Numerical Treatment of Differential Equations: Proceedings by G. Alefeld (auth.), R. Bulirsch, R. D. Grigorieff, J.

By G. Alefeld (auth.), R. Bulirsch, R. D. Grigorieff, J. Schröder (eds.)

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Extra info for Numerical Treatment of Differential Equations: Proceedings of a Conference, Held at Oberwolfach, July 4–10, 1976

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1975). C o m p a r i n g Numerical M e t h o d s for BIT 15, 10-48. (1969). K l a s s i s c h e R u n g e - K u t t a - F o r m e l n mit Schrittweiten-Kontrolle, Fehlberg, B. fOnfter und siebenter Ordnung C o m p u t i n g 4, 93-106. (1970). Klassische R u n g e - K u t t a - F o r m e l n v i e r t e r und niedriger Ordnung mit S c h r i t t w e i t e n - K o n t r o l l e und ihre A n w e n d u n g auf W ~ r m e l e i t u n g s p r o b l e m e , Com- p u t i n g 6, 61-71. Henrici, P. (1962/1968).

C) A l l e 3-stufigen f~r e i n e V e r a l l g e m e i n e r u n g zu e i n e m V R K - 3 + RK-3 Verfahren lassen sich in der folgenden Verfahren einparametrigen len: 2 3 x~y, 7: C l = 0, c 2 = x, c3 = b3= (3c2-2) / (c2-c 3)/c3/6, b2= (3c3-2) / (c3-c 2)/c2/6, a21=c2 , QRK = 1 Fflr x = ~ e r g i b t a32=i/c2/b3/6, + + + (i , 1 , 3 ), sich beispielsweise: (4c2-3) /(3c2-2) /2 ; b I = l - ( b 2 + b 3) ; a31 = c 3 - a 3 2 ; P RK = in F r a g e k o m m e n d e n (0, 0,3). Schar darstel- 45 0 1 2 1 2 1 -i 2 i 2 --1 Die von Null verschiedenen + VRK-3 Verfahren lauten: 2 ~,~: Koeffizienten ~..

Rx = Q T v . i Der V e k t o r x wird also d u r c h ein einfaches R ~ c k w & r t s e i n s e t z e n berechnet. Die bei m e h r faehen E i g e n w e r t e n m ~ g l i c h e n K o m p l i k a t i o n e n bei der B e s t i m m u n g des E i g e n s y s t e m s yon T w e r d e n sich in der g r o s s e n Konditionszahl der R e c h t s d r e i e c k s m a t r i x R ausdr~cken. Da die h e r g e l e i t e t e n V R K - q V e r f a h r e n aber f~r b e l i e b i g e T gelten, kann in einem solchen Fall das R so g e s t 6 r t werden, dass der Prozess des R ~ c k w ~ r t s e i n s e t z e n s stabil verl~uft.

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