Order conditions for RKN methods solving general second-order oscillatory systems

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• 作者：Xiong You (1) (2)
  Jinxi Zhao (1)
  Hongli Yang (3)
  Yonglei Fang (4)
  Xinyuan Wu (5)
  
• 关键词：Extended Runge ; Kutta ; Nystr?m type methods ; Extended Nystr?m trees ; Order conditions ; Second ; orderoscillatory systems ; 65L05 ; 65L06
• 刊名：Numerical Algorithms
• 出版年：2014
• 出版时间：May 2014
• 年：2014
• 卷：66
• 期：1
• 页码：147-176
• 全文大小：
• 参考文献：1. Aubry, A., Chartier, P.: Pseudo-symplectic Runge-Kutta methods. BIT. 38, 439-61 (1998). CrossRef
  2. Avdyushev, V.?A.: Special perturbation theory methods in celestial mechanics, I. Principles for the construction and substantiation of the application. Russ. Phys. J. 49, 1344-353 (2006). CrossRef
  3. Franco, J.?M.: Runge-Kutta-Nystr?m methods adapted to the numerical integration of perturbed oscillators. Comput. Phys. Comm. 147, 770-87 (2002). CrossRef
  4. García-Alonso, F., Reyes, J., Ferrádiz, J., Vigo-Aguiar, J.: Accurate numerical integration of perturbed oscillatory systems in two frequencies. ACM Trans. Math. Softw. 36, 21-4 (2009). CrossRef
  5. López, D.?J., Martín, P.: A numerical method for the integration of perturbed linear problems. Appl. Math. Comput. 96, 65-3 (1998). CrossRef
  6. Ramos, H., Vigo-Aguiar, J.: Variable-stepsize Chebyshev-type methods for the integration of second-order I.V.P.s. J. Comput. Appl. Math. 204, 102-13 (2007). CrossRef
  7. Jordan, D.?W., Smith, P.: Nonlinear ordinary differential equations, 4th edn. An introduction for scientists and engineers. Oxford University Press, Oxford, (2007)
  8. Wu, X., You, X., Li, J.: Note on derivation of order conditions for ARKN methods for perturbed oscillator. Comput. Phys. Comm. 180, 1545-549 (2009). CrossRef
  9. Wu, X., You, X., Xia, J.: Order conditions for ARKN methods solving oscillatory systems. Comput. Phys. Comm. 180, 2250-257 (2009). CrossRef
  10. Wu, X., Wang, B.: Multidimensional adapted Runge-Kutta-Nystrom methods for oscillatory systems. Comput. Phys. Comm. 181, 1955-962 (2010). CrossRef
  11. Yang, H., Wu, X., You, X., Fang, Y.: Extended RKN-type methods for numerical integration of perturbed oscillators. Comput. Phys. Comm. 180, 1777-794 (2009). CrossRef
  12. Wu, X., You, X., Shi, W., Wang, B.: ERKN integrators for systems of oscillatory secondorder differential equations. Comput. Phys. Comm. 181, 1873-887 (2010). CrossRef
  13. Wu, X., You, X., Wang, B.: Structure-Preserving Algorithms for Oscillatory Differential Equations. Springer-Verlag, Berlin, Heidelberg (2013). CrossRef
  14. Hairer, E., N?rsett, S.?P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problem. Springer-Verlag, Berlin, Heidelberg (1993).
  15. Butcher, J.?C.: The Numerical Analysis of Ordinary Differential Equations, 2nd edn. Wiley, New York, (2008).
  16. Johnson, W.?P.: The curious history of Faà di Bruno’s Formula. Math. Assoc. Am. 109, 217-34 (2002). CrossRef
  17. 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, 635-46 (1999). CrossRef
  18. Weinberger, H.?F.: A First Course in Partial Differential Equations with Complex Variables and Transform Methods. Dover Publications Inc., New York (1965).
  19. Franco, J.?M.: New methods for oscillatory systems based on ARKN methods. Appl. Numer. Math. 56, 1040-053 (2006). CrossRef
  
• 作者单位：Xiong You (1) (2)
  Jinxi Zhao (1)
  Hongli Yang (3)
  Yonglei Fang (4)
  Xinyuan Wu (5)
  
  1. State Key Laboratory for Novel Software Technology at Nanjing University, Nanjing University, Nanjing, 210093, People’s Republic of China
  2. Department of Applied Mathematics, Nanjing Agricultural University, Nanjing, 210095, People’s Republic of China
  3. Institute of Mathematics, Nanjing Institute of Technology, Nanjing, 211167, People’s Republic of China
  4. Department of Mathematics and Information Science, Zaozhuang University, Zaozhuang, 277160, People’s Republic of China
  5. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China
  
• ISSN：1572-9265

文摘

This paper proposes and investigates the multidimensional extended Runge-Kutta-Nystr?m (ERKN) methods for the general second-order oscillatory system y-+ My = f(y, y- where M is a positive semi-definite matrix containing implicitly the frequencies of the problem. The work forms a natural generalization of our previous work on ERKN methods for the special system y-+ My = f(y) (H. Yang et al. Extended RKN-type methods for numerical integration of perturbed oscillators, Comput. Phys. Comm. 180 (2009) 1777-794 and X. Wu et al., ERKN integrators for systems of oscillatory second-order differential equations, Comput. Phys. Comm. 181 (2010) 1873-887). The new ERKN methods, with coefficients depending on the frequency matrix M, incorporate the special structure of the equation brought by the term My into both internal stages and updates. In order to derive the order conditions for the ERKN methods, an extended Nystr?m tree (EN-tree) theory is established. The results of numerical experiments show that the new ERKN methods are more efficient than the general-purpose RK methods and the adapted RKN methods with the same algebraic order in the literature.