Ezplizite Runge -K u t t a -Verfahren m-ten Grades werden hergeleitet, f u r die der maximal stabile Zeitschritt pro d u swertung der rechten Seite mit m proportioniert ist, wenn diese Methoden auf semi-diskretisierte parabolisclie Anfangsrandwertprobleme angewandt werden. Dns interne Stabilitatsverhalten dieser Methoden wird mit gleichartigen, cn der Lrterntur aorgeschlagenen Runge-Kutta-Formeln verglichen. Wir zeigen sowohl mittels Analyse uiie durch numeridche ExperL-nt~nte, daJ der sn-Wert der in dieser Arbeit vorgeschlagenen Schemata nicht von internen Instabilitaten beschra,tkt i~i r d .Explicit, m-stage R u n g e -K u t t a methods are derived for which the mnximal stable integrntion step per light hnntl .$id? waluation i s proportional to m when applied to semi-discrete parabolic initial-boundary d u e problenis. Thr inirrnnl stability behaviour of these methods is compared with that of similnr R u n g e -K u t t a methods proposed in the literatior. Both by analysis and by numerical experiments we show that the value of m in the schemes proposed in this popcr 2 P not restricted by internal instabilities. mar BpeMem 3a 06pa60~1iy npaBoii w c m nponopmoHaneH c m , ecm 3TH MeTonbI npnMeHmoTcn H a nony-~H C K~~T H~O B~H H~I X napa60nariec~~x aanav c HawinLi-Imw M Kpaenmm 3navemmm. CpanimuueToi RIIYTpeHHblll xapamep Y C T Q~H B O C T I I BTUX MeTonoB c OnHoponHbmm, npennonarambma B nmepaType ~K C I I~~L I M~I I T O B , w o m -m a v e~~e cxeM npennonaraeMbIx B 3~0 i i cTaTbe He opraHmmaeTca BHyTpeHHmw BblBOnRTCII RRHbIe MeTOHbI TElna P y Ilre-Hy T T a m -O f i CTelleHLl, KOTOPblX M a I E C L l~l a~b H O -y C T O € I a r n h r i i @OpMynaMH TIIlla PyHre-)CyTTa. ~OKa3blBaeTCH C I l O M O U b H ) aHaJlkl3a KBK II C IIQMOUibM BbI~ACJlllTe.JlbIiblXHeyCTO&lHBOCTIIMEl.

Abstract. We construct explicit Runge-Kutta (-Nystrom) methods for the integration of first (and second) order differential equations having an oscillatory solution. Special attention is paid to the phase errors (or dispersion) of the dominant components in the numerical oscillations when these methods are applied to a linear, homogeneous test model. RK(N) methods are constructed which are dispersive of orders up to 10, whereas the (algebraic) order of accuracy is only 2 or 3. Application of these methods to equations describing free and weakly forced oscillations and to semidiscretized hyperbolic equations reveals that the phase errors can significantly be reduced.

No abstract

Abstract. This paper examines diagonally implicit iteration methods for solving implicit Runge-Kutta methods with high stage order on parallel computers. These iteration methods are such that after a finite number of m iterations, the iterated Runge-Kutta method belongs to the class of diagonally implicit Runge-Kutta methods (DIRK methods) using mk implicit stages where k is the number of stages of the generating implicit Runge-Kutta method (corrector method). However, a large number of the stages of this DIRK method can be computed in parallel, so that the number of stages that have to be computed sequentially is only m. The iteration parameters of the method are tuned in such a way that fast convergence to the stability characteristics of the corrector method is achieved. By means of numerical experiments it is also shown that the solution produced by the resulting iteration method converges rapidly to the corrector solution so that both stability and accuracy characteristics are comparable with those of the corrector. This implies that the reduced accuracy often shown when integrating stiff problems by means of DIRK methods already available in the literature (which is caused by a low stage order) is not shown by the DIRK methods developed in this paper, provided that the corrector method has a sufficiently high stage order.

No abstract