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BIT 16 1976), 237-240

O N T H E I M P L E M E N TAT I O N O F I M P L IC I T

R U N G E K U T T A M E T H O D S

J . C . B U T C H E R

A b s t r a c t

Th e m odi f ied lkTewton i t e ra t ions in the imp leme nta t ion of an s s tage im pl ic i tRu ng e -Ku t t a m e thod fo r an n d imen s io na l d i f fe r e n ti a l equa t i on s y s t em r e qu i r e

2s3n3/3 + O (n~) opera t ions for the L U fac tor i sa t ions and 2s2n~+ 0 n) opera t ions forthe back subs t i tu t ions . This paper descr ibes a method for t rans forming the l inearsys tem so as to redu ce these opera t ion counts .

I n t h e n u m e r i c a l s o l u t i o n o f a n n d i m e n s i o n a l s t i ff d i f f e r e n ti a l e q u a t i o ns y s t e m1) y ( x) = / ( y ( x ) ) ,

u s in g a n s s t ag e i m p l ic i t R u n g e - K u t t a m e t h o d , t h e s o l u t io n a tx m =xm_1 + h i s c o m p u t e d a s

2) Ym = Y m - t + h ~ = l b j f ( Y ~ )

w h e r e

3) Y i = Y m - t + h Z ; = l a i J ( Y j ) , i =1 , 2 , . . . . s .

To e v a l u a t e Y 1, Y ~ . . . , Y s s a t i s f y i n g t h e s y s t e m 3 ), i t is u s u a l t o u s e am od i fi ca t io n o f t h e N e w t o n R a p h s o n m e t h o d s o t h at a t t he e n d o f a

c u r r e n t i t e r a t i o n , Yi i s t o b e r e p l a c e d b yY i + w t w h e r e w l , w ~ , . . . , w sa r e g i v e n b y

4 ) w ~ - h ~ . ~ = l J w ~ - g ~ = o , ~ = 1 ,2 . . . . , s ,w i t h J , t h e n x n J a c o b i a n m a t r i x o f f , e v a l u a t e d a t a r e c e n t p o i n t o nt h e s o l u t io n t r a j e c t o r y a n d

5) Z i = - - Y, + y m _ ~ + h Z ; = l a , f ( Y i ) , i = 1 , 2 . . . . . s .

S i n ce a m a j o r p a r t o f t h e c o m p u t a t i o n t i m e i s e x p e n d e d i n t h e ev a l-u a t i o n o f J a n d t h e t r e a t m e n t o f t h e l in e a r s y s t e m 4 ), i t i s s t a n d a r dp r a c t i c e t o e v a l u a t e J a s s e l d o m a s p o s s ib l e a n d t o c a r r y o u t p r e l i m i n a r yw o r k o n th e l i n e a r s y s t e m 4 ) s o t h a t t h e a c t u a l i t e ra t i o n s c a n b e p e r -

Received March 19, 1976.

I T 1 6 - - 1 6

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2 3 8 J C B U T C H E R

f o r m e d e f f i c i e n t l y. t i s w i t h t h i s p r e l i m i n a r y w o r k t h a t t h i s p a p e r i sm a i n l y c o n c e rn e d .

L e t w, Z e Rn8 a n d t h e s x s m a t r i x A b e d e f i n e d b y

s s a s l a s~ • • a s s j

a n d l e t M = i ® I - h A ® Jb e t h e m a t r i x o f c o e f f ic i e n ts i n 4 ) w h e r e ii s t h e s × s u n i t m a t r i x a n d I t h e n x n u n i t m a t r i x . T h u s 4 ) c a n b ew r i t t e n a s

6) M w - Z = O .

T h r o u g h o u t t h i s p a p e r w e w i ll a s s u m e t h a t A i s n o n - si n g ul a r. T h i s a s-s u m p t i o n h o ld s f o r m o s t im p l ic it R u n g e - K u t t a m e t h o d s t h a t h a v eb e e n p r o p o s e d a s s u i t a b l e f o r s t i f f p r o b l e m s , a n d l e a d s t o s o m e s i m p l i f i -c a t i o n s i n t h i s p a p e r.

W e w i ll r e g a r d i t a s t h e s t a n d a r d p r a c t ic e t o c o m p u t e t h eL U f a c t o r -i s a ti o n o f M a s t h e p r e l im i n a r y t r e a t m e n t o f 4 ). I n t h i s c as e, t h e n u m b e ro f m u l t i p l i c a ti v e a n d a d d i t i v e c a l c u l at i o n s to p e r f o r m a r e e a c h C n 8 / 3 )+O n ~) fo r l a rge n ) , whereC = s a , t h e n u m b e r o f op e r at io n s i n t h e b a c k

s u b s t i t u t i o n f o r e a c h i t e r a t i o n i sD n ~ + O n )w h e r e D = s 2. W e w i ll c o n -s i d e r h o w t h e f a c t o r s C , D c a n b e l o w e r e d , e i t h e r t h r o u g h t h e c h o i ce o fp a r a m e t e r s o r e ls e t h r o u g h a s u i t a b l e o rg a n i s a t i o n o f t h e w o r k .

L e t P , Q b e n o n - s i n g u l a r s × s m a t r i c e s a n d l e t

= Q - 1 ® I ) w, ,Z = P ® I ) Z ,

.ffl = P ® I ) M Q ® I ) = P Q ) ® I - h . 4 ® Jw h e r e A =P A Q so th a t 6 ) i se q u i v a l e n t t o7) _ ~ - 2 = 0 .

S i n ce t h e c o m p u t a t i o n o f 2~ f r o m Z a n d o f w f r o m @ e a c h r e q u i r eO n )m u l t i p l i e a t i v e a n d a d d i t i v e c a l c u l a ti o n s , w e m i g h t j u s t a s w e l l u s e t h i st r a n s f o r m e d v e r s i o n o f t h e s e e q u a t i o n s i f t h i s l e a d s t o s o m e a d v a n t a g e .

W e n o w c o n s id e r h o w t o m a k e a ju d i c i o u s c h oi ce o f P a n d Q . L e t t h eJ o r d a n c a n o n i c al fo r m o f A - 1 b e

T - 1 A - 1 T = Dr1-10 0 . . . 0 ][/~1 2~. I 0

o o

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ON THE IMPLEMENTATION OF IMPLICIT RUNGE KUTTA METHODS 39

w her e each sub d ia go na l e l em en t 4 ( i = 1, 2 . . . . . s - 1 ) i s ze ro i f ~ i=~)~ i+ la n d i s e it h e r z e r o o r a n a r b i t r a r y n o n - z e r o n u m b e r i f 2~ = ~ t+ x. W h e r e i t

i s n o n - z e ro w e s u p p o s e t h a t ~ = ~t i -1 . L e tD=diag(A1,A2, . . . ,~s) . W es e l e c t P = D T-1A -1 , Q= Ts o t h a t

PQ [100. ]| ~ 0

L O Ow h e r e e a c h o f t h e s u b d i a g o n a l s e l, e 2 , . . , i s e i t h e r 0 o r 1 , a n d

PA Q = D .T h e m a t r i x ~]~ n o w c o n s i s t s o f d i a g o n a l b l o c k s o f t h e f o r mI - h ~ Jt o g e t h e r w i t h s u b d i a g o n a t b l o c k s o f 0 ( t h e z e r o m a t r i x ) o r I . T h e p r e l i m -i n a r y t r e a t m e n t o f (4) n o w c o n s is t s o f t h eL U f a c t o r i s a t i o n o f e a c ho f t h e distinct d i a g o n a l b lo c k s a n d t h e b a c k s u b s t i t u t i o n s b r e a k i n t o 8s e p a r a t e b l o c k s w i t h t h e s u b d i a g o n a l e l e m e n t s o fP Q c o n t r i b u t i n g o n l ya f u r t h e r O(n) o p e r a t i o n s .

To a s s e s s t h e f a c t o r s C a n d D , w e m u s t t a k e i n t o a c c o u n t t h e p o s s i bl ep r e s e n c e o f n o n - re a l e i g e n v al u e s o f A . L e t ~ d e n o t e t h e n u m b e r o f d is -t i n c t r e a l e i g e n v a l u e s a n d f l t h e t o t a l n u m b e r o f r e a l z e ro s o f t h e c h a r -a c t e ri s t ic p o l y n o m i a l o f A . A l s o l e t y d e n o t e t h e n u m b e r o f d i s t in c tc o n j u g a t e c o m p l e x e i g en v a l u e p a ir s a n d ~ t h e t o t a l n u m b e r o f c o n j u g a t ep a i r s o f z e r o s o f t h e c h a r a c t e r i s t i c p o l y n o m i a l o f A . T h u s~

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240 J .C . B U T C H E I ~

R E F E R E N C E S

1. J. C. Butcher, ImpHclt 2unge.Kutta P~ocessesMath. Comp. 18 1964), 50-64.2. S. P. l~orsett, Semi Explicit ~ungc-Kutta MethodsMathematics Department, Univer-

sity of Trondheim, Rep~ nt No. 6/74.

D EPAI T I ~[ EN T O F M AT H E ~ M AT I C S

T H E U NI VE I S I TY O F A U C K L A N D

A U C K L A N D , N E W Z E A L A N D