Skip to main content

Experiments with a variable-order type 1 DIMSIM code

Abstract

The issues related to the development of a new code for nonstiff ordinary differential equations are discussed. This code is based on the Nordsieck representation of type 1 DIMSIMs, implemented in a variable-step size variable-order mode. Numerical results demonstrate that the error estimation employed in the code is very reliable and that the step and order changing strategies are very robust. This code outperforms the Matlab ode45 code for moderate and stringent tolerances.

This is a preview of subscription content, access via your institution.

References

  1. J.C. Butcher, The Numerical Analysis of Ordinary Differential Equations (Wiley, Chichester/New York, 1987).

    MATH  Google Scholar 

  2. J.C. Butcher, Diagonally-implicit multi-stage integration methods, Appl. Numer. Math. 11 (1993) 347–363.

    MATH  MathSciNet  Article  Google Scholar 

  3. J.C. Butcher, A transformation for the analysis of DIMSIMs, BIT 34 (1994) 25–32.

    MATH  MathSciNet  Article  Google Scholar 

  4. J.C. Butcher and P. Chartier, The construction of DIMSIMs for stiff ODEs and DAEs, Report Series No. 308, The University of Auckland, New Zealand (July 1994).

    Google Scholar 

  5. J.C. Butcher, P. Chartier and Z. Jackiewicz, Nordsieck representation of DIMSIMs, Numer. Algorithms 16 (1997) 209–230.

    MATH  MathSciNet  Article  Google Scholar 

  6. J.C. Butcher and Z. Jackiewicz, Diagonally implicit general linear methods for ordinary differential equations, BIT 33 (1993) 452–472.

    MATH  MathSciNet  Article  Google Scholar 

  7. J.C. Butcher and Z. Jackiewicz, Construction of diagonally implicit general linear methods of type 1 and 2 for ordinary differential equations, Appl. Numer. Math. 21 (1996) 385–415.

    MATH  MathSciNet  Article  Google Scholar 

  8. J.C. Butcher and Z. Jackiewicz, Implementation of diagonally implicit multistage integration methods for ordinary differential equations, SIAM J. Numer. Anal. 34 (1997) 2119–2141.

    MATH  MathSciNet  Article  Google Scholar 

  9. J.C. Butcher and Z. Jackiewicz, Construction of high order diagonally implicit multistage integration methods for ordinary differential equations, Appl. Numer. Math. 27 (1998) 1–12.

    MATH  MathSciNet  Article  Google Scholar 

  10. J.C. Butcher, Z. Jackiewicz and H.D. Mittelmann, Nonlinear optimization approach to construction of general linear methods of high order, J. Comput. Appl. Math. 81 (1997) 181–196.

    MATH  MathSciNet  Article  Google Scholar 

  11. P. Chartier, The potential of parallel multi-value methods for the simulation of large real-life problems, CWI Quarterly 11 (1998) 7–32.

    MATH  MathSciNet  Google Scholar 

  12. J.R. Dormand and P.J. Prince, A family of embedded Runge–Kutta formulae, J. Comput. Appl. Math. 6 (1980) 19–26.

    MATH  MathSciNet  Article  Google Scholar 

  13. I. Gladwell, L.F. Shampine and R.W. Brankin, Automatic selection of the initial step size for an ODE solver, J. Comput. Appl. Math. 18 (1987) 175–192.

    MATH  MathSciNet  Article  Google Scholar 

  14. E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems (Springer, Berlin, 1993).

    MATH  Google Scholar 

  15. L.F. Shampine, Numerical Solution of Ordinary Differential Equations (Chapman Hall, New York/London, 1994).

    MATH  Google Scholar 

  16. L.F. Shampine and M.W. Reichelt, The Matlab ODE suite, SIAM J. Sci. Comput. 18 (1977) 1–22.

    MathSciNet  Article  Google Scholar 

  17. W.M. Wright, The construction of order 4 DIMSIM for ordinary differential equations, submitted.

  18. W.M. Wright, General linear methods for ordinary differential equations. Theoretical and practical investigation, M.Sc. thesis, The University of Auckland, New Zealand (1999).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Butcher, J., Chartier, P. & Jackiewicz, Z. Experiments with a variable-order type 1 DIMSIM code. Numerical Algorithms 22, 237–261 (1999). https://doi.org/10.1023/A:1019135630307

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1019135630307

  • DIMSIM methods
  • Nordsieck representation
  • local error estimation
  • step size and order changing strategy
  • 65L05