BIT Numerical Mathematics

, Volume 28, Issue 3, pp 678–700 | Cite as

Error of Runge-Kutta methods for stiff problems studied via differential algebraic equations

  • E. Hairer
  • Ch. Lubich
  • M. Roche
Part III Numerical Mathematics

Abstract

Runge-Kutta methods are studied when applied to stiff differential equations containing a small stiffness parameter ε. The coefficients in the expansion of the global error in powers of ε are the global errors of the Runge-Kutta method applied to a differential algebraic system. A study of these errors and of the remainder of the expansion yields sharp error bounds for the stiff problem. Numerical experiments confirm the results.

AMS Subject Classifications

65L05 

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References

  1. 1.
    R. C. Aiken,Stiff Computation, Oxford University Press, New York-Oxford, 1985.Google Scholar
  2. 2.
    R. Alexander,Diagonally implicit Runge-Kutta methods for stiff O.D.E.'s, SIAM J. Numer. Anal., Vol. 14 (1977), 1006–1021.Google Scholar
  3. 3.
    K. E. Brenan and L. R. Petzold,The numerical solution of higher index differential/algebraic equations by implicit Runge-Kutta methods, Preprint UCRL-95905, Lawrence Livermore National Laboratory, 1986.Google Scholar
  4. 4.
    K. Burrage and W. H. Hundsdorfer,The order of B-convergence of algebraically stable Runge-Kutta methods, BIT, Vol. 27 (1987), 62–71.Google Scholar
  5. 5.
    K. Burrage, W. H. Hundsdorfer and J. G. Verwer,A study of B-convergence of Runge-Kutta methods, Computing, Vol. 36 (1986), 17–34.Google Scholar
  6. 6.
    M. Crouzeix and P. A. Raviart,Approximation des problèmes d'évolution, Unpublished Lecture Notes, Université de Rennes, 1980.Google Scholar
  7. 7.
    K. Dekker and J. G. Verwer,Stability of Runge-Kutta Methods for Stiff Non-linear Differential Equations, North-Holland, Amsterdam - New York - Oxford, 1984.Google Scholar
  8. 8.
    P. Deuflhard, E. Hairer and J. Zugck,One-step and extrapolation methods for differential-algebraic systems, Numer. Math., Vol. 51 (1987), 501–516.Google Scholar
  9. 9.
    R. Frank, J. Schneid and C. W. Ueberhuber,Order results for implicit Runge-Kutta methods applied to stiff systems, SIAM J. Numer. Anal., Vol. 22 (1985), 515–534.Google Scholar
  10. 10.
    C. W. Gear,Differential-algebraic equation index transformations, SIAM J. Sci. Stat. Comp., Vol. 9 (1988), 39–47.Google Scholar
  11. 11.
    E. Griepentrog and R. März,Differential-algebraic Equations and their Numerical Treatment, Teubner-Texte zur Mathematik, Band 88, 1986.Google Scholar
  12. 12.
    E. Hairer,A note on D-stability, BIT, Vol. 24 (1984), 383–386.Google Scholar
  13. 13.
    E. Hairer, G. Badner and Ch. Lubich,On the stability of semi-implicit methods for ordinary differential equations, BIT, Vol. 22 (1982), 211–232.Google Scholar
  14. 14.
    E. Hairer and Ch. Lubich,Extrapolation at stiff differential equations, Numer. Math., Vol. 52 (1988), 377–400.Google Scholar
  15. 15.
    E. Hairer and Ch. Lubich,On extrapolation methods for stiff and differential algebraic equations. In:Numerical Treatment of Differential Equations, ed. by K. Strehmel, Teubner-Texte zur Mathematik, Band 104, 1988.Google Scholar
  16. 16.
    E. Hairer, Ch. Lubich and M. Roche. In preparation.Google Scholar
  17. 17.
    F. Hoppensteadt,Properties of solutions of ordinary differential equations with a small parameter, Commun. Pure Appl. Math., Vol. 24 (1971), 807–840.Google Scholar
  18. 18.
    H. O. Kreiss,Problems with different time scales. In:Recent Advances in Numerical Analysis (C. de Boor, G. H. Golub, eds.), Academic Press, 1978.Google Scholar
  19. 19.
    W. L. Miranker,Numerical Methods for Stiff Equations, D. Reidel Publishing Company, Dordrecht, Holland, 1981.Google Scholar
  20. 20.
    J. von Neumann, Eine Spektraltheorie für Operatoren eines unitären Raumes, Math. Nachr., Vol. 4 (1951), 258–281.Google Scholar
  21. 21.
    O. Nevanlinna,Matrix valued versions of a result of von Neumann with an application to time discretization, J. Comput. Appl. Math., Vol. 12–13 (1985), 475–489.Google Scholar
  22. 22.
    S. P. Nørsett and P. Thomsen,Local error control in SDIRK-methods, BIT, Vol. 26 (1986), 100–113.Google Scholar
  23. 23.
    R. E. O'Malley,Introduction to Singular Perturbations, Academic Press, New York and London, 1974.Google Scholar
  24. 24.
    J. M. Ortega and W. C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York - San Francisco - London, 1970.Google Scholar
  25. 25.
    L. R. Petzold,Order results for implicit Runge-Kutta methods applied to differential/algebraic systems, SIAM J. Numer. Anal., Vol. 23 (1986), 837–852.Google Scholar
  26. 26.
    A. Prothero and A. Robinson,On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations, Math. Comput., Vol. 28 (1974), 145–162.Google Scholar
  27. 27.
    M. Roche,Runge-Kutta methods for differential algebraic equations. To appear in SIAM J. Numer. Anal.Google Scholar
  28. 28.
    A. N. Tikhonov, A. B. Vasil'eva and A. G. Sveshnikov,Differential Equations, Springer Verlag, Berlin - Heidelberg, 1985.Google Scholar
  29. 29.
    M. van Veldhuizen,D-stability, SIAM J. Numer. Anal., Vol. 18 (1981), 45–64.Google Scholar

Copyright information

© BIT Foundations 1988

Authors and Affiliations

  • E. Hairer
    • 1
  • Ch. Lubich
    • 1
  • M. Roche
    • 1
  1. 1.Dept. de mathématiquesUniversité de GenèveGenève 24Switzerland

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