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An improved tri-coloured rooted-tree theory and order conditions for ERKN methods for general multi-frequency oscillatory systems

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Abstract

This paper develops an improved tri-coloured rooted-tree theory for the order conditions for ERKN methods solving general multi-frequency and multidimensional second-order oscillatory systems. The bottleneck of the original tricoloured rooted-tree theory is the existence of numerous redundant trees. In light of the fact that the sum of the products of the symmetries and the elementary differentials is meaningful, this paper naturally introduces the so-called extended elementary differential mappings. Then, the new improved tri-coloured rooted tree theory is established based on a subset of the original tri-coloured rooted-tree set. This new theory makes all redundant trees disappear, and thus, the order conditions of ERKN methods for general multi-frequency and multidimensional second-order oscillatory systems are reduced greatly. Furthermore, with this new theory, we present some new ERKN methods of order up to four. Numerical experiments are implemented and the results show that ERKN methods can be competitive with other existing methods in the scientific literature, especially when comparatively large stepsizes are used.

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References

  1. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. 2nd edn. Springer, Berlin (2006)

  2. Wu, X., You, X., Wang, B.: Structure-preserving algorithms for oscillatory differential equations. Springer, Berlin (2013)

    Book  MATH  Google Scholar 

  3. Wu, X., Liu, K., Shi, W.: Structure-preserving algorithms for oscillatory differential equations II. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Wu, X., Wang, B., Shi, W.: Effective integrators for nonlinear second-order oscillatory systems with a time-dependent frequency matrix. Appl. Math. Model. 37, 6505–6518 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Wu, X., Liu, C., Mei, L.: 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  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Wu, X., 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  MATH  Google Scholar 

  11. Li, J., Wang, B., You, X., Wu, X.: Two-step extended RKN methods for oscillatory systems. Comput. Phys. Commun. 182, 2486–2507 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, J., Wu, X.: Error analysis of explicit TSERKN methods for highly oscillatory systems. Numer. Algorithms 65, 465–483 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, J., Wu, X.: Adapted Falkner-type methods solving oscillatory second-order differential equations. Numer. Algorithms 62, 355–381 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  15. 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 

  16. 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 

  17. Wu, X., Wang, B., Xia, J.: Extended symplectic Runge-Kutta-Nyström integrators for separable Hamiltonian systems. In: Proceedings of the 2010 International Conference on Computational and Mathematical Methods in Science and Engineering, Vol. III, pp. 1016–1020. Spain (2010)

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  20. Yang, H., Wu, X.: Trigonometrically-fitted ARKN methods for perturbed oscillators. Appl. Numer. Math. 58, 1375–1395 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Liu, K., Wu, X.: Multidimensional ARKN methods for general oscillatory second-order initial value problems. Comput. Phys. Comm. 185, 1999–2007 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. You, X., Zhao, J., Yang, H., Fang, Y., Wu, X.: Order conditions for RKN methods solving general second-order oscillatory systems. Numer. Algorithms 66, 147–176 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. 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. Commun. 185, 2841–2850 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Boik, R.J.: Lecture notes: statistics 550 spring 2006, http://www.math.montana.edu/~rjboik/classes/550/notes.550.06.pdf, 33–35

  28. Hairer, E., Nørsett, S.P., Wanner, G.: Solving ordinary differential equations I, nonstiff problems, end edn., Springer series in Computational Mathematics. Springer, Berlin (1993)

    MATH  Google Scholar 

  29. Butcher, J.C.: An algebraic theory of integration methods. Math. Comput. 26, 79–106 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hairer, E., Wanner, G.: On the Butcher group and general multi-value methods. Computing 13, 1–15 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  31. Weinberger, H.F.: A First Course in Partial Differential Equations with Complex Variables and Transform Methods. Dover Publications Inc., New York (1965)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, People’s Republic of China

    Xianyang Zeng, Hongli Yang & Xinyuan Wu

  2. Industrial Center, Nanjing Institute of Technology, Nanjing, People’s Republic of China

    Hongli Yang

  3. Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing, People’s Republic of China

    Xianyang Zeng

  4. State Key Laboratory for Novel Software Technology at Nanjing University, Nanjing, People’s Republic of China

    Xinyuan Wu

Authors
  1. Xianyang Zeng
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  2. Hongli Yang
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  3. Xinyuan Wu
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Correspondence to Xinyuan Wu.

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Zeng, X., Yang, H. & Wu, X. An improved tri-coloured rooted-tree theory and order conditions for ERKN methods for general multi-frequency oscillatory systems. Numer Algor 75, 909–935 (2017). https://doi.org/10.1007/s11075-016-0225-5

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  • DOI: https://doi.org/10.1007/s11075-016-0225-5

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