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A study of B-convergence of Runge-Kutta methods

Eine Untersuchung über B-Konvergenz von Runge-Kutta Verfahren


This paper deals with the convergence analysis of implicit Runge-Kutta methods as applied to stiff, semilinear systems of the form\(\dot U\) (t)=QU(t)+g(t, U(t)). A criterion is developed which determines whether the order of optimalB-convergence is at least equal to the stage order or one order higher. This criterion is studied for a number of interesting classes of methods.


Dieser Aufsatz befaßt sich mit der Analyse der Konvergenz von impliziten Runge-Kutta Verfahren für steife, semi-lineare Systeme der Form\(\dot U\) (t)=QU(t)+g(t, U(t)). Ein Kriterium wird entwickelt, welches entscheidet, ob die Ordnung der optimalenB-Konvergenz mindestens gleich der Stufenordnung oder um eine Ordnung höher ist. Dieses Kriterium wird untersucht für eine Zahl von interessanten Klassen von Verfahren.

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  1. [1]

    Burrage, K.: Stability and efficiency properties of implicit Runge-Kutta methods. Ph. D. Thesis, Dept. of Math., Univ. of Auckland, 1978.

  2. [2]

    Burrage, K.: A special family of Runge-Kutta methods for solving stiff differential equations. BIT18, 22–41 (1978).

  3. [3]

    Butcher, J. C.: OnA-stable implicit Runge-Kutta methods. BIT17, 375–378 (1977).

  4. [4]

    Crouzeix, M., Raviart, P. A.: Méthodes de Runge-Kutta. Unpublished lecture notes. Université de Rennes, 1980.

  5. [5]

    Dekker, K., Kraaijevanger, J. F. B. M., Spijker, M. N.: The order ofB-convergence of the Gaussian Runge-Kutta method. Computing (this issue).

  6. [6]

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

  7. [7]

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

  8. [8]

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

  9. [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).

  10. [10]

    Hairer, E., Bader, G., Lubich, Ch.: On the stability of semi-implicit methods for ordinary differential equations. BIT22, 211–232 (1982).

  11. [11]

    Hundsdorfer, W. H.: The numerical solution of nonlinear stiffinitial value problems — an analysis of one-step methods. CWI Tract 12, Amsterdam 1985.

  12. [12]

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

  13. [13]

    Kraaijevanger, J. F. B. M.:B-convergence of the implicit midpoint rule and the trapezoidal rule. BIT (to appear).

  14. [14]

    Nørsett, S. P.: Semi-explicit Runge-Kutta methods. Report Math. and Comp. No. 6/74, Dept. of Math., Univ. of Trondheim, 1974.

  15. [15]

    Nørsett, S. P.:C-polynomials for rational approximations to the exponential function. Numer. Math.25, 39–56 (1975).

  16. [16]

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

  17. [17]

    Stetter, H. J.: ZurB-Konvergenz der impliziten Trapez- und Mittelpunktregel, unpublished note.

  18. [18]

    Verwer, J. G.: Convergence and order reduction of diagonally implicit Runge-Kutta schemes in the method of lines. Proc. Dundee 1985, D. F. Griffiths (ed.), Pitman Publ. Co. (to appear).

  19. [19]

    Wanner, G., Hairer, E., Nørsett, S. P.: Order stars and stability theorems. BIT18, 475–489 (1978).

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Burrage, K., Hundsdorfer, W.H. & Verwer, J.G. A study of B-convergence of Runge-Kutta methods. Computing 36, 17–34 (1986).

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AMS Subject Classification

  • 65L05

C R number

  • 5.17

Key words

  • Numerical analysis
  • implicit Runge-Kutta methods
  • stiff problems
  • B-convergence