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Derivation and implementation of Two-Step Runge-Kutta pairs

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Abstract

Explicit Runge-Kutta pairs are known to provide efficient solutions to initial value differential equations with inexpensive derivative evaluations. Two-step Runge-Kutta methods strive to improve the efficiency by utilizing approximations to the solution and its derivatives from the previous step. This article suggests a strategy for computing embeddedpairs of such two-step methods using a smaller number of function evaluations than that required for traditional Runge-Kutta methods of the same order. This leads to the efficient and reliable estimation of local discretization error and a robust step control strategy. The change of stepsize is achieved by a suitable interpolation of stage values from the previous step and does not require any additional function evaluations. Two examples illustrate the features of these pairs.

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References

  1. P. Albrecht, A new theoretical approach to Runge-Kutta methods. SIAM J. Numer. Anal.,24 (1987), 391–406.

    Article  MATH  MathSciNet  Google Scholar 

  2. P. Albrecht, Consequences of a linear approach to Runge-Kutta methods. Manuscript.

  3. P. Albrecht, The Runge-Kutta theory in a nutshell. SIAM J. Numer. Anal.,33 (1996), 1712–1735.

    Article  MATH  MathSciNet  Google Scholar 

  4. Z. Bartoszewski and Z. Jackiewicz, Construction of two-step Runge-Kutta methods of high order for ordinary differential equations. Numerical Algorithms,18 (1998), 51–70.

    Article  MATH  MathSciNet  Google Scholar 

  5. Z. Bartoszewski and Z. Jackiewicz, Toward a two-step Runge-Kutta code for nonstiff differential systems. To appear in Appl. Math.

  6. J.C. Butcher, The Numerical Analysis of Ordinary Differential Equations. John Wiley, New York, 1987.

    MATH  Google Scholar 

  7. J.C. Butcher and S. Tracogna, Order conditions for two-step Runge-Kutta methods. Appl. Numer. Math.,24 (1997), 351–364.

    Article  MATH  MathSciNet  Google Scholar 

  8. G.D. Byrne and R.J. Lambert, Pseudo-Runge-Kuttamethods involvingtwo points. J. Assoc. Comput. Mach.,13 (1966), 114–123.

    MATH  MathSciNet  Google Scholar 

  9. R. Caira, C. Costabile and F. Costabile, A class of pseudo Runge-Kutta methods. BIT,30 (1990), 642–649.

    Article  MATH  MathSciNet  Google Scholar 

  10. N. huu Cong and T. Mitsui, Collocation-based two-step Runge-Kutta methods. Japan J. Indust. Appl. Math.,13 (1996), 171–183.

    Article  MATH  MathSciNet  Google Scholar 

  11. E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems. Springer-Verlag, Berlin, Heidelberg, New York, 1991.

    Google Scholar 

  12. E. Hairer and G. Wanner, Order conditions for general two-step Runge-Kutta methods. SIAM J. Numer. Anal.,34 (1997), 2087–2089.

    Article  MATH  MathSciNet  Google Scholar 

  13. T.E. Hull, W.H. Enright, B.M. Fellen and A.E. Sedgwick, Comparing numerical methods for ordinary differential equations. SIAM J. Numer. Anal.,9 (1972), 603–637.

    Article  MATH  MathSciNet  Google Scholar 

  14. Z. Jackiewicz, R. Renaut and A. Feldstein, Two-step Runge-Kutta methods. SIAM J. Numer. Anal.,28 (1991), 1165–1182.

    Article  MATH  MathSciNet  Google Scholar 

  15. Z. Jackiewicz, R. Renaut and M. Zennaro, Explicit two-step Runge-Kutta methods. Appl. Math.,40 (1995), 433–456.

    MATH  MathSciNet  Google Scholar 

  16. Z. Jackiewicz and R. Vermiglio, General linear methods with external stages of different orders. BIT,36 (1996), 688–712.

    Article  MATH  MathSciNet  Google Scholar 

  17. Z. Jackiewicz and S. Tracogna, A representation formula for two-step Runge-Kutta methods. Proc. of the Second Hellenic European Conference on Mathematics and Informatics (ed. E. Lipitakis), September 22–24, 1994, Hellenic Mathematical Society, Athens, Greece, 1994, 111–120.

    Google Scholar 

  18. Z. Jackiewicz and S. Tracogna, A general class of two-step Runge-Kutta methods for ordinary differential equations. SIAM J. Numer. Anal.,32 (1995), 1390–1427.

    Article  MATH  MathSciNet  Google Scholar 

  19. Z. Jackiewicz and S. Tracogna, Variable stepsize continuous two-step Runge-Kutta methods for ordinary differential equations. Numerical Algorithms,12 (1996), 347–368.

    Article  MATH  MathSciNet  Google Scholar 

  20. Z. Jackiewicz and M. Zennaro, Variable-stepsize explicit two-step Runge-Kutta methods. Math. Comp.,59 (1992), 421–438.

    Article  MATH  MathSciNet  Google Scholar 

  21. B. Owren and M. Zennaro, Derivation of efficient, continuous, explicit Runge-Kutta methods. SIAM J. Sci. Stat. Comput.,13 (1992), 1488–1501.

    Article  MATH  MathSciNet  Google Scholar 

  22. L.F. Shampine, Numerical Solution of Ordinary Differential Equations. Chapman & Hall, New York, London, 1994.

    MATH  Google Scholar 

  23. S. Tracogna, Implementation of two-step Runge-Kutta methods for ordinary differential equations. J. Comput. Appl. Math.,76 (1996), 113–136.

    Article  MATH  MathSciNet  Google Scholar 

  24. S. Tracogna and B. Welfert, Two-step Runge-Kutta methods. Theory and practice. BIT,40 (2000), 775–799.

    Article  MATH  MathSciNet  Google Scholar 

  25. J.H. Verner, Strategies for deriving new explicit Runge-Kutta pairs. Ann. Numer. Math.,1 (1994), 225–244.

    MATH  MathSciNet  Google Scholar 

  26. J.H. Verner, High order explicit Runge-Kutta pairs with low stage order. Appl. Numer. Math.,20 (1996), 1–13.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Arizona State University, 85287-1804, Tempe, Arizona, USA

    Z. Jackiewicz

  2. Department of Mathematics and Statistics, PIMS, Simon Fraser University, V5A 1S6, Burnaby, Canada

    J. H. Verner

Authors
  1. Z. Jackiewicz
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  2. J. H. Verner
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Corresponding author

Correspondence to Z. Jackiewicz.

Additional information

The work of this author was supported by the National Science Foundation grant NSF DMS-9971164.

The work of this author was supported by the National Science and Engineering Council of Canada grant A8147.

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Jackiewicz, Z., Verner, J.H. Derivation and implementation of Two-Step Runge-Kutta pairs. Japan J. Indust. Appl. Math. 19, 227–248 (2002). https://doi.org/10.1007/BF03167454

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  • DOI: https://doi.org/10.1007/BF03167454

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