Skip to main content
SpringerLink
Log in

Monotonicity and boundedness in implicit Runge-Kutta methods

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary

This paper concerns the analysis of implicit Runge-Kutta methods for approximating the solutions to stiff initial value problems. The analysis includes the case of (nonlinear) systems of differential equations that are essentially more general than the classical test equationU′=λU (with λ a complex constant). The properties of monotonicity and boundedness of a method refer to specific moderate rates of growth of the approximations during the numerical calculations. This paper provides necessary conditions for these properties by using the important concept of algebraic stability (introduced by Burrage, Butcher and by Crouzeix). These properties will also be related to the concept of contractivity (B-stability) and to a weakened version of contractivity.

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

    Google Scholar 

  2. Burrage, K., Butcher, J.C.: Nonlinear stability of a general class of differential equation methods. BIT20, 185–203 (1980)

    Google Scholar 

  3. Butcher, J.C.: Stability properties for a general class of methods for ordinary differential equations. SIAM J. Numer. Anal.18, 37–44 (1981)

    Google Scholar 

  4. Butcher, J.C.: A short proof concerningB-stability. BIT22, 528–529 (1982)

    Google Scholar 

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

    Google Scholar 

  6. Dahlquist, G.: Error analysis for a class of methods for stiff non-linear initial value problems. Lecture Notes in Mathematics Nr. 506, pp. 60–72. Berlin, Heidelberg, New York: Springer 1976

    Google Scholar 

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

    Google Scholar 

  8. Dekker, K., Verwer, J.G.: Stability of Runge-Kutta methods for stiff non-linear differential equations. Amsterdam, New York, Oxford: North Holland 1984

    Google Scholar 

  9. 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 

  10. Hundsdorfer, W.H., Spijker, M.N.: A note onB-stability of Runge-Kutta methods. Numer. Math.36, 319–331 (1981)

    Google Scholar 

  11. Hundsdorfer, W.H., Spijker, M.N.: On the algebraic equations in implicit Runge-Kutta methods. (To appear in: SIAM J. Numer. Anal.)

  12. Kraaijevanger, J.F.B.M., Spijker, M.N.: Algebraic stability and error propagation in Runge-Kutta methods. (In preparation)

  13. Spijker, M.N.: Stability in the numerical solution of stiff initial value problems. Nieuw Archief voor Wiskunde3 XXX, 264–276 (1982)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Subfaculteit Wiskunde en Informatica, Rijksuniversiteit, Wassenaarseweg 80, Leiden, Netherlands

    M. N. Spijker

Authors
  1. M. N. Spijker
    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

Spijker, M.N. Monotonicity and boundedness in implicit Runge-Kutta methods. Numer. Math. 50, 97–109 (1986). https://doi.org/10.1007/BF01389670

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01389670

Subject Classifications

Search

Navigation