On the Internal Stability of Explicit, m‐Stage Runge‐Kutta Methods for Large m‐Values

Abstract

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.

Explicit Runge-Kutta methods for parabolic partial differential equations

“…To overcome the internal instability problem, Van der Houwen and Sommeijer [ 13] developed a class of RK methods which is based on the three-term Chebyshev recursion (3.4). They derived first and second order methods for which all coefficients are given as analytical expressions and which can be used for any (practical) value of s. These methods are called Runge-Kutta-Chebyshev (RKC) methods.…”

“…This is also very good, because by increasing s with only about I 0% the same step size as would be allowed by the optimal polynomial will yield stability, due to the quadratic behaviour. Following [13,28,39], in the remainder we will proceed with the Bakker-Chebyshev polynomial for the second order methods, since this polynomial has been extensively tested and also seems to perform even somewhat better due to smaller error constants [18]. For an even degree, /J(s) equals ~ (8 2 -1) exactly, while for an odd degree /3(s) is slightly larger.…”

Explicit stabilized integration of stiff determinisitic or stochastic problems

“…Various strategies have been proposed to approximate such polynomials. We mention DUMKA methods of order two [11] and the Runge-Kutta-Chebyshev methods [9] based on a linear combination of Chebyshev polynomials. The stability domains of the DUMKA methods include the optimal interval ( [−0, 82 · s 2 , 0]) along the negative real axis, while the stability domains of RKC methods cover only 80% of these interval.…”

“…Indeed, unlike traditional Runge-Kutta (RK) methods, Chebyshev methods can have a large number of internal stages (e.g., 100, 200). An efficient implementation of such methods (that controls also internal stability) for the ordinary differential equations dX dt = F(X), X(0) = X 0 , F : R d → R d (an autonomous form is considered here for simplicity), first given in [9], reads…”

Runge–Kutta integrators yield optimal regularization schemes

“…The approach of van der Houwen and Sommeijer [15] takes advantage of the three-term recursion for the Chebyshev polynomials and generates the Runge-Kutta-Chebyshev (RKC) family of order p = 1 (see also [3,chapter IV.2] or [9, chapter V]).…”