Generalized Runge Kutta Method

7/25/2019 Generalized Runge Kutta Method

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Num er. M ath. 33, 5 5- 6 8 (1979)

Numerische

athemaUk

9 by Springer-Verlag 1979

General ized Runge Kutta Methods

of Order Four with Stepsize Control

for Stiff Ordinary Differential Equations

1 P e t e r K a p s , 2 P e t e r R e n t r o p

x Institut fiir Math em atik I, Universit~it Inns bruck , Tech nikerstr. 13, A -6020 Innsb ruck , Austria

z Institut fiir Ma them atik der Techn ischen Un iversit~t Miinchen,

Arcisstr. 21, D-800 0 Miinchen 2, Germ any (Fed . Rep.)

S u m m a r y . G e n e r a l i z e d A (c~ )-stab le R u n g e - K u t t a m e t h o d s o f o r d e r f o u r w i t h

s t ep s i ze c o n t r o l a r e s t u d ie d . T h e e q u a t i o n s o f c o n d i t i o n f o r t h is c l a ss o f s e m i -

i m p l i c i t m e t h o d s a r e s o l v e d t a k i n g t h e t r u n c a t i o n e r r o r i n t o c o n s i d e r a t i o n .

F o r a p p l i c a t io n a n A - s t ab l e a n d a n A ( 8 9 .3 ~ m e t h o d w i t h s m a ll

t r u n c a t i o n e r r o r a r e p r o p o s e d a n d t e s t re s u l ts f o r 25 s ti ff i n i ti a l v a l u e

p r o b l e m s f o r d if f er e n t to l e r a n c e s a r e d i s c u ss e d .

Subjec t c lass i f ica t ions .

A M S ( M O S ) : 6 5 L 0 5 ; C R : 5 . 1 7 .

1. Introduct ion

I n i t i a l v a l u e p r o b l e m s w i t h s t r o n g l y d e c r e a s i n g a n d i n c r e a s i n g s o l u t i o n c o m p o -

n e n t s a r e c a l l e d s ti ff p r o b l e m s . T h e y m a i n l y a p p e a r i n c h e m i c a l k i n e ti c s, e l e c tr i c

c i r c u i t s a n d c o n t r o l t h e o r y . U s u a l i n t e g r a t i o n r o u t i n e s a s c o m p a r e d i n D i e k h o f f

e t a l. [ 6 ] , E n r i g h t e t a l . [ 7 ] f ai l, b e c a u s e o f t h e d i f f e r e n t g r o w t h o f t h e s o l u t i o n

c o m p o n e n t s . N e w s t ab i li ty r e q u i r e m e n t s l ik e A - s ta b i li ty h a v e b e e n i n t r o d u c e d t o

o v e r c o m e th e s e p r o b l e m s , s ee D a h l q u i s t [-5 ], G r i g o r i e f f [ 1 0 ] .

T h e p r e s e n t r e p o r t is c o n c e r n e d w i th g e n e r a li z e d R u n g e - K u t t a m e t h o d s o f

o r d e r f o u r w i t h t h r e e f u n c t i o n e v a l u a t i o n s p e r s t e p . A s t e p s i z e c o n t r o l i s

i m p l e m e n t e d b y e m b e d d i n g a t h ir d o r d e r m e t h o d . O n e e v a l u a ti o n o f t h e Ja c o b i

m a t r ix a n d t h e s o l u t i o n o f a l i n e a r e q u a t i o n s y s t e m o f o r d e r n is n e c e s s a r y p e r

s te p . A n A - s t a b l e a n d a n A ( 8 9 . 3 ~ a l g o r i t h m a r e t e s te d b y s o l v i n g 2 5 s ti ff

i n i t i a l v a l u e p r o b l e m s f r o m B e d e t , E n r i g h t , H u l l [ 2 ] a n d E n r i g h t , H u l l , L i n d b e r g

[ 8 ]

2 . G e n e r a l i z e d R u n g e K u t t a M e t h o d s

R O W - M e t h o d s .

T h e a u t o n o m o u s i ni t ia l v al u e p r o b l e m :

y ( x ) = f ( y ( x ) ) ,

Y( Xo) Yo

(2.1)

0 0 2 9 - 5 9 9 X / 7 9 / 0 0 3 3 / 0 0 5 5 / $ 0 2 . 8 0


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56 P. K aps and P. Re ntrop

i s c o n s i d e r e d i n a n - d i m e n s i o n a l r e a l o r c o m p l e x s p a c e . A n u m e r i c a l s o l u t i o n o f

t h e f o l l o w i n g t y p e i s s t u d i e d :

yh(Xo + h)

=Y 0 + ~

ci ki

(2.2)

i=

( I - ; ~ h f ' ( Y o ) ) k i = h Y o + ~ c ~jk j + h f ' ( y o ) ~ 7 ij kj i = l , . . . , s .

j = l j = l

Th e co e f f i c i en t s 7 , ~ ij, ~ ij a r e r ea l n u m b e r s , h d e n o te s t h e s t ep s i ze ,

f ' (Y o )

t h e

J a c o b i - , I t h e n x n i d e n t i t y m a t r i x a n d s t h e n u m b e r o f st a ge s . T h e v e c t o r s k~ (i

= 1 , . . . , s ) a r e c o m p u t e d b y s o l v i n g a s y s te m o f l i n e a r e q u a t i o n s o f o r d e r n fo r s

d i f f e r e n t r i g h t h a n d s i d e s .

M e t h o d ( 2 . 2 ) i s c a l l e d R o s e n b r o c k - W a n n e r m e t h o d , s h o r t R O W - m e t h o d .

T h e f i r s t w h o s e e m e d t o h a v e s t u d i e d s i m i l a r f o r m u l a s w a s R o s e n b r o c k [ 1 8 ] .

W a n n e r [ 2 0 ] i n t r o d u c e d t h e c o ef fi ci en ts ~ u a n d p r o p o s e d t h e t h e o r y o f B u t c h e r

s e ri e s [1 1 ] f o r d e r i v a t i o n o f t h e e q u a t i o n s o f c o n d i t i o n . I n [ 1 3 ] a n d W o l f b r a n d t

[ 2 1 ] t h e s e m e t h o d s a r e c a l l e d m o d i f i e d R o s e n b r o c k m e t h o d s , i n N o r s e t t ,

W o l f b r a n d t [ 1 7 ] R O W - m e t h o d s .

F o r 7 = 7 u = 0 t he R O W - m e t h o d s r ed u c e t o u su al R u n g e - K u t t a m et ho d s.

T h e r e f o r e R O W - m e t h o d s c a n b e c o n si d e re d a s g e n er a li ze d R u n g e - K u t t a m e t h-

o d s .

Stab i l i ty P roper ties o f RO W -M ethods . T o s t u d y t h e s ta b il it y p r o p e r t i e s o f R O W -

m e t h o d s , t h e s c a l a r t e s t d i f f e r e n t i a l e q u a t i o n i s u s e d :

y ' = 2 y , y ( X o )= Y o ; 2 ~ l U , y o O l 2 , y : I R - ~ C

S i nc e f ' ( y ) = 2 , it h o ld s : k~ =R z( z ) yo , z=2h , w h e r e Ri(z ) a r e r a t i o n a l f u n c t i o n s

w i th d e n o m i n a t o r ( 1 - 7 z ) ~ a n d d e g r e e o f n u m e r a t o r < i . T h u s t he n u m e r i ca l

s o l u t i o n

Yh

is :

Yn = R (z) Yo

(2.3)

~,, - , , P (z )

w i t h t h e s t a b i l i t y f u n c t i o n : R ( z ) = 1 +

~= 1 Q ~ A z J - ~ z )

F o r a r a t i o n a l a p p r o x i -

m a t i o n (2 .3 ) o f o r d e r p h o l d s :

P r o p o s i t i o n ( 2 .4 ) . T h e s t a b i l i t y f u n c t i o n o f a R O W - m e t h o d w i t h o r d e r p >__s is

g i v e n b y

R ( z ) = (1 - y z)- - - ~ k= o

w h e r e

( n + ~ ] x '

/ J ~ , ) ( x ) = i ~ 0 ( - 1 ) i \ n - i i ~ -


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G e n e r a l i z e d R u n g e K u t t a M e t h o d s 5 7

s t a n ds f o r t h e g e n e r a l iz e d L a g u e r r e p o l y n o m i a l s , s e e A b r a m o w i t z , S t e g u n [ 1 ] .

R z) is a r a t i o n a l a p p r o x i m a t i o n t o e ~ o f o r d e r > s .

Proof

A p p l y i n g T h e o r e m 4 o r P r o p o s i t i o n 6 o f N o r s e t t , W a n n e r [ 1 6 ] o n e c a n

s h o w t h a t P z) i s d e t e r m i n e d u n i q u e l y b y Q z ) . I n o u r c a s e t h e s y m m e t r i c

p o l y n o m i a l s S , a r e :

S , = ( ~ ) 7 ' . I t f o l l o w s :

e z)= s

k=o i=o k i

B y m e a n s o f t h e s t a b i li t y f u n c t i o n o n e c a n c h a r a c t e r i z e s o m e s t a b i li t y p r o p e r t i e s

v e r y c o n v e n i e n t l y .

On e h as s t ab i l i t y a t i n f in i ty , i f f :

l i m I R ( z ) ] = L s 1 ) < 1 , w h e r e L s : = / 3 ~ (2 .5 )

Z ~ O 9

\ y ]

F or y > 0 a m e th o d (2.2 ) is A-s tab le , i f f:

I R i y ) l< l fo r ye lR (2.6 )

o r e q u i v a l e n t l y , i ff t h e E - p o l y n o m i a l ( s e e N o r s e t t [ 1 5 ] ) s a t is f ie s :

E y )=lQ i y )12-1P i y ) lZ>O

Vy e lR. ( 2 .7 )

Embedded ROW -Methods . E r r o r e s t i m a t i o n a n d s t e p s i z e c o n t r o l i s p e r f o r m e d

u sin g t w o e m b e d d e d m e t h o ds . A R O W - m e t h o d o f o r d e r 4 :

Yh Xo + h) = Yo + ~ ci ki (2.8)

i = 1

a nd a R O W - m e t h o d o f o r d e r 3:

Y h X o + h ) = y o + ~ ~ i k i g < s)

i = 1

a r e c o m b i n e d , w h e r e t h e c o e f f i c ie n t s 7, Y , j, cq~ ( i = l , . . . , s , j = l , . . . , i - 1 ) a n d

t h e r e f o r e t h e k , a r e t h e s a m e f o r b o t h f o r m u l as . T h e r e s u l t o f t h e f o u r t h o r d e r

m e t h o d i s t a k e n a s i n i ti a l g u e s s f o r t h e n e x t s te p . T h e d i f f e r e n t o r d e r s o f t h e t w o

f o r m u l a s l e a d t o a n e s t i m a t i o n o f t h e lo c a l t r u n c a t i o n e r r o r E S T o f t h e t h i r d

o r d e r m e t h o d , i n a n a l o g y t o [ 6 ] a n d [ 7 ] . U s i n g t h i s i n f o r m a t i o n t h e f o l l o w i n g

s t e p s i z e c o n t r o l f o r a g i v e n t o l e r a n c e T O L i s p r o p o s e d .

hne w:= 0. 9 hold ( T O L ~i- (2 .9)

\ E S T /

if hnew g r e a t e r 1 .5 h ola then hnew. '= 1.5 ho~d

if h .e w le ss 0.5 ho~d t h e n h ,~w :=0 .5 ho~d.


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58 P. Kap s and P. Rentrop

T h e s a f e t y f a c t o r 0 . 9 s e r v e s t o k e e p h n ew s m a l l e n o u g h t o b e a c c e p t e d , i f t h e

t r u n c a t i o n e r r o r i n th e n e x t s t ep i s g r o w i n g . T h e b o u n d s 0 .5 a n d 1.5 f o r t h e r a t i o

o f t w o s t e p s a r e i n t r o d u c e d t o p r e v e n t a s te p s i z e p r e d i c t i o n , w h i c h i s h i g h l y z ig -

z a g i n c h a r a c t e r . T h e v a l u e s o f t h e t h r e e c o n s t a n t s a r e f i xe d b y e x p e r i e n c e . E S T

i s d e f i n e d b y

n

E S T : = m x

[ Y i h ( X ~ - - f ; i h ( X ~

i = 1 S i

w h e r e S = ( S 1 . . . . S ,,) T s t a n d s f o r a s u i t a b l e s c a l i n g v e c t o r .

S i : = m ax C , ly /,h X/)[ ) i = 1 , . . . , n 2 .10)

C > 0 in t he f ol lo w i ng C = I )

x o < x j < Xo la , x j r e p r e s e n t s t h e d i s c r e t e a b s c i s s a .

h , e w is a c c e p t e d , i f E S T < T O L , o t h e r w i s e f o r m u l a 2 .9 ) is a p p l i e d o n c e m o r e

w i t h t h e v a l u e E S T b e l o n g i n g t o t h e f a i l e d s t e p s i z e p r e d i c t i o n . T h i s y i e l d s a

s m a l l e r h . . . . w h i c h m a y b e s u c ce s sf u l. R e p e a t e d a p p l i c a t i o n o f 2 .9) is t e r m i n a -

t ed , i f hnew=


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Generalized

o r d e r 1 :

o r d e r 2 :

o r d e r 3 :

o r d e r 4 :

o r d e r 5 :

Runge Kutta Methods

Z c i = l ,

~ . c i f li = 8 9 Y = P : ( Y ) ,

Z Ci f l i j f l j _____1 _ 7 + 72 = P 4 ( 7 ) ,

- - g - - ~ 7 - - - - P 6 (7 ),

Z Ci f l ik 0~2 1 1

p

= ~ - ~ 7 = 7(7),

( fo r t r u n c a t i o n e r r o r i n v e s t i g a t io n s )

C O~4 - 1

i i - - 5

Z c l a~ O~,k l k = ~ - - 88 = P1 o(Y ) ,

2 C i O ~ i k f l k O ~ i l f l l = 2 ~

1 1 2

~ 7 + ~ - 7 = P l y ( Y ) ,

2 1

Z C i O~iO~ikO~k - - T 5 ,

Z C i Xi O~ik lk l f l l 1 1. - - 1 .2

Z c i f l i k c ~ 3 =T6 7 -1 8 8 = p 1 4 (7 ) ,

f l l - - T 6 - - ~ 7 + ~ 7

= P 1 5 ( 7 ) ,

~ = ~ - - g r ' - 5 ~ = P ~ 6( 7 ) ,

59

3 . 1 )

3 . 2 )

3 . 3 )

3 . 4 )

3 . 5 )

3 . 6 )

3 . 7 )

3 . 8 )

3 . 9 )

3 . 1 0 )

3 . 1 1 )

3 . 1 2 )

3 . 1 3 )

3 . 1 4 )

3 . 1 5 )

3 . 1 6 )

3 . 1 7 )

. C , f l , k f l k , f l , m B m - - 1 . - - 2

- l ~ g - - g ~, -t- y - - 2 7 3 + 7 4 = P 1 7 ( 7 )

s u m m a t i o n i n d i c e s i, j , k , l , m = 1 . . . , s ,

a b b r e v i a t i o n s

f l i j = a l j + 71 j , ~ = 71 i = 0 f o r i < j .

R e m a r k . T h e o r d e r c o n d i t i o n s f o r Yh a r e o b t a i n e d b y r e p l a c i n g c i b y c i a n d s b y g

in t h e a b o v e e q u a t i o n s . T h e y a r e d e n o t e d b y (? ). F r o m ( 3.4 ) a n d ( 3.8 ) i t f o l lo w s

i m m e d i a t e ly , t h a t t h e r e a r e n o m e t h o d s o f o r d e r 4 w i th s = 2 . A - s t a b l e m e t h o d s

o f o r d e r 4 e x i s t fo r s = 3 , s e e [ 1 3 , 1 4] . T h e r e is n o e m b e d d e d m e t h o d (2 .8 ) w i t h

s = 3 , ~_< 3 , s e e L e m m a ( 3 .t 8 ) . N e v e r t h e l e s s , o n e c a n c o n s t r u c t m e t h o d s w i t h s = 4,

g = 3 a n d o n l y t h r e e f u n c t i o n e v a l u a t i o n s p e r s te p , se e P r o p o s i t i o n s ( 3.1 9) , (3 .2 0) .

I n [1 3 , 1 4 ] i t i s s h o w n t h a t t h e r e e x i s ts n o f i v e o r d e r m e t h o d ( 2.2 ) w i t h s - - 4 .

Lemma ( 3 .1 8 ). T h e r e e x i s ts n o e m b e d d e d m e t h o d ( 2. 8) w i t h s = 3, ~_< 3 .

P r o o f

g = 2 i s i m p o s s i b l e , b e c a u s e t h e z e r o s o f P 4 (7 ) a n d P s (7 ) a re d i f fe r e n t.

L e t ~ = 3 , t h e n ( 3 . 1 ) , ( 3 . 3 ) a n d ( 3 . 4 ) f o r

Y h

a n d Yh d e f in e t h e s a m e l i n e a r s y s t e m

f o r t h e q , r e s p . cl ( i = 1 , 2 ,3 ) . D u e t o ( 3 .7 ) a n d ( 3. 8) t h e s y s t e m h a s a u n i q u e

s o lu t io n c ~ = ~ . 9

P r o p o s i t io n ( 3.1 9) . T h e r e e x i st e m b e d d e d m e t h o d s ( 2.8 ) w i t h s = 4 , g = 3 a n d t h r e e

f u n c t i o n e v a l u a t i o n s p e r s t e p . T h e p a r a m e t e r s 7 , ~ 2 , a a , c 4 a n d fl4 3 a r e f r e e

( e x ce p t f o r s p e c i a l v a l u e s l e a d i n g t o n o n s o l v a b l e l i n e a r s y s te m s ) .

P r o o f

C h o o s i n g ~ 4 = ~ 3 , w i t h a , ,1 = c % 1 , a 4 2 = a 3 2 , c q~ 3 = 0 , o n e g e t s a fo u r s t a g e

m e t h o d w i t h o n l y t h r e e f u n c t i o n e v a l u a t i o n s . E q u a t i o n s (3 .1 ), (3 .3 ), (3 .5 ) d e t e r m i -


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60 P . Kap s and P . Ren t rop

n e C l ,

C2

C 3 J - C 4 :

~ ~

c2 =

-~.

~ a~/ c 3 + c 4 88

F r o m (3 .8 ) f o l l o w s : 1 33 2/3 2= P s ( Y ) / c 4 / 3 4 3 = u . E q u a t i o n s ( ~ 1 ) , ( 3. 3), (3 7 4) d e f i n e C a ,

c2 , Ca :

0 c 3 / 4 ( 7 )

( 3. 7) l e a ds t o c 3 /332 + c 4 /342 = ( P 7 ( 7 ) - c 4 /343 a 2)/c ~2 = : v . F r o m ( 3.4 ) a n d (3 . 2 ) on e

o b t a i n s / 3 2 a n d / 3 3 :

v c / 3 3 1

e 2 , 3 ,

t h e r e f o r e /3 3 2 , / 3 4 2 a r e g i v e n b y :

% 2 c a n b e c a l c u l a t e d f r o m ( 3.6 ):

~ 3 2 = P 6 ( 7 ) / ( ( c 3 + c 4 ) ~ 3 / 3 2 ) . / 3 4

f o l l o w s f r o m

( 3 . 2 ) : / 3 4 = ( P 2 ( 7 ) - c 2 / 3 2 - c 3 / 3 3 ) / c 4 9 #

T h e r e m a i n i n g f re e p a r a m e t e r s , e x c e p t 7 , c a n b e c h o s e n s o t h a t s e v e ra l e q u a t i o n s

o f c o n d i t i o n o f o r d e r f iv e a r e s a t is f ie d . T h e c o e f f ic i e n t 7 d e t e r m i n e s e s s e n t i a ll y

t h e s t a b i l i t y p r o p e r t i e s , s e e (2 .5 ), (2 .6 ). c 4 h a s n o i n f l u e n c e o n t h e t r u n c a t i o n

e r r o r , i ts v a l u e c a n b e c h o s e n t o c 3 = 0 .

P r o p o s i t i o n ( 3 .2 0 ). T h e f re e p a r a m e t e r s o f P r o p o s i t i o n ( 3 .1 9 ) a r e c h o s e n a s :

~2 = 2 y,

51 I Z 2

~ 3 - - 1 1

4 3-~2

c 4 a n d / 3 4 3 a r e s o l u t i o n s o f th e l i n e a r s y s t e m :

T h e n E q s . (3 .9 ), (3 . 1 0 ), (3 .1 1 ), ( 3 .1 4 ) a n d ( 3 . 1 5 ) a r e s a t i s f i e d , t o o . I t h o l d s c 3 = 0 .

P r o o f

F o r a 2 : # ~ 3 t h e E q s . (3 .1 ), (3 .3 ), ( 3.5 ) a n d ( 3 .9 ) p o s s e s s a s o l u t i o n , i ff :

: : I i l

e t a 2 a ~

= 0 .


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Generalized R unge-K uttaMethods 61

T h u s t h e c h o i c e o f % i m p l ie s (3 .9 ). D u e t o t h e l in e a r s y s t e m i t h o l d s c 3 = 0 a n d

(3 .14). T h e c h o i c e o f % c o m bi n e d w i t h ( 3 .6 ) l e a ds t o t he s i m p l i f y i ng a s s um p-

t i ons :

k l

Th ere for e (3 .9) im pl ies (3 . t0) , (3 .11). (3 .14) impl ies (3 .15) , see a l so [13 ] , Pr op os i -

t io n 5, p. 20. 9

R e s t r i c t i o n . E v a l u a t i o n o f th e r i g h t h a n d s id e o f t h e d i f fe r e n ti a l e q u a t i o n m e r e l y

i n t he i n t e g r a t i on i n t e r va l r e qu i r e s :

0= 0 a nd b > 0,

s t a b l e a t i n f in i t y , iff: a > 0

( a n a lo g u e r e s u l t fo r t h e m e t h o d o f o r d e r 3 w i t h & /~ ). C o m p u t a t i o n o f th e z e r o s

of a a nd b , r e sp . f i an d ~ l eads t o t he s t ab i l i t y i n t e rv a l s o f ~, l i s ted i n T able (3.24).


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62 P. K aps and P. Ren trop

T a b l e 3 . 2 4 ) . Stabil i ty Intervals

A -stability Stab ility at infinity,

see als o sk etch (3.25)

p = 3 [ 89 [0.15332,~] u [89 oo[

p = 4 [ 0 . 3 9 4 3 4 ,1 .2 80 57 ] [0 .10 56 7 , .10727]

u[0.20385 , 0.25] ~[0.3943 4, oo [

T h e i n t e r v a l s o f A - s ta b i l it y c o r r e s p o n d t o th e r e c e n t l y p u b l i s h e d v a l u e s o f

B u r r a g e [ 4 ] , T a b l e 1 .

S k e t c h ( 3 , 2 5 ) . L a g u e r r e P o l y n o m i a l s L 3 ( ~ ) , L 4 ( ~ )

14.00

12.00

10.00

8.00

6.00

4.00

2.00

0.00

2.00

4.00

6.00

8 .00

10.00

12. O0

14.00

_

L J 1 / ~

i / 1_~(1/~,)

F o r t h e c h o i c e o f ? , b e s i d e s t a b il i t y c o n s i d e r a t i o n s t h e t r u n c a t i o n e r r o r w a s

t a k e n i n to a c c o u n t . F o r a m e t h o d o f o r d e r p th e l o c a l t r u n c a t i o n e r r o r i s g iv e n

b y :

Np l

hp+I ~ TiP+l~zf+lDf+l+O hp+2),

s e e [ 3 ] .

i =


9/14

G e n e r a l i z e d R u n g e - K u t t a M e t h o d s 6 3

T he num e r i c a l c ons t a n t s T ~p+ ~ a r e d e t e r m i n e d b y t h e p a r a m e t e r s o f t h e m e t h o d ,

D~+ ~ a r e t h e e l e m e n t a r y d i f fe r e n ti a ls o f o r d e r p + 1, e ~ + ~ t h e c o r r e s p o n d i n g

c oe f f i c i e n t s o f B u t c he r a nd

Nv ~

t h e n u m b e r o f tr ee s o f o r d e r p + l . T h e

f o l low i ng e xp r e s s i on de f i ne s t he e r r o r c on s t a n t :

6 = m ax [T/p+ 11 (3.2 6)

T w o m e t h od s w i t h d i f f e r e n t 7 a n d d i f f e re n t s t a b i l it y p r ope r t i e s a r e p r op os e d . I f

b o t h R O W - m e t h o d s ( 2 .8 ) s h o u l d b e A - s ta b le , s m a l l v a lu e s o f 7 l e a d t o s m a l l

t r u n c a t i o n e r r o rs . T h e r e f o r e 7 = 0 . 3 9 5 is p r o p o s e d . T h e h y p o t h e s e s o f P r o p o s i -

t i on (3.20) g ive e3 < 0 , co n t r ad i c t i n g re s t r i c t i on (3.21). In Ta ble (3.27) a coe f f i c i en t

set fo r 7 = 0 .395 w i th s m a l l t ru nc a t io n e r rors is l is t ed , which do n t s a t i s fy (3.20).

T a b l e (3 .2 7) . G R K 4 A , ~ = 0 . 3 9 5

= 0 . 3 9 5 ~ 21 = - 0 . 7 6 7 6 7 2 3 9 5 4 8 4

Y31 = - 0 . 8 5 1 6 7 5 3 2 3 7 4 2 ~ 32 = 0 . 5 2 2 9 6 7 2 8 9 1 8 8

7 41 = 0 . 2 8 8 4 6 3 1 0 9 5 4 5 ~ 42 = 0 . 8 8 0 2 1 4 2 7 3 3 8 1 E - 1

~43 = - 0 . 3 3 7 3 8 9 8 4 0 6 2 7

~ 2 1 = 0 . 4 3 8

~ 3 1 = 0 .7 9 6 92 0 4 57 9 3 8 e 32 = 0 . 7 3 0 7 9 5 4 2 0 6 1 5 E - 1

c l = 0 . 3 4 6 3 2 5 8 3 3 7 5 8 c 2 = 0 . 2 8 5 6 9 3 1 7 5 7 1 2

c3 = 0 . 3 6 7 9 8 0 9 9 0 5 3 0

c 1 = 0 . 1 9 9 2 9 3 2 7 5 7 0 1 c 2 = 0 . 4 8 2 6 4 5 2 3 5 6 7 4

c a = 0 . 6 8 0 6 1 4 8 8 6 2 5 6 E - 1 c 4 = 0 .2 5

6 N 0 . 9 4 2 / 5 f o r t h e m e t h o d o f o r d e r 4 , s e e ( 3 .2 6 )

S N 1 . 0 8 /4 f o r t h e m e t h o d o f o r d e r 3 , s e e ( 3 .2 6 )

T h e s e c o n d m e t h o d i s c o n s t r u c t e d a c c o r d i n g t o P r o p o s i t i o n ( 3 . 2 0 ) .

7 e [ 0 .1 0 5 6 7 ,0 . 1 0 7 2 7] p r o d u c e g r e a t v a l u e s o f L 3 ( 1 ) . F o r

7e [0 .20385 , 0 .25]

L 3 ( ~ ) i s s m a l t , s e e s k e t c h ( 3 . 2 5 ) , a n d t h e s t a b i l i t y r e g i o n o f t h e f o u r t h o r d e r

m e t h o d i s v e r y l a r g e . F o r 7 = 0 . 2 3 1 t h e f o u r t h o r d e r m e t h o d i s A ( 8 9 . 3 ~

and the h yp oth ese s o f (3 .20) and re s t r i c t i on (3.21) a re sa t i s fi ed . A co e f f i c i en t se t is

l is t ed i n Tab le (3.28). A fur th e r re l a t ed co e f f i c i en t se t wi th 7=0 .22 042 841 can b e

found in Stoer , Bul i rsch 1-19] .

ab l e

(3 .2 8) . G R K 4 T , ~ = 0 . 2 3 1

7 = 0 .2 3 1 Y21 = - 0 . 2 7 0 6 2 9 6 6 7 7 5 2

~ al = 0 . 3 1 1 2 5 44 8 3 2 9 4 T32 = 0 . 8 5 2 4 4 5 6 2 8 4 8 2 E - 2

741 = 0 .2 8 2 8 1 6 8 3 2 0 4 4 Y 42 = - 0 . 4 5 7 9 5 9 4 8 3 2 8 1

743 = - 0 . 1 1 1 2 0 8 3 3 3 3 3 3

% ~ = 0 . 4 6 2

% 1 = - 0 . 8 1 5 6 6 8 1 6 8 3 2 7 E - 1 ~ a2 = 0 .9 6 1 77 5 1 50 1 6 6

6~ = - 0 . 7 1 7 0 8 8 5 0 4 4 9 9 62 = 0 . 1 7 7 6 1 7 9 1 2 1 7 6 E + 1

~3 = - 0 . 5 9 0 9 0 6 1 7 2 6 1 7 E - 1

c~ = 0 . 2 1 7 4 8 7 3 7 1 6 5 3 c 2 = 0 . 4 8 6 2 2 9 0 3 7 9 9 0

c a = 0 . c 4 = 0 . 2 9 6 2 8 3 5 9 0 3 5 7

6 ~ 0 . 1 9 9 / 5 f o r t h e m e t h o d o f o r d e r 4 , s e e ( 3. 26 )

8 ~ 0 . 4 6 1 / 4 f o r t h e m e t h o d o f o r d e r 3 , s e e ( 3. 26 )


10/14

6 4 P . K a p s a n d P . R e n t ro p

4 Nu m erical Implementation

T o c o m p u t e t h e v e c t o r s k i 2 .2 ), a li n e a r s y s t e m o f o r d e r n fo r f o u r r ig h t h a n d

s i d e s m u s t b e s o l v e d . I n o r d e r t o a v o i d m a t r i x - v e c t o r m u l t i p l i c a t i o n s , t h e

e q u i v a l e n t f o r m d u e t o [ 2 1 ] i s u s e d :

( I -- h Y f ( Y o ) ) k , = h f ( Y o ) ,

( I - h y f ( y o ) ) ( k 2 + ~ 2 , k l ) = h f ( Y o + a21 k 1) ~- ~21 k , ,

( I - h y f ( y o ) ) (k 3 +

~3~ k~ + ~32 k2 ))

= h f ( Y o + C t 3 1 k ~

+ a 32 k2 ) + ~ 3 1k l + Y 32k2),

( I - h v f ( y o ) ) ( k 4 + 74~ k~ + ~42 k2 + ~43 k3))

= h f ( Y o + a 3 a k ~ - ] - a 3 2

k2 ) -b (~41 k 1 1- ~42 k2 -I - ~43 k3 )

w h e r e

T h e J a c o b i a n f ( Y o ) is c o m p u t e d b y d i ff e re n c e a p p r o x i m a t i o n a n d s h o u l d b e

r e p l a c e d b y a n a n a l y t i c v e r s i o n f o r v e r y s e n s i t i v e p r o b l e m s . T h e m a t r i x

1 - h

7 f ( Y o ) )

is d e c o m p o s e d b y L U - f a c t o r iz a t io n . C o m p u t a t i o n o f t h e k is e q u iv a -

l e n t to b a c k s u b s t i tu t i o n s . F o r l a r g e s p a r s e s y s t e m s th e s t r u c t u r e o f t h e J a c o b i a n

i s s a v e d a n d t h e s t a n d a r d r o u t i n e f o r L U - d e c o m p o s i t i o n s h o u l d b e e x c h a n g e d

b y s u b ro u t in e s fo r s p ar se s ys te m s . B o t h p r o g r am s G R K 4 A 3.27) a n d G R K 4 T

3 . 28 ) ha ve a s t r uc t u r e a s s i m p l e a s t he R K F m e t hods [ 6 , 7 ] a nd c a n be e a s i l y

i m p l e m e n t e d . E x c e p t f o r g e n e r a t i o n o f t h e J a c o b i a n n o n e s t e d lo o p s a r e n e ce s-

s a r y . T he c a l l i ng s e que nc e i s i n a c c o r da nc e w i t h [ 6 , 7 ] .

5 Test Exam ples

T h e p r o p o s e d m e t h o d s w e r e t e s t ed o n 2 5 st if f d i f fe r e n ti a l e q u a t i o n s [ 8 ]. T h e

p r op e r t i e s o f t he d i f f e r e n t ia l e qu a t i on s a r e on l y b r ie f l y de s c r i be d i n t he f o l low -

i ng , f u r t he r i n f o r m a t i ons c a n be f ound i n [ 8 ] . T he t e s t s e t i s d i v i de d i n t o f i ve

c lasses :

C l a s s A : L i ne a r w i t h r e a l e i ge nva l ue s

C l a s s B : L i n e a r w i t h n o n - r e a l e i g e n v a lu e s .

C l a s s C : N o n l i n e a r c o u p l i n g w i t h r e a l e i g en v a t u es .

C l a s s D : N on l i ne a r w i t h r e a l e i ge nva l ue s .

C l a s s E : N o n l i n e a r w i t h n o n - r e a l e i g e n v a l u e s .

T h e f o l lo w i n g a b b r e v i a t io n s a r e u s e d :

T Z : T o t a l c o m p u t i n g t i m e i n s e c o n d s t o s o l v e a p r o b l e m . C o m p u t a t i o n s

w e r e p e rf o r m e d i n F O R T R A N s in g le p r e c is io n w i th a 38 b i t m a n ti ss a

11 d e c im a l s ) o n t h e T R 4 4 0 o f t h e L e i b n i z R e c h e n z e n t r u m d e r B a ye ri-

s c h e n A k a d e m i e d e r W i s s e n s c h a f t e n .


11/14

G e n e r a l i z ed R u n g e - K u n a M e t h o d s 6 5

F CN N u m b e r o f f u n c ti o n c a ll s

FJAC N u m b e r o f J a c o b i a n e v a lu a ti o n s. O n e e v a l u a t io n o f t h e J a c o b i a n c o s t s

n fu n c t io n ca l l s .

TF

T o t a l n u m b e r o f f u n c t i o n c a ll s:

T F = F C N + n - FJ A C .

LU N u m b e r o f L U - d e c o m p o s i t i o n s , e q u iv a l e n t to t h e n u m b e r o f s te ps .

ERR M a x i m u m e r r o r o f s o l u ti o n c o m p o n e n t s a t t h e e n d o f t h e in t er v al . T h e

r ef e re n c e s o lu t io n w a s c o m p u t e d b y t h e p r o c e d u r e D R I V E w i th T O L

= 1 . E - 8 . D R I V E i s t h e i m p r o v e d G E A R v e r s i o n f r o m 1 3 . 1 . 1 9 7 5 , d u e t o

G e a r 1 -9 ] a n d H i n d m a r s h [ 1 2 ] .

F o r a l l e x a m p l e s t h e i n i t i a l s t e p s i z e H I = 1 . E - 3 a n d t he st ep s iz e c o n t r o l

fo rm u la 2 .9 ) we re u sed . Th e t e s t r e su l t s a r e l i s t ed in Tab le 5 .1 ) an d Tab le 5 .2 ) .

B o t h m e t h o d s s o l v e a l l e x a m p l e s r e l i a b l y . G R K 4 A l o o s e s p r e c i s i o n i n D 5

a n d E 2 . A c c o r d i n g t o p re c i s i o n a n d f a s tn e ss , G R K 4 T is t h e s u p e r i o r m e t h o d , in

s p i te o f i ts w e a k e r s t a b i li ty c o n d i t io n s . O n l y i n E 4 , th e c o m p u t i n g t i m e o f

G R K 4 T is e n la r ge d . A n o v e r a ll s u m m a r y f o r b o t h m e t h o d s a n d f o r t h r e e

t o l e r a n c e s i n a c c o r d a n c e w i t h [ -8 ] i s g i v e n i n T a b l e 5 .3 ).

T a b l e 5 .1 ). S t a t i s t i c s f o r e a c h p r o b l e m , T O L = 1, E - 4

P r o b l e m G R K 4 A

T Z L U F C N F J A C T F E R R

A I 0 . 3 1 4 0 1 2 0 4 0 2 8 0 2 .1 E - 7

A 2 1 .2 2 5 3 1 55 4 9 5 9 6 4 . 8 E - 8

A 3 0 . 4 4 6 0 1 7 5 5 5 3 9 5 1 .1 E - 6

A 4 2 .0 1 7 4 2 1 3 6 5 8 6 3 1 . 5 E - 6

B 1 1 . 4 4 1 8 3 5 4 2 1 7 6 1 , 2 4 6 4 . 0 E - 5

B 2 0 . 4 9 4 0 1 2 0 4 0 3 6 0 1 . 0 E - 6

B 3 0 .5 3 4 3 1 2 9 4 3 3 8 7 7 . 5 E - 7

B 4 0 . 77 6 2 1 86 6 2 5 5 8 1 . 1 E - 6

B 5 2 . 0 4 1 6 4 4 9 2 1 6 4 1 , 4 7 6 2 . 4 E - 6

C 1 0 . 3 6 4 5 1 3 5 4 5 3 1 5 1 . 7 E - 7

C 2 0 . 37 4 3 1 2 9 4 3 3 01 3 . 6 E - 7

C 3 0 . 4 5 5 3 1 5 9 5 3 3 7 1 1 .1 E - 5

C 4 1 . 0 4 1 2 2 3 6 6 1 2 2 8 5 4 9 . 7 E - 6

C 5 1 . 3 2 1 5 4 4 6 2 1 5 4 2 , 9 1 9 1 . 2 E - 5

O 1 1 . 1 9 2 0 7 6 2 1 2 0 7 1 , 2 4 2 5 . 9 E - 6

D 2 0 . 4 6 7 8 2 3 1 7 5 4 5 6 7 . 2 E - 5

D 3 0 . 3 8 4 9 1 4 0 4 2 3 0 8 1 . 5 E - 7

D 4 0 . 1 4 2 5 7 5 2 5 1 5 0 1 . 8 E - 5

D 5 0 .1 1 2 8 8 4 2 8 1 4 0 8 . 7 E - 3

D 6 0 . 13 2 3 6 9 2 3 1 3 8 1 . 8 E - 4

E 1 1 .8 1 2 8 8 5 5 2 1 8 4 1 , 28 8 3 . 4 E - 1 0

E 2 0 . 33 8 7 2 5 0 7 6 4 0 2 8 . 9 E - 4

E 3 0 . 4 8 8 0 2 3 8 7 8 4 7 2 4 . 7 E - 5

E 4 0 . 79 7 8 2 2 9 7 3 5 2 1 3 . 5 E - 4

E 5 0 . 2 6 3 3 9 9 3 3 2 3 1 3 . 0 E - 8


12/14

Table 5.2). Statistics for each probem, TO L= 1. E- 4

P. Kaps and P. Rentrop

Problem GRK4T

TZ LU FCN FJAC TF ERR

A1 0.25 35 105 35 245 1.6E-6

A2 1.14 51 148 46 562 2 .1E-7

A3 0.39 54 157 49 353 2 .0E- 5

A4 1.76 65 186 56 756 2.0E - 5

B 1 1.24 160 473 153 1,085 3.6 E - 5

B2 0.45 36 108 36 324 1.2E-6

B 3 0.46 38 114 38 342 1.6E- 6

B 4 0.66 53 159 53 477 2.0 E - 6

B 5 1.75 140 420 140 1,260 1.7 E - 6

C1 0.31 41 123 41 287 1.8E-7

C2 0.32 39 117 39 273 5 .0E- 8

C 3 0.44 53 159 53 371 1.5 E - 7

C4 0.94 111 333 111 777 1.2E - 7

C5 1.19 135 405 135 945 4 .5E- 8

D 1 1.29 231 658 196 1,246 3 .8E- 6

D2 0.36 63 182 56 350 4 .9E-5

D3 0.45 57 164 50 364 3 .2E- 8

D4 0.14 25 75 25 150 2 .2E-6

D5 0.14 36 104 32 168 1.1E-4

D6 0.10 17 51 17 102 2 .9E-6

E 1 1.08 168 327 109 763 3 .4E- 10

E 2 0.35 96 268 76 420 4.6 E - 4

E 3 0.49 89 249 71 462 4.3 E - 6

E4 3.27 354 942 234 1,878 9 .7E- 5

E5 0.27 33 99 33 231 3 .3E-8

Table 5.3). Overall Summary

Method TOL TZ LU FCN FJAC TF

GRK 4 A 1.E - 2 9.39 1,066 2,851 927 6,859

1.E-4 18.85 2,112 5,971 1,955 14,428

1 .E 6 65.35

7,271 21,189 7,088 49,324

GRK4T 1.E - 2 8.40 960 2,666 864 6,419

I . E - 4 19.23 2,180 6,126 1,884 14,181

1.E - 6 58.07 6,676 19,179 6,320 43,336

B o t h m e t h o d s a r e l o w o rd e r m e t h o d s a n d w o rk v e ry w el l f o r l o w t o le r a n ce s .

Th e y s h o u l d b e u s e d o n l y for t o l e r a n c e s b e t w e e n 1 .E -2 a n d 1 .E -4 .

Com par i so n wi th DRI VE [9, 12] :

Th e i m p ro v e d G EA R v e r s i o n D R IV E is a v a i l a b l e t o t h e a u t h o r s . C o m p u t i n g

t ime and func t ion ca l ls fo r a ll exa mple s a re l i s t ed in Tab le 5.4).


13/14

G e n e r a l i z e d R u n g e - K u n a M e t h o d s

able

5 .4 ). O v e r a l l S u m m a r y f o r T O L = 1 .E - 4

G R K 4 A G R K 4 T D R I V E - G E A R

Al l ex am ple s T Z 18 .85 19 .23 44 .21

T F 14,428 14,181 9,099

Al l ex am ple s T Z 16 .02 14 .21 21 .57

exc ept B5, E4 T F 12,43 t 11,043 5 ,423

67

Comparing computing time, both methods are competitive with DRIVE,

although the number of function calls TF is enlarged by a factor two. DRIVE

runs very efficient in the classes D and E, but produces heavy difficulties in B5,

where precision is lost and computing time reaches 21.23 seconds. The great

number of evaluations of the Jacobian and LU-decompositions in GRK 4 A and

GRK4T are disadvantageous for large complicated systems, which are not

included in the test set [8].

6 Applicat ion o f G RK 4A and G RK 4T

to the restricted Three Body Problem

To give some information how both methods will work for non-stiff differential

equations, the restricted Three Body Problem earth-moon-spaceship) tested in

Bulirsch, Stoer [3] and [7] was solved. Results for the non-stiff differential

equation solvers DIFSY1, VOAS, RKF7, and RKF4 from [6] together with

GR K4 A and GRK4T are listed in Table 6.1).

able

6.1). T h r ee B o d y P r o b l em , T O L = 1 . E - 4 ,

HI

1 . E - - 3

S ta tis ti c s D I F S Y 1 V O A S R K F 7 R K F 4 G R K 4 A G R K 4 T

TZ 1.11 2.36 1.28 1.44 1.89 2.48

TF 1,215 669 1,233 1,398 1,048 1,339

This difficult example, which requires a robust and reliable stepsize controll,

was solved precisely by both methods. The more complicated structure of

GRK4A and GRK4T enlarged the computing time by a factor 1.5, although

the number of function calls is comparable to the related routine RKF 4.

Conclusion

With GRK4A and GRK4T two reliable, fast and precise algorithms for the

numerical solution of stiff systems of ordinary differential equations are avail-

able. The loworder methods should be applied for low tolerances up to TOL

t .E-4 . One would prefer these methods for problems with n < 10, because of


14/14

6 8 P . Ka p s a n d P . Re n t r o p

t h e la r g e n u m b e r o f L U d e c o m p o s i t i o n s . I f s t a b il it y r e q u i r e m e n t s a re w e a k e r

G R K 4 T s e e m s t o b e t h e f as te r a n d m o r e p r e c is e r o u t in e .

Acknowledgement

T h e a u th o r s w i s h t o t h a n k P r o f . R . Bu l i r s c h . Dr . E . Ha i r e r a n d P r o f . G . W a n n e r

fo r insp i r ing and he lp fu l d i scuss ions .

R e f e r e n c e s

1. Ab r a m o w i t z , M ., S te g u n , I.A . : H a n d b o o k o f m a th e m a t i c a l f u n c ti o n s . Ne w Yo r k : D o v e r P u bi .

Inc. 1970

2 . Be d e t , R .A ., E n r ig h t , W .H . , Hu l l , T .E .: S T I F F DE T E S T : A p r o g r a m f o r c o m p a r in g n u m e r i c a l

m e th o d s f o r s t if f o r d in a r y d i f f e re n t i a l e q u a t i o n s . T e c h . Re p . N o . 8 1 , Un iv e r s i t y o f T o r o n to 1 97 5

3 . Bu l i r s c h , R . , S to e r , J . : Nu m e r i c a l t r e a tm e n t o f o r d in a r y d i f f e r e n t i a l e q u a t i o n s b y e x t r a p o l a t i o n

me thods . Num er . Ma th . 8 , 1 -13 1966)

4 . Bu r r a g e , K . : A s p e c ia l f a m i ly o f R u n g e - K u t t a m e th o d s f o r s o lv in g s t if f d i f fe r e n t ia l eq u a t io n s .

B IT 18, 22-4 1 1978)

5 . Da h lq u i s t , G . : A s p e c ia l p r o p e r ty f o r l i n e a r m u l t i s t e p m e th o d s . B1 T 3 , 2 7 - 4 3 1 96 3)

6. Diek hoff , H.-J . , Lory, P . , O berle , H.J ., P esch, H.-J. , R ent rop , P . , Seydel , R.: Co m pa rin g rou tines

f o r t h e n u m e r i c a l s o lu t i o n o f i n it i a l v a lu e p r o b l e m s o f o r d in a r y d i f fe r e n ti a l e q u a t i o n s i n m u l t i p l e

shoo t ing . Nu me r . Ma th . 27 , 449-46 9 1977)

7 . Enr ig h t , W.H., B ede t , R , Fa rk as , I ., Hu l l , T .F . : Tes t re su l t s on in i t ia l va lu e meth ods fo r non-s t i f f

o r d in a r y d i f f e re n t i a l e q u a t i o n s . T e c h n . Re p . N o . 6 8 , Un iv e r s i t y o f T o r o n to 1 97 4

8 . E n r ig h t , W .H . , Hu l l , T .E. , L in d b e r g , B . : Co m p a r in g n u m e r i c a l m e th o d s f o r st i ff s y s t e m s o f

o rd ina ry d i f fe ren t ia l equa t ion s . Tech . Rep . No . 69 , 1974 Univers i ty o f To ron to , see a l so BIT 15 ,

10-48 1975)

9 . Ge a r , C .W . : N u m e r i c a l i n i t ia l v a lu e p r o b l e m s i n o r d in a r y d i f fe r e n ti a l e q u a t i o n s . N .Y . : P r e n ti c e

Hal l 1970

10. Gr igor ie f f , R .D. : N um er ik gew/Shnl icher Di f fe ren t ia lg le ich ungen . S tu t tga r t : Teu bn er 1972

11 . Ha i r e r , E ., W a n n e r , G . : O n t h e B u tc h e r g r o u p a n d g e n e r a l m u t iv a lu e m e th o d s . Co m p u t in g 1 3 , 1 -

15 1974)

1 2. H in d m a r s h , A .C . : G E A R - o r d in a r y d i f fe r e n t ia l e q u a t i o n s y s t e m s o lve r . UC I D- 3 0 0 0 1 , Re v. 2 ,

Un iv e r s i t y o f Ca l i f o r n i a : L a wr e n c e L iv e r m o r e L a b o r a to r y 1 97 2

1 3. Ka p s , P . : M o d i f i z i e r te R o s e n b r o c k m e th o d e n d e r Or d n u n g 4 , 5 u n d 6 z u r n u m e r i s c h e n I n t eg r a -

t i o n s t e if e r D i f f e r e n t i a lg l e i c h u n g e n . D i s s e r t a t i o n , U n iv er s it~ it I n n s b r u c k , S e p t e m b e r 1 97 7

14. Kap s , P . , Wa:nner, G. : Ro sen bro ck- t ype me tho ds o f h igh o rder . In p ress 1979)

1 5. No r s e t t , S .P .: C - p o ly n o m ia l s f or r a t i o n a l a p p r o x im a t io n t o t h e e x p o n e n t i a l f u n c t io n . Nu m e r .

M ath . 25 , 39 -56 1975)

1 6. No r s e t t , S .P. , W a n n e r , G . : T h e r e a l - p o l e s a n d w ic h f o r r a t i o n a l a p p r o x im a t io n s a n d o s c i ll a ti o n

equ a t ions . BIT 19 , 79 -94 1979)

1 7. No r s e t t , S .P ., W o l f b r a n d t , A . : Or d e r c o n d i t i o n s fo r Ro s e n b r o c k - ty p e m e th o d s . N u m e r . M a th . 32 ,

1-15 1979)

18. Ro senb rock , H .H. : Som e gene ra l imp l ic i t p rocesses fo r the num er ica l so lu t ion o f d if fe ren tial

equa tons . Co mp . J . 5 , 329 331 1963)

19. S toe r , J. , Bu l i r sch , R. : E in f t ih rung in d ie Num er isc he M ath em at i k I I. Ber l in , He id e lbe rg , New

Yo rk : Spr in ger Ver lag , 2. Auf lage , 1978

2 0. W a n n e r , G . : L e t t e r t o S.P. N o r s e t t a n d u n p u b l i s h e d c o m m u n ic a t i o n s t o P . Ka p s a n d A .

W o l f b r a n d t , 1 9 7 3

2 1. W o l f b r a n d t , A . : A s tu d y o f Ro s e n b r o c k p r o c e s se s w i th r e s p e c t t o o r d e r c o n d i t i o n s a n d s ti ff

s tab i l i ty . Dis se r ta t ion , Res ea rch Pep . 77 .01 R, Univ ers i ty o f G/S teborg, M arc h 1977

Re c e iv e d S e p t e m b e r 2 2 , 1 9 7 8 /Re v i s e d Ap r i l 9 , 1 97 9

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