Skip to main content
SpringerLink
Log in

An algebraic characterization ofB-convergent Runge-Kutta methods

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary

In the analysis of discretization methods for stiff intial value problems, stability questions have received most part of the attention in the past.B-stability and the equivalent criterion algebraic stability are well known concepts for Runge-Kutta methods applied to dissipative problems. However, for the derivation ofB-convergence results — error bounds which are not affected by stiffness — it is not sufficient in many cases to requireB-stability alone. In this paper, necessary and sufficient conditions forB-convergence are determined.

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. Barker, G.P., Berman, A., Plemmons, R.J.: Positive diagonal solutions to the Lyapunov equations. Linear and Multilinear Algebra5, 249–256 (1978)

    Google Scholar 

  2. Berman, A., Ward, R.C.:A L P I classes of stable and semipositive matrices. Lin. Alg. Appl.21, 163–174 (1978)

    Google Scholar 

  3. Burrage, K., Butcher, J.C.: Stability criteria for implicit Runge-Kutta methods. SIAM J. Numer. Anal.16, 46–57 (1979)

    Google Scholar 

  4. Burrage, K., Hundsdorfer, W.H.: The order ofB-convergence of algebraically stable methods. BIT27, 62–71 (1987)

    Google Scholar 

  5. Crouzeix, M.: Sur laB-stabilité des méthodes de Runge-Kutta. Numer. Math.32, 75–82 (1979)

    Google Scholar 

  6. Crouzeix, M., Hundsdorfer, W.H., Spijker, M.N.: On the existence of solutions to the algebraic equations in implicit Runge-Kutta methods. BIT23, 84–91 (1983)

    Google Scholar 

  7. Dahlquist, G.: Error analysis for a class of methods for stiff nonlinear initial value problems. In: Watson, G.A. (ed.) Lecture Notes in Mathematics 506, Berlin Heidelberg New York: Springer 1976

    Google Scholar 

  8. Dahlquist, G., Jeltsch, R.: Generalized disks of contractivity for explicit and implicit Range-Kutta methods. Report TRITA-NA-7906, Dept. Comp. Sci., Roy. Inst. Techn., Stockholm (1979)

    Google Scholar 

  9. Dekker, K., Hairer, E.: A necessary condition forBSI-stability. BIT25, 285–288 (1985)

    Google Scholar 

  10. Dekker, K., Kraaijevanger, J.F.B.M., Schneid, J.: On the relation between algebraic stability andB-convergence for Runge-Kutta methods, Report 88-39, Faculty Tech. Math. Inf., Delft Univ. Techn. 1988 (to appear in Numer. Math.)

  11. Dekker, K., Verwer, J.G.: Stability of Runge-Kutta methods for stiff nonlinear differential equations, Amsterdam: North Holland 1984

    Google Scholar 

  12. Frank, R., Schneid, J., Ueberhuber, C.W.: The concept ofB-convergence. SIAM J. Numer. Anal.18, 753–780 (1981)

    Article  Google Scholar 

  13. Frank, R., Schneid, J., Ueberhuber, C.W.: Stability properties of implicit Runge-Kutta methods. SIAM J. Numer. Anal.22, 497–514 (1985)

    Google Scholar 

  14. Frank, R., Schneid, J., Ueberhuber, C.W.: Order results for implicit Runge-Kutta methods applied to stiff systems. SIAM J. Numer. Anal.22, 515–534 (1985)

    Google Scholar 

  15. Hundsdorfer, W.H.: The numerical solution of stiff initial value problems. CWI-Tract 12, Centre for Math. and Comput. Sc., Amsterdam 1985

    Google Scholar 

  16. Hundsdorfer, W.H., Schneid, J.: On the equivalence ofBS-stability andB-consistency. Report NM-R88, Centre for Math. and Comput. Sc., Amsterdam 1988 (to appear in BIT)

    Google Scholar 

  17. Kraaijevanger, J.F.B.M.:B-convergence of the implicit midpoint rule and the trapezoidal rule. BIT25, 652–666 (1985)

    Google Scholar 

  18. Liu, M.Z., Kraaijevanger, J.F.B.M.: On the solvability of the systems of equations arising in implicit Runge-Kutta methods. Report 87-02, Univ. of Leiden 1987 (to appear in BIT)

  19. Prothero, A., Robinson, A.: On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations. Math. Comput.28, 145–162 (1974)

    Google Scholar 

  20. Schneid, J.: Characterization ofB-convergent Runge-Kutta methods for strictly dissipative initial value problems. Computing42, 61–67 (1989)

    Google Scholar 

  21. Spijker, M.N.: The relevance of algebraic stability in implicit Range-Kutta methods. In: Strehmel, K. (ed.) Numerical treatment of differential equations, third seminar, Halle, pp. 158–164, Teubner Texte zur Mathematik 82, Leipzig 1986

Download references

Author information

Authors and Affiliations

  1. Centre for Mathematics and Computer Science, P.O. Box 4079, 1009, AB Amsterdam, The Netherlands

    Willem H. Hundsdorfer

  2. Institut für Angewandte und Numerische Mathematik, Technical University of Vienna, Wiedner Hauptstrasse 8-10, A-1040, Vienna, Austria

    Josef Schneid

Authors
  1. Willem H. Hundsdorfer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Josef Schneid
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This paper was written while J. Schneid was visiting the Centre for Mathematics and Computer Science with an Erwin-Schrödinger stipend from the Fonds zur Förderung der wissenschaftlichen Forschung

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hundsdorfer, W.H., Schneid, J. An algebraic characterization ofB-convergent Runge-Kutta methods. Numer. Math. 56, 695–705 (1989). https://doi.org/10.1007/BF01405197

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01405197

Subject Classifications

Search

Navigation