Step-control stability of diagonally implicit multistage integration methods (Q2708034)

The author studies the behaviour of several step size selection schemes when some explicit multistage multivalue methods are applied to the numerical solution of stiff equations. As it is well known [see e.g. \textit{E. Hairer} and \textit{G. Wanner}, Solving ordinary differential equations. II, Second Revised Edition, Springer Verlag (1996; Zbl 0859.65067), p. 25-26] when explicit Runge-Kutta methods are used for the solution of stiff problems the size of the step is usually restricted by the stability of the method, and in many problems the step size determined by the standard selection scheme shows a ragged behaviour. This phenomenon has been explained by considering the performance of the method for the test equation \( y' = \lambda y \) for complex \( \lambda \) as a discrete dynamical system and studying the stability of their fixed points. Moreover a new step size selection technique, the so called PI step size control, introduced by \textit{K. Gustafsson, M. Lundh} and \textit{G. Söderlind} [BIT 28, 270-287 (1988; Zbl 0645.65039)] has proved to be very efficient to smooth the step sizes for stiff problems reducing considerably the number of failed steps. NEWLINENEWLINENEWLINEIn the paper under consideration the author applies these ideas to some multistage multivalue methods. More specifically he considers methods with \( r=s=q=p \), where \(r\) is the number of external stages, \(s\) the number of internal stages, \(q\) the stage order and \(p\) the order. First of all for \( p=2\) and a particular choice of the free parameters he studies the SC-stability of the resulting method and justifies the behaviour of the standard step size selection scheme for a stiff problem. Next he considers the behaviour of the PI step size technique for the same method showing that, as in standard Runge-Kutta methods, an improvement over the standard technique is detected.