for ordinary differential equations with extrinsic oscillation. Appl. Math. Modell. 39, 2528–2538 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Wang, B., Wu, X.: A highly accurate explicit symplectic ERKN method for multi-frequency and multidimensional oscillatory Hamiltonian systems. Numer. Algo. 65, 705–721 (2014)

    Article  MATH  Google Scholar 

  23. Wang, B., Wu, X.: Explicit multi-frequency symmetric extended RKN integrators for solving multi-frequency and multidimensional oscillatory reversible systems. CALCOLO 52, 207–231 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang, B., Wu, X., Xia, J.: Error bounds for explicit ERKN integrators for systems of multi-frequency oscillatory second-order differential equations. Appl. Numer. Math. 74, 17–34 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wang, B., Wu, X., Zhao, H.: Novel improved multidimensional Strömer-Verlet formulas with applications to four aspects in scientific computation. Math. Comput. Modell. 57, 857–872 (2013)

    Article  MATH  Google Scholar 

  26. Wu, X.: A note on stability of multidimensional adapted Runge-Kutta-Nyström methods for oscillatory systems. Appl. Math. Modell. 36, 6331–6337 (2012)

    Article  MATH  Google Scholar 

  27. Wu, X., Wang, B., Liu, K., Zhao, H.: ERKN methods for long-term integration of multidimensional orbital problems. Appl. Math. Modell. 37, 2327–2336 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Wu, X., Wang, B., Xia, J.: Explicit symplectic multidimensional exponential fitting modified Runge-Kutta-Nyström methods. BIT 52, 773–795 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  32. Yang, H., Wu, X., You, X., Fang, Y.: Extended RKN-type methods for numerical integration of perturbed oscillators. Comput. Phys. Comm. 180, 1777–1794 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. Yang, H., Zeng, X., Wu, X., Ru, Z.: A simplified Nyström-tree theory for extended Runge-Kutta-Nyström integrators solving multi-frequency oscillatory systems. Comput. Phys. Comm. 185, 2841–2850 (2014)

    Article  MATH  Google Scholar 

  34. Yoshida, H.: Construction of higher order symplectic integrators. Phys. Lett. A 150, 262–268 (1990)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The authors are sincerely thankful to two anonymous reviewers for their valuable suggestions, which help improve the presentation of the manuscript significantly.

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Authors and Affiliations

  1. School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, 273165, China

    Bin Wang & Fanwei Meng

  2. Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing, 211167, China

    Hongli Yang

Authors
  1. Bin Wang
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  2. Hongli Yang
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  3. Fanwei Meng
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Correspondence to Bin Wang.

Additional information

The research of the first author was supported by the National Natural Science Foundation of China under Grant 11401333, by the Natural Science Foundation of Shandong Province under Grant ZR2014AQ003 and by the China Postdoctoral Science Foundation under Grant 2015M580578. The research of the second author was supported in part by the Science Foundations of the Nanjing Institute of Technology under Grant YKJ201114 and under Grant QKJB2011022. The research of the third author was supported by the National Natural Science Foundation of China under Grant 11171178.

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Wang, B., Yang, H. & Meng, F. Sixth-order symplectic and symmetric explicit ERKN schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations. Calcolo 54, 117–140 (2017). https://doi.org/10.1007/s10092-016-0179-y

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  • DOI: https://doi.org/10.1007/s10092-016-0179-y

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