Skip to main content
SpringerLink
Log in

A note on convergence concepts for stiff problems

Über Konvergenzkonzepte für steife Probleme

  • Published:
Computing Aims and scope Submit manuscript

Abstract

Most convergence concepts for discretizations of nonlinear stiff initial value problems are based on one-sided Lipschitz continuity. Therefore only those stiff problems that admit moderately sized one-sided Lipschitz constants are covered in a satisfactory way by the respective theory. In the present note we show that the assumption of moderately sized one-sided Lipschitz constants is violated for many stiff problems. We recall some convergence results that are not based on one-sided Lipschitz constants; the concept of singular perturbations is one of the key issues. Numerical experience with stiff problems that are not covered by available convergence results is reported.

Zusammenfassung

Die meisten Konvergenzkonzepte für Diskretisierungen nichtlinearer steifer Anfangswertprobleme basieren auf dem Begriff der einseitigen Lipschitz-Stetigkeit. Folglich sind durch diese theoretischen Konzepte nur steife Probleme mit moderater einseitiger Lipschitzkonstante abgedeckt. In der vorliegenden Arbeit zeigen wir, daß die Annahme moderater einseitiger Lipschitzkonstanten für viele steife Probleme verletzt ist. Wir weisen auf einige Konvergenzresultate hin, die nicht auf einseitigen Lipschitzkonstanten basieren; die Konzepte der singulären Störungstheorie sind hier von wesentlicher Relevanz. Wir berichten über einige numerische Erfahrungen mit steifen Problemen, die durch keine existierende Konvergenztheorie abgedeckt sind.

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. W. Auzinger, R. Frank, F. Macsek, Asymptotic error expansions for stiff equations: The implicit Euler scheme, Report Nr. 72/87, Institut für Angewandte und Numerische Mathematik, TU Wien, 1987. (To appear in SIAM J. Numer. Anal.)

  2. W. Auzinger, R. Frank, Asymptotic error expansions for stiff equations: An analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case, Numer. Math.56, 469–499 (1989).

    Google Scholar 

  3. W. Auzinger, R. Frank, Asymptotic error expansions for stiff equations: The implicit midpoint rule, Report Nr. 77/88, Institut für Angewandte und Numerische Mathematik, TU Wien, 1988. (Submitted.)

  4. W. Auzinger, R. Frank, G. Kirlinger, Asymptotic error expansions for stiff equations: Applications, Report Nr. 78/89, Institut für Angewandte und Numerische Mathematik, TU Wien, 1989. (To appear in Computing.)

  5. J. C. Butcher, A stability property of implicit Runge-Kutta methods, BIT15, 358–361 (1975).

    Google Scholar 

  6. G. Dahlquist, Error analysis for a class of methods for stiff nonlinear initial value problems, in: Lecture Notes in Mathematics 506, G. A. Watson (Ed.), Springer-Verlag, Berlin, 1976.

    Google Scholar 

  7. K. Dekker, J. G. Verwer, Stability of Runge-Kutta methods for stiff nonlinear differential equations, North-Holland Publ, Amsterdam, New York, Oxford 1984.

    Google Scholar 

  8. R. Frank, J. Schneid, C. W. Ueberhuber, Einseitige Lipschitzbedingungen für gewöhnliche Differentialgleichungen, Report Nr. 33/78, Institut für Numerische Mathematik, TU Wien, 1978.

  9. R. Frank, J. Schneid, C. W. Ueberhuber, The concept of B-convergence, SIAM J. Numer. Anal.18, 753–780 (1981).

    Google Scholar 

  10. R. Frank, J. Schneid, C. W. Ueberhuber, Stability properties of implicit Runge-Kutta methods, SIAM J. Numer, Anal.22 497–515 (1985).

    Google Scholar 

  11. R. Frank, J. Schneid, C. W. Ueberhuber, Order results for implicit Runge-Kutta methods, SIAM J. Numer. Anal.22, 515–534 (1985).

    Google Scholar 

  12. E. Hairer, Ch. Lubich, Extrapolation at stiff differential equations, Numer. Math.52, 377–400 (1988).

    Google Scholar 

  13. E. Hairer, Ch. Lubich, M. Roche, Error of Runge-Kutta methods for stiff problems studied via differential algebraic equations, Report, Dept. de Mathématiques, Université de Genève, 1987.

  14. E. Kamke, Differentialgleichungen reeller Funktionen, Chelsea Publishing Company, New York, 1947.

    Google Scholar 

  15. H.-O. Kreiss, Difference methods for stiff ordinary differential equations, SIAM J. Numer. Anal.15, 21–58 (1978).

    Google Scholar 

  16. M. van Veldhuizen, D-Stability, SIAM J. Numer. Anal.18, 45–64 (1981).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Angewandte und Numerische Mathematik, Technische Universität Wien, Wiedner Hauptstrasse 6-10, A-1040, Wien, Austria

    W. Auzinger, R. Frank & G. Kirlinger

Authors
  1. W. Auzinger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Frank
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. Kirlinger
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Dedicated to Professor Hans J. Stetter on the occasion of his 60th birthday.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Auzinger, W., Frank, R. & Kirlinger, G. A note on convergence concepts for stiff problems. Computing 44, 197–208 (1990). https://doi.org/10.1007/BF02262216

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02262216

AMS Subject Classifications

Key words

Search

Navigation