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

Summary 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 . Spijker 19] proved for the case <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.This study was completed while this author was visiting the Oxford University Computing Laboratory with a stipend from the Netherlands Organization for Scientific Research (N.W.O.)