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High-Order Symplectic and Symmetric Composition Integrators for Multi-frequency Oscillatory Hamiltonian Systems

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Recent Developments in Structure-Preserving Algorithms for Oscillatory Differential Equations

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Abstract

This chapter presents symplectic and symmetric composition methods based on Adapted Runge–Kutta–Nyström (ARKN) and extended Runge–Kutta–Nyström (ERKN) integrators for solving multi-frequency and multi-dimensional oscillatory Hamiltonian systems with the Hamiltonian \(H(p,q)=\dfrac{1}{2}p^{\intercal }p+\dfrac{1}{2}q^{\intercal }Kq+U(q)\), where \(p=q'\) and K is a symmetric and positive semi-definite matrix. We first consider the symplecticity conditions for multi-frequency and multi-dimensional ARKN integrators. We then analyse the symplecticity of the adjoint integrators of the multi-frequency and multi-dimensional symplectic ARKN and ERKN integrators, respectively. On the basis of the theoretical analysis, and using the idea of composition methods, we derive four new high-order symplectic and symmetric integrators. The numerical results quantitatively show the advantage and efficiency of the high-order symplectic and symmetric integrators.

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References

  1. Blanes, S., Casas, F., Murua, A.: Splitting and composition methods in the numerical integration of differential equations. Bol. Soc. Esp. Mat. Apl. 45, 89–145 (2008)

    MathSciNet  MATH  Google Scholar 

  2. Blanes, S., Moan, P.C.: Practical symplectic partitioned Runge-Kutta and Runge-Kutta(-Nyström) methods. J. Comput. Appl. Math. 142, 313–330 (2002)

    Article  MathSciNet  Google Scholar 

  3. 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  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Calvo, M.P., Chartier, P., Murua, A., Sanz-Serna, J.M.: Numerical stroboscopic averaging for ODEs and DAEs. Appl. Numer. Math. 61, 1077–1095 (2011)

    Article  MathSciNet  Google Scholar 

  6. Calvo, M.P., Chartier, P., Murua, A., Sanz-Serna, J.M.: A stroboscopic numerical method for highly oscillatory problems. In: Engquist, B., Runborg, O., Tsai, R. (eds.) Numerical Analysis and Multiscale Computations. Lecture Notes in Computational Science and Engineering, vol. 82, pp. 73–87. Springer, Berlin (2011)

    MATH  Google Scholar 

  7. Calvo, M.P., Sanz-Serna, J.M.: Heterogeneous multiscale methods for mechanical systems with vibrations. SIAM J. Sci. Comput. 32, 2029–2046 (2011)

    Article  MathSciNet  Google Scholar 

  8. Deuflhard, P.: A study of extrapolation methods based on multistep schemes without parasitic solutions. Z. angew. Math. Phys. 30, 177–189 (1979)

    Article  MathSciNet  Google Scholar 

  9. Engquist, W.E.B., Li, X., Ren, W., Vanden-Eijnden, E.: Heterogeneous multiscale methods: a review. Appl. Numer. Math. 2, 367–450 (2007)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  11. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd edn. Springer, Berlin (1993)

    MATH  Google Scholar 

  12. Jimánez, S., Vázquez, L.: Analysis of four numerical schemes for a nonlinear Klein-Gordon equation. Appl. Math. Comput. 35, 61–93 (1990)

    MathSciNet  Google Scholar 

  13. Leimkuhler, B., Reich, S.: Simulating Hamiltonian Dynamics. Cambridge Monographs on Applied and Computational Mathematics, vol. 14. Cambridge University Press, Cambridge (2005)

    Google Scholar 

  14. Liu, K., Wu, X.Y.: High-order symplectic and symmetric composition methods for multi-frequency and multi-dimensional oscillatory Hamiltonian systems. J. Comput. Math. 33, 355–377 (2015)

    MathSciNet  MATH  Google Scholar 

  15. McLachlan, R.I.: On the numerical integration of ordinary differential equations by symmetric composition methods. SIAM J. Sci. Comput. 16, 151–168 (1995)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Reich, S.: Symplectic integration of constrained Hamiltonian systems by composition methods. SIAM J. Numer. Anal. 33, 475–491 (1996)

    Article  MathSciNet  Google Scholar 

  18. Sanz-Serna, J.M.: Modulated Fourier expansions and heterogeneous multiscale methods. IMA J. Numer. Anal. 29, 595–605 (2009)

    Article  MathSciNet  Google Scholar 

  19. Sanz-Serna, J.M., Calvo, M.P.: Numerical Hamiltonian Problem. Applied Mathematics and Mathematical Computation, 2nd edn. Springer, Chapman & Hall, London (1994)

    Google Scholar 

  20. Shi, W., Wu, X.Y.: On symplectic and symmetric ARKN methods. Comput. Phys. Commun. 183, 1250–1258 (2012)

    Article  MathSciNet  Google Scholar 

  21. Shi, W., Wu, X.Y.: A note on symplectic and symmetric ARKN methods. Comput. Phys. Commun. 184, 2408–2411 (2013)

    Article  MathSciNet  Google Scholar 

  22. 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 

  23. Suzuki, M.: General theory of higher-order decomposition of exponential operators and symplectic integrators. Phys. Lett. A 165, 387–395 (1992)

    Article  MathSciNet  Google Scholar 

  24. 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 

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

    Article  Google Scholar 

  26. Wu, X.Y., Wang, B.: Multidimensional adapted Runge-Kutta-Nyström methods for oscillatory systems. Comput. Phys. Commun. 181, 1955–1962 (2010)

    Article  Google Scholar 

  27. 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 

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

    Article  MathSciNet  Google Scholar 

  29. Wu, X.Y., You, X., Li, J.: Note on derivation of order conditions for ARKN methods for perturbed oscillators. Comput. Phys. Commun. 180, 1545–1549 (2009)

    Article  MathSciNet  Google Scholar 

  30. 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 

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

    Book  Google Scholar 

  32. 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 

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

    Article  MathSciNet  Google Scholar 

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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
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  2. Bin Wang
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Correspondence to Xinyuan Wu .

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Wu, X., Wang, B. (2018). High-Order Symplectic and Symmetric Composition Integrators for Multi-frequency Oscillatory Hamiltonian Systems. In: Recent Developments in Structure-Preserving Algorithms for Oscillatory Differential Equations. Springer, Singapore. https://doi.org/10.1007/978-981-10-9004-2_5

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