BIT Numerical Mathematics

, Volume 15, Issue 1, pp 10–48 | Cite as

Comparing numerical methods for stiff systems of O.D.E:s

  • W. H. Enright
  • T. E. Hull
  • B. Lindberg
Article

Abstract

This paper describes a technique for comparing numerical methods that have been designed to solve stiff systems of ordinary differential equations. The basis of a fair comparison is discussed in detail. Measurements of cost and reliability are made over a collection of 25 carefully selected problems. The problems have been designed to show how certain major factors affect the performance of a method.

The technique is applied to five methods, of which three turn out to be quite good, including one based on backward differentiation formulas, another on second derivative formulas, and a third on extrapolation. However, each of the three has a weakness of its own, which can be identified with particular problem characteristics.

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References

  1. R. Bedet, W. H. Enright and T. E. Hull (1974),STIFF DETEST: a program for comparing numerical methods for stiff ordinary differential equations, Dept. of Computer Science Tech. Rep., in preparation, University of Toronto, Toronto.Google Scholar
  2. G. Bjurel, G. Dahlquist, B. Lindberg, S. Linde and L. Odén (1970),Survey of stiff ordinary differential equations, Report NA 70.11, Dept. of Information Processing, Royal Inst. of Tech., Stockholm.Google Scholar
  3. J. C. Butcher (1964),Implicit Runge-Kutta processes, Math. Comp. 18, pp. 50–64.Google Scholar
  4. G. Dahlquist (1963),A special stability problem for linear multistep methods, BIT 3, pp. 27–43.Google Scholar
  5. G. Dahlquist and B. Lindberg (1973),On some implicit one-step methods for stiff differential equations, TRITA-NA-7302, Dept. of Information Processing, Royal Inst. of Tech., Stockholm.Google Scholar
  6. A. K. Datta (communicated by H. H. Robertson) (1967),An evaluation of the approximate inverse algorithm for numerical integration of stiff differential equations, Technical Report MSH/67/84, Imperial Chemical Industries Ltd., Cheshire.Google Scholar
  7. H. T. Davis (1962),Introduction to Nonlinear Differential and Integral Equations, Dover, New York.Google Scholar
  8. E. J. Davison (1971), Private communication.Google Scholar
  9. B. L. Ehle (1968),High order A-stable methods for the numerical solution of differential equations, BIT 8, pp. 276–278.Google Scholar
  10. W. H. Enright (1972),Studies in the numerical solution of stiff ordinary differential equations, Dept. of Computer Science Tech. Rep. No. 46, University of Toronto, Toronto.Google Scholar
  11. W. H. Enright (1974a),Second derivative multistep methods for stiff ordinary differential equations, SIAM J Numer. Anal. 11, pp. 321–331.Google Scholar
  12. W. H. Enright (1974b),Optimal second derivative methods for stiff systems, in Stiff Differential Systems (R. A. Willoughby ed.), Plenum Press, pp. 95–111.Google Scholar
  13. W. H. Enright, R. Bedet, I. Farkas and T. E. Hull (1974),Test results on initial value methods for non-stiff ordinary differential equations, Dept. of Computer Science Tech. Rep. No. 68, University of Toronto, Toronto.Google Scholar
  14. C. W. Gear (1969),The automatic integration of stiff ordinary differential equations, Proceedings of IFIP Congress 1968, North Holland Publishing Company, Amsterdam, pp. 187–193.Google Scholar
  15. C. W. Gear (1971a),Algorithm 407, DIFSUB for solution of ordinary differential equations, C.A.C.M. 14, pp. 185–190.Google Scholar
  16. C. W. Gear (1971b),Numerical intial value problems in ordinary differential equations, Prentice-Hall, Englewood Cliffs, N.J.Google Scholar
  17. C. W. Gear (1971c),Simultaneous numerical solution of differential-algebraic equations, IEEE Trans. on Circuit Theory 18, pp. 89–95.Google Scholar
  18. G. D. Hachtel, R. K. Brayton and F. G. Gustavson (1971),The sparse tableau approach to network analysis and design, IEEE Trans. on Circuit Theory 18, pp. 101–113.Google Scholar
  19. G. Hall, W. H. Enright, T. E. Hull and A. E. Sedgwick (1973),DETEST: a program for comparing numerical methods for ordinary differential equations, Dept. of Computer Science Tech. Rep. No. 60, University of Toronto, Toronto.Google Scholar
  20. A. C. Hindmarsh (1972),GEAR: Ordinary differential equation system solver, UCID-30001, Rev. 2, Lawrence Livermore Laboratory, University of California, Livermore.Google Scholar
  21. T. E. Hull, W. H. Enright, B. M. Fellen and A. E. Sedgwick (1972),Comparing numerical methods for ordinary differential equations, SIAM J. Numer. Anal. 9, pp. 603–637.Google Scholar
  22. R. W. Klopfenstein (1970), Private communication.Google Scholar
  23. R. W. Klopfenstein and C. B. Davis (1970),PECE algorithms for the solution of stiff equations, unpublished manuscript.Google Scholar
  24. F. T. Krogh (1973),On testing a subroutine for the numerical integration of ordinary differential equations, JACM 20, pp. 545–562.Google Scholar
  25. J. D. Lawson (1967),Generalized Runge-Kutta processes for stable systems with large Lipschitz constants, SIAM. J. Numer. Anal. 4, pp. 372–380.Google Scholar
  26. J. D. Lawson and B. L. Ehle (1972),Improved generalized Runge-Kutta, Proceedings of Canadian Computer Conference, Session 72, pp. 223201–223213.Google Scholar
  27. B. Lindberg (1971),On smoothing and extrapolation for the trapezoidal rule, BIT 11, pp. 29–52.Google Scholar
  28. B. Lindberg (1972),IMPEX—a program package for solution of systems of stiff differential equations, Report NA72.50, Dept. of Information Processing, Royal Inst. of Tech., Stockholm.Google Scholar
  29. B. Lindberg (1974),Optimal stepsize sequences and requirements for the local error for methods for stiff differential equations, Dept. of Computer Science Tech. Rep. No. 67, University of Toronto, Toronto.Google Scholar
  30. W. Liniger and R. A. Willoughby (1967),Efficient numerical integration of stiff systems of ordinary differential equations, Technical Report RC-1970, IBM Thomas J. Watson Research Center, Yorktown Heights, N.Y.Google Scholar
  31. H. H. Robertson (1966),The solution of a set of reaction rate equations, in Numerical Analysis, An Introduction (J. Walsh ed.), Academic Press, London, pp. 178–182.Google Scholar

Copyright information

© BIT Foundations 1975

Authors and Affiliations

  • W. H. Enright
    • 1
  • T. E. Hull
    • 1
  • B. Lindberg
    • 1
  1. 1.Department of Computer ScienceUniversity of TorontoTorontoCanada

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