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

  • Karol Gajda
  • Małgorzata Jankowska
  • Andrzej Marciniak
  • Barbara Szyszka
Part of the Lecture Notes in Computer Science book series (LNCS, 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.

Keywords

the initial value problem one-step interval methods symplectic interval methods multistep interval methods floating-point interval arithmetic 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Butcher, J.C.: The Numerical Analysis of Ordinary Differential Equations. Runge–Kutta and General Linear Methods. J. Wiley & Sons, Chichester (1987)MATHGoogle Scholar
  3. 3.
    Eijgenraam, P.: The Solution of Initial Value Problems Using Interval Arithmetic. Mathematical Centre Tracks 144 (1981)Google Scholar
  4. 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. 5.
    Gajda, K., Marciniak, A.: Symplectic Interval Methods for Solving the Hamiltonian Problem. Pro. Dialog. 22, 27–38 (2007)Google Scholar
  6. 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. 7.
    Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I - Nonstiff Problems. Springer, Heidelberg (1987)MATHGoogle Scholar
  8. 8.
    Hammer, R., Hocks, M., Kulisch, U., Ratz, D.: Numerical Toolbox for Verified Computing I. Basic Numerical Problems. Springer, Berlin (1993)MATHGoogle Scholar
  9. 9.
    Hill, G.W.: Researches in the Lunar Theory. Am. J. Math. I (1878)Google Scholar
  10. 10.
    Hoefkens, J., Berz, M., Makino, K.: Controlling the Wrapping Effect in the Solution of ODEs for Asteroids. Reliable Computing 8, 21–41 (2003)CrossRefGoogle Scholar
  11. 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. 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. 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. 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. 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. 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. 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. 18.
    Jaulin, L., Kieffer, M., Didrit, O., Walter, É.: Applied Interval Analysis. Springer, London (2001)MATHGoogle Scholar
  19. 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. 20.
    Krckeberg, F.: Ordinary differential equations. In: Hansen, E. (ed.) Topics in Interval Analysis, pp. 91–97. Clarendon Press, Oxford (1969)Google Scholar
  21. 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. 22.
    Marciniak, A.: Finding the Integration Interval for Interval Methods of Runge–Kutta Type in Floating-Point Interval Arithmetic. Pro. Dialog. 10, 35–46Google Scholar
  23. 23.
    Marciniak, A.: Implicit Interval Methods for Solving the Initial Value Problem. Numerical Algorithms 37, 241–251 (2004)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Marciniak, A.: Multistep Interval Methods of Nystrm and Milne-Simpson Types. Computational Methods in Science and Technology 13 (1), 23–39 (2007)MathSciNetGoogle Scholar
  25. 25.
    Marciniak, A.: Numerical Solutions of the N-body Problem. Reidel, Dordrecht (1985)MATHGoogle Scholar
  26. 26.
    Marciniak, A.: On Multistep Interval Methods for Solving the Initial Value Problem. Journal of Computational and Applied Mathematics 199, 229–237 (2007)MATHCrossRefMathSciNetGoogle Scholar
  27. 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. 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. 29.
    Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1966)MATHGoogle Scholar
  30. 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)MATHCrossRefMathSciNetGoogle Scholar
  31. 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. 32.
    Rihm, R.: On a class of enclosure methods for initial value problems. Computing 53, 369–377 (1994)MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Sanz–Serna, J.M., Calvo, M.P.: Numerical Hamiltonian Problems. Chapman & Hall, London (1994)MATHGoogle Scholar
  34. 34.
    Shokin, J.I.: Interval Analysis [in Russian]. Nauka, Novosibirsk (1981)Google Scholar
  35. 35.
    Szyszka, B.: Implicit Interval Methods of Runge-Kutta Type [in Polish], Ph.D. Thesis, Poznan University of Technology, Pozna (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Karol Gajda
    • 1
  • Małgorzata Jankowska
    • 2
  • Andrzej Marciniak
    • 3
    • 4
  • Barbara Szyszka
    • 1
  1. 1.Institute of MathematicsPoznan University of TechnologyPoznańPoland
  2. 2.Institute of Applied MechanicsPoznan University of TechnologyPoznańPoland
  3. 3.Institute of Computing SciencePoznan University of TechnologyPoznańPoland
  4. 4.Faculty of Mathematics and Computer ScienceAdam Mickiewicz UniversityPoznańPoland

Personalised recommendations