Skip to main content
SpringerLink
Log in

Contractivity of Runge-Kutta methods

  • Part II Numerical Mathematics
  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

In this paper we present necessary and sufficient conditions for Runge-Kutta methods to be contractive. We consider not only unconditional contractivity for arbitrary dissipative initial value problems, but also conditional contractivity for initial value problems where the right hand side function satisfies a circle condition. Our results are relevant for arbitrary norms, in particular for the maximum norm.

For contractive methods, we also focus on the question whether there exists a unique solution to the algebraic equations in each step. Further we show that contractive methods have a limited order of accuracy. Various optimal methods are presented, mainly of explicit type. We provide a numerical illustration to our theoretical results by applying the method of lines to a parabolic and a hyperbolic partial differential equation.

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. Berman, A. and Plemmons, R. J.:Nonnegative Matrices in the Mathematical Sciences. New York San Francisco London: Academic Press 1979.

    Google Scholar 

  2. Bolley, C. and Crouzeix, M.: Conservation de la positivité lors de la discrétisation des problèmes d'évolution paraboliques. RAIRO Anal. Numér. 12, 237–245 (1978).

    Google Scholar 

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

    Article  Google Scholar 

  4. Butcher, J. C.:Implicit Runge-Kutta processes. Math. Comp. 18, 50–64 (1964).

    Google Scholar 

  5. Butcher, J. C.:The Numerical Analysis of Ordinary Differential Equations. Chichester New York Brisbane: John Wiley 1987.

    Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

  8. Dahlquist, G. and Jeltsch, R.:Shifted Runge-Kutta methods and transplanted differential equations. In: Strehmel, K. (ed.)Numerical Treatment of Differential Equations, proceedings, Halle, 1987, pp. 47–56. Leipzig: Teubner 1988.

    Google Scholar 

  9. Dekker, K. and Verwer, J. G.:Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations. Amsterdam New York Oxford: North-Holland 1984.

    Google Scholar 

  10. Fehlberg, E.: Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Wärmeleitungsprobleme. Computing 6, 61–71 (1970).

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  13. Frank, R., Schneid, J. and Ueberhuber, C. W.:B-convergence: a survey. Appl. Numer. Math. 5, 51–61 (1989).

    Article  Google Scholar 

  14. Friedman, A.:Partial Differential Equations of Parabolic Type. Englewood Cliffs, N.J.: Prentice-Hall 1964.

    Google Scholar 

  15. Griend, J. A. van de and Kraaijevanger, J. F. B. M.:Absolute monotonicity of rational functions occurring in the numerical solution of initial value problems. Numer. Math. 49, 413–424 (1986).

    Article  Google Scholar 

  16. Hairer, E.:Highest possible order of algebraically stable diagonally implicit Runge-Kutta methods. BIT 20, 254–256 (1980).

    Google Scholar 

  17. Hairer, E., Nørsett, S. P. and Wanner, G.:Solving Ordinary Differential Equations I. Berlin Heidelberg New York: Springer 1987.

    Google Scholar 

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

    Article  Google Scholar 

  19. Hundsdorfer, W. H. and Spijker, M. N.:On the algebraic equations in implicit Runge-Kutta methods. SIAM J. Numer. Anal. 24, 583–594 (1987).

    Article  Google Scholar 

  20. John, F.:Partial Differential Equations. Appl. Math. Sciences, Vol. 1. New York Heidelberg Berlin: Springer 1971.

    Google Scholar 

  21. Kraaijevanger, J. F. B. M.:Absolute monotonicity of polynomials occurring in the numerical solution of initial value problems. Numer. Math. 48, 303–322 (1986).

    Article  Google Scholar 

  22. Kraaijevanger, J. F. B. M., Lenferink, H. W. J. and Spijker, M. N.:Stepsize restrictions for stability in the numerical solution of ordinary and partial differential equations. J. Comp. Appl. Math. 20 67–81 (1987).

    Article  Google Scholar 

  23. Kraaijevanger, J. F. B. M. and Spijker, M. N.:Algebraic stability and error propagation in Runge-Kutta methods. Appl. Numer. Math. 5, 71–87 (1989).

    Article  Google Scholar 

  24. Kraaijevanger, J. F. B. M.:Contractivity in the maximum norm for Runge-Kutta methods. Report TW-89-06, Dept. of Math. and Comp. Sci., Leiden Univ. 1989. To appear in the proceedings of the 1989 IMA conference on computational ordinary differential equations, London. Oxford Univ. Press (editors: Cash, J. R., Gladwell, I.).

  25. Lancaster, P., Tismenetsky, M.:The Theory of Matrices, 2nd ed.. Orlando San Diego New York London: Academic Press 1985.

    Google Scholar 

  26. Lenferink, H. W. J.:Contractivity preserving explicit linear multistep methods. Numer. Math. 55, 213–223 (1989).

    Article  Google Scholar 

  27. Lenferink, H. W. J.:Contractivity preserving implicit linear multistep methods. Math. Comp. 56, 177–199 (1991).

    Google Scholar 

  28. Martin, R. H., Jr.:Nonlinear Operators and Differential Equations in Banach Spaces. New York London Sydney: John Wiley 1976.

    Google Scholar 

  29. Nevanlinna, O. and Liniger, W.:Contractive methods for stiff differential equations. BIT 18, 457–474 (1978); BIT 19, 53–72 (1979).

    Google Scholar 

  30. Sand, J.:Circle contractive linear multistep methods. BIT 26, 114–122 (1986).

    Google Scholar 

  31. Sand, J.:Choices in contractivity theory. Appl. Numer. Math. 5, 105–115 (1989).

    Article  Google Scholar 

  32. Schönbeck, S. O.:On the extension of Lipschitz maps. Arkiv för Matematik 7, 201–209 (1967).

    Google Scholar 

  33. Spijker, M. N.:Contractivity of Runge-Kutta methods. In: Dahlquist, G., Jeltsch, R. (editors)Numerical Methods for Solving Stiff Initial Value Problems, proceedings, Oberwolfach, 28.6–4.7.1981. Institut für Geometrie und Praktische Mathematik der RWTH Aachen, Bericht Nr. 9, 1981.

  34. Spijker, M. N.:Contractivity in the numerical solution of initial value problems. Numer. Math. 42, 271–290 (1983).

    Article  Google Scholar 

  35. Spijker, M. N.:Numerical contractivity in the solution of initial value problems. In: Strehmel, K. (ed.)Numerische Behandlung von Differentialgleichungen, proceedings, Halle, 1983, pp. 118–124. Martin-Luther-Universität, Halle-Wittenberg 1984.

    Google Scholar 

  36. Spijker, M. N.:On the relation between stability and contractivity. BIT 24, 656–666 (1984).

    Google Scholar 

  37. Spijker, M. N.:Stepsize restrictions for stability of one-step methods in the numerical solution of initial value problems. Math. Comp. 45, 377–392 (1985).

    Google Scholar 

  38. Stetter, H. J.:Analysis of Discretization Methods for Ordinary Differential Equations. Berlin Heidelberg New York: Springer 1973.

    Google Scholar 

  39. Vanselow, R.:Nonlinear stability behaviour of linear multistep methods. BIT 23, 388–396 (1983).

    Article  Google Scholar 

  40. Wakker, P. P.:Extending monotone and non-expansive mappings by optimization. Cahiers du C.E.R.O. 27, 141–149 (1985).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Leiden, P.O. Box 9512, 2300, RA Leiden, The Netherlands

    J. F. B. M. Kraaijevanger

Authors
  1. J. F. B. M. Kraaijevanger
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research supported by the Netherlands Organization for Scientific Research (N.W.O.) and the Royal Netherlands Academy of Arts and Sciences (K.N.A.W.)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kraaijevanger, J.F.B.M. Contractivity of Runge-Kutta methods. BIT 31, 482–528 (1991). https://doi.org/10.1007/BF01933264

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01933264

Subject Classifications

Search

Navigation