Skip to main content
SpringerLink
Log in
Book cover

Structure-Preserving Algorithms for Oscillatory Differential Equations II pp 211–227Cite as

Multidimensional ARKN Methods for General Multi-frequency Oscillatory Second-Order IVPs

  • Chapter
  • First Online:

  • 572 Accesses

Abstract

Based on B-series theory, the order conditions of the multidimensional ARKN methods were presented by Wu et al. [18] for general multi-frequency oscillatory second-order initial value problems where the functions on right-hand side depend on both position and velocity. The class of physical problems which fall within its scope is broader.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Ascher UM, Reich S (1999) On some difficulties in integrating highly oscillatory Hamiltonian systems. In: Proceedings of the computational molecular dynamics, Springer Lecture Notes, pp 281-296

    Google Scholar 

  2. Cohen D, Hairer E, Lubich C (2003) Modulated Fourier expansions of highly oscillatory differential equations. Found Comput Math 3:327–450

    Article  MathSciNet  MATH  Google Scholar 

  3. Garcá-Archilla B, Sanz-Serna J, Skeel R (1998) Long-time-step methods for oscillatory differential equations. SIAM J Sci Comput 30:930–963

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Fang Y, Wu X (2007) A new pair of explicit ARKN methods for the numerical integration of general perturbed oscillators. Appl Numer Math 57:166–175

    Article  MathSciNet  MATH  Google Scholar 

  6. Fang Y, Wu X (2008) A trigonometrically fitted explicit Numerov-type method for second-order initial value problems with oscillating solutions. Appl Numer Math 58:341–351

    Article  MathSciNet  MATH  Google Scholar 

  7. García-Alonso F, Reyes J, Ferrádiz J, Vigo-Aguiar J (2009) Accurate numerical integration of perturbed oscillatory systems in two frequencies. ACM Trans Math Softw 36:21–34

    Article  MathSciNet  Google Scholar 

  8. Hairer E, Nørsett SP, Wanner G (1993) Solving ordinary differnetial equations I, Nonstiff problems, 2nd edn., Springer series in computational mathematicsSpringer, Berlin

    MATH  Google Scholar 

  9. Iserles A (2002) Think globally, act locally: solving highly-oscillatory ordinary differential equations. Appl Numer Math 43:145–160

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Petzold L, Jay L, Yen J (1997) Numerical solution of highly oscillatory ordinary differential equations. Acta Numer 6:437–484

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Van der Houwen PJ, Sommeijer BP (1987) Explicit Runge-Kutta(-Nyström) methods with reduced phase errors for computing oscillating solution. SIAM J Numer Anal 24:595–617

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Weinberger HF (1965) A first course in partial differential equations with complex variables and transform methods. Dover Publications, Inc, New York

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Wu X, You X, Wang B (2013) Structure-preserving integrators for oscillatory ordinary differential equations. Springer, Heidelberg (jointly published with Science Press Beijing)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, China

    Xinyuan Wu

  2. Nanjing University of Finance and Economics, Nanjing, China

    Kai Liu

  3. Nanjing University, Nanjing, China

    Wei Shi

Authors
  1. Xinyuan Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kai Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wei Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xinyuan Wu .

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer-Verlag Berlin Heidelberg and Science Press, Beijing, China

About this chapter

Cite this chapter

Wu, X., Liu, K., Shi, W. (2015). Multidimensional ARKN Methods for General Multi-frequency Oscillatory Second-Order IVPs. In: Structure-Preserving Algorithms for Oscillatory Differential Equations II. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48156-1_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-48156-1_10

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-48155-4

  • Online ISBN: 978-3-662-48156-1

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation