Diagonal implicit symplectic extended RKN methods for solving oscillatory Hamiltonian systems

  • Mingxue Shi
  • Hao Zhang
  • Bin WangEmail author


This paper studies diagonal implicit symplectic extended Runge–Kutta–Nyström (ERKN) methods for solving the oscillatory Hamiltonian system \(H(q,p)=\dfrac{1}{2}p^\mathrm{T}p+\dfrac{1}{2}q^\mathrm{T}Mq+U(q)\). Based on symplecticity conditions and order conditions, we construct some diagonal implicit symplectic ERKN methods. The stability of the obtained methods is discussed. Three numerical experiments are carried out to show the performance of the methods. It follows from the numerical results that the new diagonal implicit symplectic methods are more effective than RKN methods when applied to the oscillatory Hamiltonian system.


Diagonal implicit methods Symplectic methods ERKN methods Oscillatory Hamiltonian systems 

Mathematics Subject Classification

65L05 65L20 65P10 



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


  1. Cohen D, Hairer E, Lubich C (2005) Numerical energy conservation for multi-frequency oscillatory differential equations. BIT 45:287–305MathSciNetCrossRefGoogle Scholar
  2. García A, Martín P, González AB (2002) New methods for oscillatory problems based on classical codes. Appl Numer Math 42:141–157MathSciNetCrossRefGoogle Scholar
  3. García-Archilla B, Sanz-Serna JM, Skeel RD (1999) Long-time-step methods for oscillatory differential equations. SIAM J Sci Comput 20:930–963MathSciNetCrossRefGoogle Scholar
  4. Hochbruck M, Lubich C (1999) A Gautschi-type method for oscillatory second-order differential equations. Numer Math 83:403–426MathSciNetCrossRefGoogle Scholar
  5. Hairer E, Lubich C (2000) Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J Numer Anal 38:414–441MathSciNetCrossRefGoogle Scholar
  6. Hairer E, Lubich C, Wanner G (2006) Geometric numerical integration: structure-preserving algorithms for ordinary differential equations. Springer-Verlag, Berlin, HeidelbergzbMATHGoogle Scholar
  7. Kevorkian J, Cole JD (1981) Perturbation Methods in Applied Mathematics. Applied Mathematical Sciences, 34th edn. Springer, New YorkCrossRefGoogle Scholar
  8. Kevorkian J, Cole JD (1996) Multiple scale and singular perturbation methods, applied mathematical sciences, vol 114. Springer, New YorkCrossRefGoogle Scholar
  9. Okunbor D, Skeel RD (1994) Canonical Runge-Kutta-Nyström methods of order 5 and 6. J Comput Appl Math 51:375–382MathSciNetCrossRefGoogle Scholar
  10. Ruth RD (1983) A canonical integration technique. IEEE Trans Nuclear Sci NS 30:2669–2671CrossRefGoogle Scholar
  11. Simos TE, Vigo-Aguiar J (2003) Exponentially fitted symplectic integrator. Phys Rev E 67:016701–7MathSciNetCrossRefGoogle Scholar
  12. Van der Houwen PJ, Sommeijer BP (1987) Explicit Runge-Kutta (-Nyström) methods with reduced phase errors for computing oscillating solutions. SIAM J Numer Anal 24:595–617MathSciNetCrossRefGoogle Scholar
  13. Wang B (2018) Triangular splitting implementation of RKN-type Fourier collocation methods for second-order differential equations. Math Meth Appl Sci 41:1998–2011MathSciNetCrossRefGoogle Scholar
  14. Wang B, Iserles A, Wu X (2016) Arbitrary order trigonometric Fourier collocation methods for second-order ODEs. Found Comput Math 16:151–181MathSciNetCrossRefGoogle Scholar
  15. Wang B, Li T, Wu Y (2018) Arbitrary-order functionally fitted energy-diminishing methods for gradient systems. Appl Math Lett 83:130–139MathSciNetCrossRefGoogle Scholar
  16. Wang B, Meng F, Fang Y (2017a) Efficient implementation of RKN-type Fouier collocation methods for second-order differential equations. Appl Numer Math 119:164–178MathSciNetCrossRefGoogle Scholar
  17. Wang B, Wu X (2015) Explicit multi-frequency symmetric extended RKN integrators for solving multi-frequency and multidimensional oscillatory reversible systems. Calcolo 52:207–231MathSciNetCrossRefGoogle Scholar
  18. Wang B, Wu X (2018) The formulation and analysis of energy-preserving schemes for solving high-dimensional nonlinear Klein-Gordon equations. IMA J Numer Anal.
  19. Wang B, Wu X, Meng F (2017b) Trigonometric collocation methods based on Lagrange basis polynomials for multi-frequency oscillatory second order differential equations. J Comput Appl Math 313:185–201MathSciNetCrossRefGoogle Scholar
  20. Wang B, Wu X, Xia J (2013) Error bounds for explicit ERKN methods for systems of multi-frequency oscillatory second-order differential equations. Appl Numer Math 74:17–34MathSciNetCrossRefGoogle Scholar
  21. Wang B, Yang H, Meng F (2017c) Sixth order symplectic and symmetric explicit ERKN schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations. Calcolo 54:117–140MathSciNetCrossRefGoogle Scholar
  22. Wu X (2012) A note on stability of multidimensional adapted Runge-Kutta-Nyström methods for oscillatory systems. Appl Math Modell 36:6331–6337CrossRefGoogle Scholar
  23. Wu X, Liu K, Shi W (2015) Structure-preserving algorithms for oscillatory differential equations ll. Springer-Verlag, HeidelbergCrossRefGoogle Scholar
  24. Wu X, Wang B, Xia J (2012) Explicit symplectic multidimensional exponential fitting modified Runge-Kutta-Nyström methods. BIT 52:773–795MathSciNetCrossRefGoogle Scholar
  25. Wu X, You X, Shi W, Wang B (2010) ERKN integerators for systems of oscillatory second-order differential equations. Comput Phys Comm 181:1873–1887MathSciNetCrossRefGoogle Scholar
  26. Wu X, You X, Wang B (2013) Structure-preserving algorithms for oscillatory differential equations. Springer-Verlag, Berlin, HeidelbergCrossRefGoogle Scholar
  27. Zhang H, Shi M, Li J, Wang B (2017) Diagonal implicit symmetric ERKN integrators for solving oscillatory reversible systems. Inter J Appl Comput Math 3:1229–1247MathSciNetCrossRefGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesQufu Normal UniversityQufuChina

Personalised recommendations