Numerische Mathematik

, Volume 57, Issue 1, pp 249–262 | Cite as

On the relation between algebraic stability andB-convergence for Runge-Kutta methods

  • K. Dekker
  • J. F. B. M. Kraaijevanger
  • J. Schneid


This paper is concerned with the numerical solution of stiff initial value problems for systems of ordinary differential equations using Runge-Kutta methods. For these and other methods Frank, Schneid and Ueberhuber [7] introduced the important concept ofB-convergence, i.e. convergence with error bounds only depending on the stepsizes, the smoothness of the exact solution and the so-called one-sided Lipschitz constant β. Spijker [19] proved for the case β<0 thatB-convergence follows from algebraic stability, the well-known criterion for contractivity (cf. [1, 2]). We show that the order ofB-convergence in this case is generally equal to the stage-order, improving by one half the order obtained in [19]. Further it is proved that algebraic stability is not only sufficient but also necessary forB-convergence.

Subject Classifications

AMS(MOS): 65L05, 65L20 CR: G1.7 


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Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • K. Dekker
    • 1
  • J. F. B. M. Kraaijevanger
    • 2
  • J. Schneid
    • 3
  1. 1.Faculty of Technical Mathematics and InformaticsDelft University of TechnologyDelftThe Netherlands
  2. 2.Deparment of Mathematics and Computer ScienceUniversity of LeidenLeidenThe Netherlands
  3. 3.Institut für Angewandte und Numerische MathematikTechnische Universität WienWienAustria

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