Skip to main content
SpringerLink
Log in

Experiments with a variable-order type 1 DIMSIM code

  • Published:
Numerical Algorithms Aims and scope Submit manuscript

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, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

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.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  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.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  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.

    Article  MATH  MathSciNet  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.

    Article  MATH  MathSciNet  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.

    Article  MATH  MathSciNet  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.

    Article  MATH  MathSciNet  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.

    Article  MATH  MathSciNet  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.

    Article  MATH  MathSciNet  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.

    Article  MathSciNet  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

  1. Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand

    J.C. Butcher

  2. INRIA, Campus Universitaire de Beaulieu, F-35042, Rennes Cedex, France

    P. Chartier

  3. Department of Mathematics, Arizona State University, Tempe, AZ, 85287, USA

    Z. Jackiewicz

Authors
  1. J.C. Butcher
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Chartier
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Z. Jackiewicz
    View author publications

    You can also search for this author in PubMed Google Scholar

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

Search

Navigation