Skip to main content
SpringerLink
Log in

A Survey of Interval Runge–Kutta and Multistep Methods for Solving the Initial Value Problem

  • Conference paper
Parallel Processing and Applied Mathematics (PPAM 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4967))

Abstract

The paper is dealt with a number of one– and multistep interval methods developed by our team during the last decade. We present implicit interval methods of Runge–Kutta type, interval versions of symplectic Runge–Kutta methods and interval multistep methods of Adams–Bashforth, Adams–Moulton, Nyström and Milne–Simpson types.

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

Access this chapter

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 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Berz, M., Makino, K.: Verified integration of ODEs and flows with differential algebraic methods on Taylor models. Reliable Computing 4 (4), 361–369 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. Butcher, J.C.: The Numerical Analysis of Ordinary Differential Equations. Runge–Kutta and General Linear Methods. J. Wiley & Sons, Chichester (1987)

    MATH  Google Scholar 

  3. Eijgenraam, P.: The Solution of Initial Value Problems Using Interval Arithmetic. Mathematical Centre Tracks 144 (1981)

    Google Scholar 

  4. Gajda, K.: Interval Methods of a Symplectic Runge-Kutta Type [in Polish], Ph.D. Thesis, Poznan University of Technology, Pozna (2004)

    Google Scholar 

  5. Gajda, K., Marciniak, A.: Symplectic Interval Methods for Solving the Hamiltonian Problem. Pro. Dialog. 22, 27–38 (2007)

    Google Scholar 

  6. Gajda, K., Marciniak, A., Szyszka, B.: Three- and Four-Stage Implicit Interval Methods of Runge-Kutta Type. Computational Methods in Science and Technology 6, 41–59 (2000)

    Google Scholar 

  7. Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I - Nonstiff Problems. Springer, Heidelberg (1987)

    MATH  Google Scholar 

  8. Hammer, R., Hocks, M., Kulisch, U., Ratz, D.: Numerical Toolbox for Verified Computing I. Basic Numerical Problems. Springer, Berlin (1993)

    MATH  Google Scholar 

  9. Hill, G.W.: Researches in the Lunar Theory. Am. J. Math. I (1878)

    Google Scholar 

  10. Hoefkens, J., Berz, M., Makino, K.: Controlling the Wrapping Effect in the Solution of ODEs for Asteroids. Reliable Computing 8, 21–41 (2003)

    Article  Google Scholar 

  11. Jankowska, M.: Interval Multistep Methods of Adams Type and their Implementation in the C++ Language, Ph.D. Thesis, Poznan University of Technology, Pozna (2006)

    Google Scholar 

  12. Jankowska, M., Marciniak, A.: An Interval Version of the Backward Differentiation (BDF) Method. In: SCAN 2006 Conference Post-Proceedings IEEE-CPS Product No. E2821 (2007)

    Google Scholar 

  13. Jankowska, M., Marciniak, A.: On the Interval Methods of the BDF Type for Solving the Initial Value Problem. Pro. Dialog. 22, 39–59 (2007)

    Google Scholar 

  14. Jankowska, M., Marciniak, A.: On Two Families of Implicit Interval Methods of Adams-Moulton Type. Computational Methods in Science and Technology 12 (2), 109–113 (2006)

    Google Scholar 

  15. Jankowska, M., Marciniak, A.: Implicit Interval Multistep Methods for Solving the Initial Value Problem. Computational Methods in Science and Technology 8 (1), 17–30 (2002)

    Google Scholar 

  16. Jankowska, M., Marciniak, A.: On Explicit Interval Methods of Adams-Bashforth Type. Computational Methods in Science and Technology 8 (2), 46–57 (2002)

    Google Scholar 

  17. Jankowska, M., Marciniak, A.: Preliminaries of the IMM System for Solving the Initial Value Problem by Interval Multistep Methods [in Polish]. Pro. Dialog. 10, 117–134 (2005)

    Google Scholar 

  18. Jaulin, L., Kieffer, M., Didrit, O., Walter, É.: Applied Interval Analysis. Springer, London (2001)

    MATH  Google Scholar 

  19. Kalmykov, S.A., Shokin, J.I., Juldashev, E.C.: Solving Ordinary Differential Equations by Interval Methods [in Russian]. Doklady AN SSSR 230 (6) (1976)

    Google Scholar 

  20. Krckeberg, F.: Ordinary differential equations. In: Hansen, E. (ed.) Topics in Interval Analysis, pp. 91–97. Clarendon Press, Oxford (1969)

    Google Scholar 

  21. Lohner, R.J.: Enclosing the solutions of ordinary initial and boundary value problems. In: Kaucher, E.W., Kulisch, U.W., Ullrich, C. (eds.) Computer Arithmetic: Scientific Computation and Programming Languages. Wiley-Teubner Series in Computer Science, Stuttgart (1987)

    Google Scholar 

  22. Marciniak, A.: Finding the Integration Interval for Interval Methods of Runge–Kutta Type in Floating-Point Interval Arithmetic. Pro. Dialog. 10, 35–46

    Google Scholar 

  23. Marciniak, A.: Implicit Interval Methods for Solving the Initial Value Problem. Numerical Algorithms 37, 241–251 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Marciniak, A.: Multistep Interval Methods of Nystrm and Milne-Simpson Types. Computational Methods in Science and Technology 13 (1), 23–39 (2007)

    MathSciNet  Google Scholar 

  25. Marciniak, A.: Numerical Solutions of the N-body Problem. Reidel, Dordrecht (1985)

    MATH  Google Scholar 

  26. Marciniak, A.: On Multistep Interval Methods for Solving the Initial Value Problem. Journal of Computational and Applied Mathematics 199, 229–237 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  27. Marciniak, A., Szyszka, B.: One- and Two-Stage Implicit Interval Methods of Runge-Kutta Type. Computational Methods in Science and Technology 5, 53–65 (1999)

    Google Scholar 

  28. Marciniak, A., Szyszka, B.: On Representation of Coefficients in Implicit Interval Methods of Runge-Kutta Type. Computational Methods in Science and Technology 10 (1), 57–71 (2004)

    Google Scholar 

  29. Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1966)

    MATH  Google Scholar 

  30. Nedialkov, N.S., Jackson, K.R.: An interval Hermite-Obreschkoff method for computing rigorous bounds on the solution of an initial value problem for an ordinary differential equation. Reliable Computing 5 (3), 289–310 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  31. Rihm, R.: Interval methods for initial value problems in ODE’s. In: Topics in Validated Computations, Proceedings of the IMACS-GAMM International Workshop on Validated Computations, Oldenburg, Germany, J. Herzberger (1994)

    Google Scholar 

  32. Rihm, R.: On a class of enclosure methods for initial value problems. Computing 53, 369–377 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  33. Sanz–Serna, J.M., Calvo, M.P.: Numerical Hamiltonian Problems. Chapman & Hall, London (1994)

    MATH  Google Scholar 

  34. Shokin, J.I.: Interval Analysis [in Russian]. Nauka, Novosibirsk (1981)

    Google Scholar 

  35. Szyszka, B.: Implicit Interval Methods of Runge-Kutta Type [in Polish], Ph.D. Thesis, Poznan University of Technology, Pozna (2003)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Poznan University of Technology, Piotrowo 3a, 60-965, Poznań, Poland

    Karol Gajda & Barbara Szyszka

  2. Institute of Applied Mechanics, Poznan University of Technology, Piotrowo 3, 60-965, Poznań, Poland

    Małgorzata Jankowska

  3. Institute of Computing Science, Poznan University of Technology, Piotrowo 2, 60-965, Poznań, Poland

    Andrzej Marciniak

  4. Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614, Poznań, Poland

    Andrzej Marciniak

Authors
  1. Karol Gajda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Małgorzata Jankowska
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andrzej Marciniak
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Barbara Szyszka
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Roman Wyrzykowski Jack Dongarra Konrad Karczewski Jerzy Wasniewski

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Gajda, K., Jankowska, M., Marciniak, A., Szyszka, B. (2008). A Survey of Interval Runge–Kutta and Multistep Methods for Solving the Initial Value Problem. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Wasniewski, J. (eds) Parallel Processing and Applied Mathematics. PPAM 2007. Lecture Notes in Computer Science, vol 4967. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68111-3_144

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-68111-3_144

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-68105-2

  • Online ISBN: 978-3-540-68111-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation