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J O U R N L O F

CO M P UT T I O N L ND

PPLIED M THEM T ICS

ELSEVIER Journal of Com putational and Applied Mathem atics 79 (1997) 189-205

A finite difference m ethod for the num erical solution o f the

Schrrdinger equation

T . E . S i m o s a P . S .

W i l l ia m s b

Laboratory of Applied M athematics an d C ompu ters, Technical University of Crete, Kounoupidiana, 73100 Hania,

Crete, Greece

b Department of Computing, Information Systems and Mathematics, Faculty o f Hum an Sciences, Londo n Guildhall

University, 100 M inories, Lon don EC3 N 1JY, United Kingdom

Received 20 May 1996; revised 29 October 1996

bs trac t

A new approach, which is based on a new property of phase-lag fo r computing eigenvalues o f Schrrdinger equations

with potentials, is developed in this paper. We investigate two cases: (i) The specific case in which the potential

V(x)

is an even function with respect to x. It is assumed, also, that the w ave functions tend to zero for x ~ -4-oo. (ii) T he

general case o f the M orse potential and o f the W ood s-S axo n or optical potential. Num erical and theoretical results show

that this new approach is more efficient compared to previously derived methods.

Keywords. Schrrdinger equation; Eigenvalue problem; Finite differences; Phase-lag

A M S classification.

65L05

1 . I n t r o d u c t i o n

I n r e c e n t y e a r s th e S c h r r d i n g e r e q u a t i o n h a s b e e n t h e s u b j e c t o f g r e a t a ct i v it y , t h e a i m b e i n g

t o a c h i e v e a f a s t a n d r e l i a b l e a l g o r i t h m t h a t g e n e r a t e s a n u m e r i c a l s o l u t io n ( s e e [ 4, 7 , 1 2 - 1 8 , 2 7 -

3 5 , 2 3 , 3 , 3 7 - 3 9 ] ) .

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

y ( x )

= [ V ( x ) -

E ] y ( x ) .

1 )

E q u a t i o n s o f th i s ty p e o c c u r v e r y f r e q u e n t l y i n th e o r e t i c a l p h y s i c s , f o r e x a m p l e [ 2 6] , a n d t h e r e i s

a r e al n e e d t o b e a b l e t o s o l v e t h e m b o t h e f f ic i e n tl y a n d r e l ia b l y b y n u m e r i c a l m e t h o d s . I n ( 1 ) , E

i s a r e a l n u m b e r d e n o t i n g

t h e e n e r g y

a n d V is a g i v e n f u n c t i o n w h i c h d e n o t e s t h e p o te n t ia l . W e

i n v e s t i g a t e t w o c a s e s .

0377-0427/97/ 17.00 (~) 1997 Elsevier Science B .V. All rights reserved

P H

S 0 3 7 7 - 0 4 2 7 ( 9 6 ) 0 0 1 5 6 - 2


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190 T.E. Simo s, P.S. Wil liams~Journal of Com putational and Applied Mathematics 79 1997) 189-205

I n t h e f i r s t c a s e ,

V x )

i s a n e v e n f u n c t i o n a n d

y x ) ~ 0

f o r x ~ + o c . A s e x a m p l e s o f p o t e n t i a ls

w h i c h s a t i s f y t h e s e p r o p e r t i e s w e p r e s e n t t h e f o l l o w i n g p o t e n t i a l s , w h i c h a r e w e l l k n o w n i n s e v e r a l

a r e a s o f p h y s i c s :

( i ) T h e o n e - d i m e n s i o n a l a n h a r m o n i c o s c i l l a t o r p o t e n t i a l ,

)ox2

w i t h 2 a n d 7 p a r a m e t e r s . ( 2 )

i(x) = x 2 q- 1 + yx 2

( i i ) T h e s y m m e t r i c d o u b l e - w e l l p o t e n t i a l ,

V , x )

= x 6 - f ix2 w i t h f l a p a r a m e t e r . ( 3 )

( i i i ) T h e R a z a v y p o t e n t i a l ,

Viii x) = ½ m 2 ( c o s h ( 4 x ) - 1 ) - m n + 1 ) c o s h ( 2 x ) w i t h n a n d m p a r a m e t e r s . ( 4 )

F o r t h e n u m e r i c a l s o l u t i o n o f t h e s p e c i fi c e i g e n v a l u e S c h r 6 d i n g e r e q u a t i o n ( 1 ) t h e f o l l o w i n g p r o -

c e d u r e s h a v e b e e n o b t a i n ed : ( 1 ) R a y l e i g h - R i t z m e t h o d s ( se e [ 2 7 ] ), p e r t u r b a t i o n m e t h o d s ( se e [ 4 ] ),

m e t h o d s u s i n g P a d 6 a p p r o x i m a n t s ( s e e [ 2 3] ), d i r e c t n u m e r i c a l i n t e g r a t io n t e c h n i q u e s o r b o u n d a r y

v a l u e t e c h n i q u e s ( s e e [ 1 3 - 1 5 , 3 1 ] ) a n d a n o p e r a t o r m e t h o d b a s e d u p o n t h e S O ( 2 , 1 ) d y n a m i c g r o u p

( s e e [ 1 3 ]) . W e n o t e h e r e t h a t t h e m e t h o d b a s e d o n S O ( 2 , 1 ) d y n a m i c g r o u p g i v e s m u c h m o r e a c c u -

r a t e r e s u l t s t h a n a l l f i n i te - d i f f e re n c e m e t h o d s u s e d . A n a l y t i c a l a p p r o a c h e s t o th e S c h r 6 d i n g e r e q u a t i o n

( 1 ) h a v e b e e n o b t a i n e d f o r V i (x ) g i v e n b y ( 2 ) b y F l e s s a s [ 1 7 , 1 8 ], V a r m a [ 3 8 ], W h i t e h e a d e t al . [3 9 ]

f o r Vii x) g i v e n b y ( 4 ) b y R a z a v y [ 30 ] . W e m u s t n o t e h e re t h a t in th e s e c a s e s th e m a t r i x m e t h o d s

a r e m o r e u s e f u l t h a n t h e s h o o t i n g t e c h n i q u e s .

F a c k a n d V a n d e n B e r g h e [ 1 6 ] h a v e s h o w n n u m e r i c a l l y t h a t t h e i r m e t h o d i s t h e m o s t a c c u r a t e

d i r e c t f in i te - d if f er e n c e m e t h o d f o r th e n u m e r i c a l s o l u t i o n o f t h e e i g e n v a l u e p r o b l e m ( 1 ) .

I n t h e s e c o n d c a s e V x ) i s a g e n e r a l f u n c t i o n . A s e x a m p l e s , w e p r e s e n t t h e f o l l o w i n g p o t e n t i a l s ,

w h i c h a re w e l l k n o w n i n s e v e ra l a r e a s o f p h y s i c s :

( i ) T h e M o r s e p o t e n t i a l ( s e e [ 1 , 2 8 ] ) :

Voi x) = D t t -

2 ) , t = e x p ( a X ) , ( 5 )

w h e r e X = x e - x , x e = 1 . 9 97 5 , a = 0 . 7 1 1 2 4 8 , a n d D = 1 8 8 . 4 3 5 5 .

( i i) T h e w e l l - k n o w n W o o d s - S a x o n p o t e n t i a l ( s e e [ 1] ):

Uo Uot

V° (x) - 1 + t a0(1

q - t ) 2

( 6 )

w h e r e t = e x p ( ( x - x ~ ) / a o ) , u 0 = - 5 0 , x ¢ = 7 a n d a 0 = 0 . 6 .

F o r t h e n u m e r i c a l c o m p u t a t i o n o f th e e i g e n v a l u es o f t h e M o r s e p o t e n t ia l ( 5 ) a n d t h e W o o d s - S a x o n

p o t e n t ia l ( 6 ) t h e r e a re , a l so , o t h e r n u m e r i c a l m e t h o d s w h i c h g i v e v e r y a c c u r a t e r e s u l ts ( s e e [ 37 , 2 0 -

2 2 ] ) . T h e s e m e t h o d s a r e b a s e d o n t h e s h o o t i n g te c h n i q u e , w h i c h i s c o m p l e t e l y d i ff e re n t f r o m t h e

t e c h n i q u e u s e d b y t h e m a t r i x m e t h o d s .

I n S e c t i o n 2 w e w i l l d e v e l o p t h e b a s ic t h e o r y f o r th e p h a s e - l a g a n a l y s i s o f th e s y m m e t r i c f o u r-

s t e p m e t h o d s . A s i m p l e f o u r - s t e p m e t h o d w i t h m i n i m a l p h a s e - l a g h a s b e e n c o n s t r u c t e d i n S e c t i o n 3 .

F i n a ll y , i n S e c t i o n 4 t h e a p p l i c a t io n o f th e d e v e l o p e d m e t h o d t o th e p r o b l e m ( 1 ) h a s b e e n p r e s e n t e d

a n d e x t e n d e d n u m e r i c a l r e s u l t s b a s e d o n t h e p o t e n t i a l s V~, V ii, V ii;, Vgi a n d I 10 , a r e p r o d u c e d t o s h o w

t h e e f f i ci e n c y o f th e n e w a p p r o a c h .


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T.E. Simo s, P.S. Williams~Journal of Computational and Applied M athematics 79 (1997) 189-205 191

2. Ph ase-la g analysis of general symm etric 2k-step k E N methods

I n r e c e n t y e a r s t h e r e h a s b e e n c o n s i d e r a b l e i n t e re s t in t h e n u m e r i c a l s o l u t i o n o f s e c o n d - o r d e r

p e r i o d i c i n i t i a l - v a l u e p r o b l e m s s e e [ 6 , 8 - 1 1 , 2 5 , 2 9 , 3 6 ] ) :

y = f ( x , y ) , y ( x o ) = Y0, y' (xo) = Y'o. 7 )

T o i n v e s t i g a t e t h e a b s o l u t e s t a b i l i t y p r o p e r t i e s w e i n t r o d u c e t h e s c a l a r t e s t e q u a t i o n

y = - - w 2 y . 8 )

W h e n w e a p p l y a sy m m e t r i c 2 k m e t h o d t o t h e sc a l ar t e s t e q u a t i o n 8 ) w e o b t a i n a d i f fe r e n c e

e q u a t i o n o f th e f o r m

A k ( H )y n + k + ' + A l ( H ) y n + l + A o ( H ) y n + A I ( H ) y n - 1 + ' + A k ( H ) y n - k ---- 0, 9 )

w h e r e H =

wh, h

i s t h e s t e p l e n g t h a n d

A o ( H ) , A 1 ( H ) , . . . , A k ( H )

a r e p o l y n o m i a l s o f H a n d y , i s

t h e c o m p u t e d a p p r o x i m a t i o n t o y ( n h ) , n = 0 , 1 , 2 , . . .

T h e c h a r a c t e ri s ti c e q u a t i o n a s so c i a t e d w i t h 9 ) i s

A k ( H ) s k + . . . + A I ( H ) s + A o ( H ) + A I ( H ) s -1 + ' + A k ( H ) s - k

= 0 . 1 0 )

B a s e d o n L a m b e r t a n d W a t s o n [ 25 ] w e h a v e t h e f o l l o w i n g d e fi n it i o n.

Definition 1 . A s y m m e t r i c 2 k - s te p m e t h o d w i t h t h e ch a r a c te r is t ic e q u a t i o n g i v e n b y 1 0 ) i s s a i d t o

h a v e a n i n t e rv a l o f p e r i o d i c i t y [ H 0 , H 1 ] i f , f o r a l l H C [ H 0 , H 1 ] , t h e r o o t s s i , i = 1 , . . . , 2 k o f 1 0 )

s a t i s f y

S l = e i 0 H ) ,

s 2 = e - iO (n ), s i l

~ < 1 , i = 3 , . . . , 2 k , 1 1 )

w h e r e

O ( H )

i s a re a l f u n c t i o n o f H .

Definition 2 . F o r a n y m e t h o d c o r r e s p o n d i n g to th e c h a r a c te r i st i c e q u a t i o n 1 0 ) t h e p h a s e - l a 9 i s

d e f i n e d a s t h e l e a d i n g t e r m i n t h e e x p a n s i o n o f

t = H - O(H ) . 1 2 )

T h e n i f t h e q u a n t i t y t =

O H q+l)

a s H ~ 0 , t h e o r d e r o f p h a s e - l a g i s q .

T h e p h a s e - l a g t h e o r y d e v e l o p e d i n t h e p a p e r i s v a l i d f o r t h e c a s e w h e r e w i s i m a g i n a r y a s w e l l .

I n t h a t c a s e w e h a v e t h e

e x p o n e n t i a l e r r o r

w h i c h i s e q u i v a l e n t t o

p h a s e - l a g .

Theorem 1 . F o r a l l H i n th e i n t er v a l o f p e r i o d i c i ty , t h e s y m m e t r i c 2 k - s t e p m e t h o d w i t h c h a r a c -

t e r i s t i c eq u a t i o n 9 i ven b y 1 0 ) h a s p h a s e - l a 9 o r d e r q a n d p h a s e - l a g c o n s t a n t c 9 i v e n b y

_cHq+2 + O H q + 3 ) : 2 A k H ) c o s ( k H ) + . . . + 2 A j H ) co s ( j H ) + . . . + A o ( H ) ( 1 3 )

2 k 2 A k ( H ) + . . . + 2 j 2 A j ( H ) + . . . + 2 A ~ H )


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192 T.E. Simo s, P. S. Williams~Journal o f C omputational and App lied Ma thematics 79 (1997) 189-205

P r o o f . I f w e p u t s = e ~° ~/) t h e n 1 0 ) b e c o m e s

2 A k H ) c o s ( k O ( H )) + . . . + 2 A j H ) c o s ( j O ( H ) ) + . . . + A o ( H ) = 0 . 14 )

B y d e f i n i t io n t h e p h a s e - l a g o r d e r q a n d p h a s e - l a g c o n s t a n t c a r e g i v e n b y

t = H - c o s 0 H ) ) =

cH q+ +

O H q + 2 ) . 1 5 )

T h e n s i n c e

O( H) = H - t , 1 6 )

w e m a y s h o w f r o m t r i g o n o m e t r i c e x p a n s i o n s t h a t

c o s 0 H ) ) = c o s H - t ) = c o s H + c H q+2 + O H q + 3 ) , 1 7 )

s i n

( O ( H ) )

= s i n H - t ) = s i n

H

- c H q+l q-

O H q+2). 18 )

B y a n i n d u c t i v e a r g u m e n t u s i n g t h e f a m i l i a r i d e n t i t i e s

c o s

( j O ( H ) )

= c o s j - 1

) O ( H ) )

c o s

( O ( H ) )

- s in j - 1

) O ( H ) )

s i n

(O (H ) ) ,

1 9 )

s i n ( j O ( H ) ) = s i n j - 1 ) O ( H ) ) c o s ( O ( H ) ) + c o s j - 1 ) O ( H ) ) s i n (O (H ) ) . 2 0 )

I t i s n o w s t r a i g h t f o r w a r d t o s h o w t h a t

c o s ( j O ( H ) ) = c o s ( jH ) + c jZH q+2 + O H q+ 3) , 21 )

s i n

( j O ( H ) )

= s i n

( j H ) - c j H q+l +

O H q +2 ). 2 2 )

T h e n s u b s t i tu t i n g 2 1 ) a n d 2 2 ) i n t o 1 4 ) f o r j = 1 , 2 , . . . , k w i l l g i v e t h e r e s u lt .

T h e c o n v e r s e o f t h e th e o r e m m a y a l s o b e e a s il y s h o w n . T h e c o n v e r s e s t at e s t h a t i f 1 3 ) i s t r u e

t h e n t h e m e t h o d w i l l h a v e p h a s e - l a g o r d e r q a n d p h a s e - l a g c o n s t a n t c . T o p r o v e t h i s w e s u p p o s e

t h a t t h e m e t h o d h a s p h a s e -l a g o r d e r p a n d p h a s e - l a g c o n s t a n t d . T h i s m e a n s

t = H - O(H ) = d H q+ -k

O H p+I).

T h e n w e m a y s h o w b y t r i g o n o m e t r i c e x p a n s i o n s t h a t

c o s ( j O ( H ) ) = c o s ( j H ) + dj 2H p+2 + O H p+3).

S u b s t it u ti n g f o r c o s j H ) i n 1 3 ) a n d u s i n g 1 4 ) w e h a v e

k k

- 2 d H p+I ~ A j J 2 = - 2 c H q+l ~ A j J 2 +

O H q + 2 ) .

j= j=

E q u a t i n g h i g h e s t p o w e r s g i v e s

p = q a nd d = c ,

i .e . , t h e t h e o r e m i s p r o v e d . [ ]

2 3 )

24 )

2 5 )

2 6 )


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T.E. S imos, P.S. Williams/Journal of Computational and Applied M athematics 79 (1997) 189-205 193

T h e f o r m u l a p r o p o s e d f r o m t h e a b o v e t h e o r e m g i v e s u s a d i r e c t m e t h o d t o c a l c u l a t e t h e p h a s e -

l a g o f a n y s y m m e t r i c 2 k - s te p m e t h o d . I t i s o b v i o u s t h a t f o r th e s y m m e t r i c fo u r - s te p m e t h o d s t h e

p h ase - l ag o rd e r an d t h e p h ase - l ag co n s t an t a r e g i v en b y (1 3 ) w i t h k = 2 .

T h e f o r m u l a f o r t h e c a l c u l a t io n o f th e e x p o n e n t ia l e r r o r is e x a c t l y th e s a m e a s t h e f o r m u l a f o r

t h e ca l cu l a t i o n o f t h e p h ase - l ag , i .e ., f o rm u l a (1 3 ) b u t fo r w i m a g i n a r y t h e co s an d s in a r e r ep l ac ed

b y co sh an d s i n h .

3 D e r i v a t i o n o f t h e n e w m e t h o d

B a s e d o n t h e m e t h o d d e v e l o p e d b y H e n r i c i [ 19 ], w e i n tr o d u c e th e f o l l o w i n g f o u r - st e p m e t h o d

( w i t h o n e f r e e p a r a m e t e r ) :

I I

l

I I I

= y . - a h 2 ( y + 2 - 4 y , , l 6 Y n - - 4 y._ + Y . - 2 ) ,

( 2 7 )

| t 2 t ~ II I/ It It

Y.+2 - 2y.+t + 2 y. - 2 y . - i + Y.-2 = l-Tdn t~Y.+2 + 104y.+1 + 14Y~n + 10 4y .-I + 9Y .-2),

w h e r e -

Yn = f ( x . , y . ) , Y .- 1 = f ( x . _ l , Y .-1 ) , Y n - 2 = f ( X n - 2 , Y.-2) ,

. + z - - f ( x . + 2 , Y.+2), Y .+ I = f ( x . + l , Y .+ I ) ,

y = f ( x . , y . )

an d a i s a co n s t an t t o b e su i t ab l y ch o sen .

I I I I I I I I I I

A p p l y i n g t h e T a y l o r s e r i e s e x p a n s i o n s o f Y n + 2 , Y n + b Y . , Y . - 1 , Y . - 2 , Y . + z , Y n + I , Y . , Y . - 1 an d Y . -2 i n

( 2 7 ) w e h a v e t h e f o l l o w i n g r e s u lt f o r t h e l o c al tr u n c a t i o n e rr o r ( L T E ) o f th e m e t h o d ( 2 7 ) :

h 8

LT E - 36 2--40 [-95y(~8) + 3528ay(~6)F~] + O (h l° )' (2 8)

w h e r e F =

~f/~?x.

A p p l y i n g t h e n e w m e t h o d ( 2 7 ) t o t h e s c a la r te s t e q u a ti o n ( 8 ) w e h a v e t h e d i f fe r e n c e s c h e m e ( 9 )

an d t h e a s so c i a t ed ch a rac t e r i s t i c eq u a t i o n (1 0 ) w i t h

A z ( H ) = 1 + 3 H 2 + V a i l 4,

1 3 H 2 7 a l l 4, ( 2 9 )

A ~ ( H ) = - 2 + 7g - - -

A o ( H ) = 2 + 7 H 2 7 a n 4.

A p p l y i n g ( 1 3 ) in T h e o r e m 1 t o ( 2 9 ) ( w i th k = 2 ) a n d e x pa n d i n g c o s ( 2 H ) a n d c o s ( H ) v i a T a y lo r

se r i e s , w e h av e t h e fo l l o w i n g ex p re s s i o n fo r t h e p h ase - l ag :

H 7 ( 3 5 2 8 a - 9 5 ) H 9 ( 4 2 0 0 a - 12 1 )

t = +

120 960 345 600

T o h a v e m i n i m a l p h a s e - l a g

9 5

a

3 5 2 8 ~

an d fo r t h i s a t h e p h ase - l ag i s g i v en b y

8 3 H 9

t - - + O ( H t l ) .

3 628 800

T h e ab o v e an a l y s i s l e ad u s t o t h e fo l l o w i n g t h eo rem .

+ O (H 11 ) . (3 0)

31)

( 3 2 )


6/17

194 T.E. Simos, P.S. Will iams~Journal of Computational and App lied Mathematics 79 (199 7) 189-205

T h e o r e m 2.

T h e m e t h o d

2 7 )

w i t h a 9 i v e n b y

3 1 )

& a s i xt h - o r d e r m e t h o d w i t h p h a s e - l a 9 o f o r d e r

e i o h t .

T h e r e s u lt p r o d u c e d f o r m i n i m a l e x p o n e n t i a l e r r o r is e x a c t l y th e s a m e a s th a t p r o d u c e d f o r m i n i m a l

p h a s e - la g . T h e p a r a m e t e r a i s t h e sa m e a s i n 3 1 ) a n d w e r e c o v e r E q. 3 2 ) .

I n A p p e n d i x A w e p r e s en t t h e p h as e - l ag an a l y s i s o f t h e m e t h o d o f F ack e t a l. [ 1 6] . I n A p p e n d i x B

w e i n v e s ti g a t e t h e i n te r v al o f p e r i o d i c it y o f t h e p r e s e n t m e t h o d a n d o f t h e m e t h o d o f F a c k e t al . [ 1 6] .

4 N u m e r i c a l i l lu s t r a ti o n s

4 .1 . C o m p u t e r i m p l e m e n t a t i o n

C a s e i: W e m u s t n o te th a t a l th o u g h t h e s o lu t io n s o f 1 ) a r e d e f i n e d i n t h e i n te r v al - c ~ , + c ~ ) ,

t h e s e s o l u t i o n s a r e e i t h e r o f ev en o r o d d p a r i t y , i .e . , w e h av e y ( x ) - - + y ( - x ) . S o , t h e n u m er i ca l

s o l u ti o n o f 1 ) c a n b e r e s tr i ct e d i n to th e r e g i o n [ 0 , + c ~ ) . F u r t h e rm o r e , i t i s a s s u m e d t h a t t h e w a v e

f u n c t i o n s s a t i s f y t h e D i r i ch l e t b o u n d a r y co n d i t i o n y ( x ) = 0 a t s o m e x = R , t o w h i ch a v a l u e o f R i s

s p ec i f ied . F o r t h e n u m e r i ca l i n t eg r a t io n o f 1 ) t h e i n t e r v a l [ 0 , R ] i s d i v i d ed i n t o N p a r t s o f l en g t h

h = R / N ,

s u ch t h a t y 0 = y 0 ) , y l =

y ( h ) , . . . , y N = y ( N h ) = y ( R ) .

C a s e ii: F o r t h e s eco n d ca s e w e h av e t h e n u m er i ca l i n t eg r a t i o n o f 1 ) o v e r th e i n t er v a l [ - R , R ] .

This in te rva l i s d iv ide d in to N par t s o f l eng th h = 2 R / N , suc h tha t Y0 = y 0) , Yl = y ( h ) . . . . , Y N =

y ( N h ) = y ( R ) .

A s s u m i n g t h a t Yn = - y ( n h ) a n d a p p l y i n g 2 7 ) t o 1 ) w e h a v e th e f o l lo w i n g d if fe r e n ce e q u a ti o n :

12 0 - 9h2Vn+2 + 14ah4VnVn+2

120

30 + 13h2Vn+l + 7ah4VnVn+l

Y,+2 - 15 Y,+l

120 - 7h2Vn + 42 ah 4 V~

q - y ~ -

6 0

30 + 13h2Vn_l + 7ah4VnVn_l

15 Yn- 1

120

9 h 2 V n _ 2

+ 14ah4V n V,_2

120 Y . - 2

- _ h2E

9 - 14ah2 V~ + V,+2)

120

7ah2 V~ + V n+ l ) - ] - 13

Yn+2 + Y,+I

15

7 1 - 1 2 a h 2 V ~ ) 7 a h 2 V ~ V n_ l ) -'{ - 13 9 - 14ah2 V~ + V~_2)

6 0 Y + 15 Yn- 1 + 120

y 2 ]

3 3 )

7 a h 4 E 2

6 0

( Y n + 2 4 yn + l + 6 y n - - 4 y , - I +

Yn-2),

w he re Vk = V ( k h ) , k = n + 2 , n + 1, n, n - 1, n - 2.

C o n s i d e r i n g t h a t w e h av e ev en o r o d d p a r i t y w av e f u n c t i o n s , i . e. , Yk = + Y - k , i t i s e a s y to o b t a i n

t h e f o l l o w i n g d i s c re t i z a t io n o f 1 ) :

A Y = - h 2 E C a l C B y + h 4 ( E c a l c ) 2 c y 3 4 )


7/17

T.E. Simos , P.S . Wil liams~Journal of Computational and A ppl ied Mathematics 79 (1 997 ) 189-205 195

w i t h m a t r i c e s A , B , and C hav i ng t he fo l l ow i ng pen t ad i agon a l fo rm :

(SI ,bS1 ,2 ,S1 ,3)

( a , , l , a l , 2 , a l , 3 ) = (S , , , ,O ,O)

fo r e v e n pa r i t y s o l u t i o n s ,

fo r

o d d

pa r i t y

s o l u t i o n s ,

w i t h

S l , 1

120 - 7h 2 Vo + 4 2 ah 4//02

60

7ah4VoV l q- 13h2Vl + 30

8 1 , 2 z - - 2

15

- 9 h 2 V2 + 14ah4VoVz + 12

$1,3 = 60

f

Q , , Q , . 2 , 0 , . 3 )

l ,

(bl, 1, bl,2, bl,3) --

(Q I , I , 0 , 0 )

fo r

e v e n

pa r i t y

s o l u t i o n s ,

fo r o d d pa r i t y s o l u t i o n s ,

w h e r e

a l , 1

7(1 - 12ahZVo)

60

7ah2(V o + V1) + 13

, , 2 ~ 2

15

9 - 1 4 a h 2 ( V o ÷ / I 2 ) .

Q1,3 = 60

{ 4

~ 3 a , - 7 a ) f o r e v e n pa r i t y s o l u t i o n s ,

(C1,1,Cl,2, C l,3)---- ( _ 7 a , 0 , 0 ) f o r o d d pa r i t y s o l u t i o n s ,

(a2,1,a2,2,a2,3,a2.4) = 8 2 , 1 , 5 2 , 2 , 8 2 , 3 , 8 2 , 4 ) ,

w h e r e

7ah4VoVl + 13h2Vo + 30

2,1 = 15 '

8 2 , 2

120 - 7h 2 V1 + 42 ah 4

V I2

60

-9h2V 1 + 14ah4V12 + 120

120

7ah4V1112 + 13h 2 V2 + 30

8 2 3 - - - ~

1 5

-9hZV 3 + 14ah4V l V3 + 120

8 2 , 4 ~

120

b 2 , 1 , b 2 . 2 , b 2 , 3 , b 2 , 4 ) = T 1 , T 2 , Z 3 , T 4 ) ,

w h e r e

7ahZ(Vo + V1) + 13

35)

( 3 6 )

( 3 7 )

3 8 )

( 3 9 )

T 1 ~--

15


8/17

196 T.E. Simo s, P.S. Wil l iams~Journal o f Com putat ional and Appl ied Ma thematics 79 1997) 189-205

7 h 2 1 - 1 2 ah 2 V 1) + h 2 9 - 2 8 a h 2 V 1

T

6 0 1 2 0

7ah2 Vl ÷ 112)+ 13

T3 =

15

9 - 14ah2 V ~ + V 3)

T4 =

1 2 0

C 2 .1 , C 2,2 , C 2.3 , c 2 , 4 ) = 7 a , - - 7 a + 7 a , 7 a , - 7 a )

4 0 )

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

a n , . - 2 , a . , n - l , an, n,

a n , n + l , a n, n + 2 ) = G 1 , G 2 , G 3 , G 4 ,

G s ) ,

4 1 )

w h e r e

G 1 z

-9h2Vn_2 q- 14ah4VnVn_2q-- 120

1 2 0

G 2 ~ - -

7ah4V. Vn_l +

13h2V n_l + 30

15

12 0 - 7h2V~ + 42 ah 4 V~

G3 =

6 0

7ah4VnVn+l + 13h2Vn+ l + 30

G4 =

15

-9hZV~+2 + 14ahaVnV.+z + 120

G 5 ~

1 2 0

bn ,n-2 , bn , . -1 , bn , . , bn, n+ l ,

b n , n + 2 ) = H 1 , 1 -1 2, 1 -1 3 , H 4 , H 2 , 5 ) ,

w h e r e

H 1

9 - 14ah2 V ~ + V ~ _2)

1 2 0

7 a h 2 V ~ + V , _ l ) + 13

H 2 = 15

t =

7 1 - 12ahZV,)

6 0

H 4 =

7ah2 Vn

+ Vn+l) + 13

15

9 - 14ah2 V ~ + V .+2)

/ - / 5 =

1 2 0

4 2 )

¢n.n--2, Cn, n -1 , Cn , n , Cn .n l e n . . + 2 ) = - ~ o a , 7 a , - ~ o a , 7 a , - 7 a ) 4 3 )


9/17

T.E. S imos , P .S . W illiams~Journal of Computational and Applied Ma thematics 79 199 7) 189-205 197

f o r n = 3 1 ) N - 2 :

aN-1 ,N-3, aN--I ,N-2, aN-1,N--1, aN- -1, N) = F1 ,

F 2 , F 3 , F 4 ) ,

44)

w h e r e

F1 z

-9h2VN _4 n - 14ah4VN_2VN_4 q- 1 2 0

1 2 0

7ah4VN_2VN_3 q- 13h2VN_3 q-

3 0

15

F =

F3 =

1 2 0 - - 7h2V N_ 2 + 42ah4VN2_2

6 0 1 2 0

--9h2VN _2 + 14ah4VN2_2 + 1 2 0

4-

7ah4VN_2VN_I q- 13h2VN_l + 3 0

F4 =

15

bN-- I ,N--3 , b N - I , N - 2 , b N -- 1,N -- 1, b N - 1 , N ) = E 1 ,

E2

E3, E 4 ) ,

4 5 )

w h e r e

E1 z

9 - 14ah2 VN_2 + VN_4

1 2 0

7ah2 VN-2 + VN-3) + 13

15

E2 z

E3 z

7 1 - 12ah2VN_2) 9 - 28ah2VN_2

4-

6 0 1 2 0

7ah2 VN-2 + V N - I ) + 1 3

E 4 =

15

CN-I ,N-3 , CN-1,N--2, CN-I,N-1,

C N - -I ,N ) = - -7 a , 7 a , - - 7 a + 7 a , ~ a )

4 6 )

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

aN.N-2~ aN, N--l,

D 1 , 3 , D 1 , 2 , D l , le )

aN,N) = 0 , O, D l , lo)

f o r even p a r i t y solutions,

f o r o d d p a r i t y solutions,

4 7 )

w h e r e

-9h 2V N-3 q- 14ah4VN-1VN-3 q- 1 2 0

0 1 le

6 0

_ 2 7ah4Vu_lVu_2 + 13hZVu_2 + 3 0

D1

2

• 1 5

1 2 0 - - 7h2Vu_l + 42ah4V2_ 1

O1, 3 ~

6 0

1 2 0 - 7 h 2 VN- 1 + 42 ah 4 VN _ 1

D1, lo = 60


10/17

198

f

( M 1 , 3 ,

MI,2,

M l , l e )

bN, N--2, bN N -1, bN,N ) =

( 0 , O , M l , l o )

w h e r e

9 - 14ah2(VN _l + VN_3)

Ml, le = 60

2 7 a h 2 ( V u - I + V u - 2 ) + 13

M 1 , 2

15

7h2(1 - 1 2 a h 2 V N _ l )

M l , 3 =

60

7h2(1 - 1 2 a h 2 V N _ l )

M1,1o = 60

T.E. Simos, P.S. Will iams~Journal of C omputational and Applied M athematics 79 1997) 189-205

fo r e v e n p a r i t y s o l u t i o n s ,

fo r o d d p a r i t y s o l u t i o n s ,

( 4 8 )

T o t e s t th e v a l i d it y o f t h e p r o p o s e d n e w m e t h o d w e h a v e a p p l i e d t h e n e w m e t h o d t o t h e p o t e n ti a ls

( 2 ) - ( 4 ) f o r s p ec if ic c h o i c e s o f th e p a r a m e t e rs , f o r th e f ir s t c a s e. B a s e d o n t h e w o r k o f F a c k a n d

V a n d e n B e r g h e [ 15 ], w e h a v e c h o s e n t h e a p p r o p r ia t e v a l u e s o f R w h i c h a r e s h o w n i n t ab l es . F o r t h e

s e c o n d c a s e w e h a v e a p p l i e d t h e n e w m e t h o d t o t h e p o t e n t i a l s ( 5 ) - ( 6 ) . F o r c o m p a r i s o n p u r p o s e s

w e h a v e u s e d t h e e x te n d e d N u m e r o v m e t h o d o f F a c k a n d V a n d e n B e r g h e [ 16 ].

4 .2 . N u m e r i c a l r e s u lt s

{ 4

] 3a , - 7 a ) f o r e v e n p a r i t y s o l u t i o n s ( 4 9 )

(CN N--2 CN N--I CN N )= 0 , 0 , - 7 a ) f o r o d d p a r i t y s o l u t i o n s

a l l o t h e r m a t r i x e l em en t s o f A , B a n d C a r e e q u a l t o z e r o a n d Y = ( yo , Y l , . . . , y N _ l ) T.

S i n ce C i s a n o n - s i n g u l a r m a t r i x , t h e p ro b l em (3 4 ) i s eq u i v a l en t t o t h e p ro b l em

(hZECa lc ) 2

Y -

h 2 E C a l c C - 1B y - C - 1 A Y = 0• (50 )

B y i n t r o d u c i n g t h e v a r i a b le W =

h2EC I¢Y

t h e a b o v e e q u a t io n c a n b e t r a n s f o r m e d i nt o t h e s t a n d a r d

e i g e n v a l u e p r o b l e m

It i s ea sy to see tha t t h e m at r i ces A , B , C an d

[ c 7 C o

a r e n o t s y m m e t r i c • T r a n s f o r m i n g t h e m a t r i x

- C - I B I C - I A

t o a H essen b e rg fo rm , i t s e i g en v a l u e s can b e o b t a i n ed u s i n g , fo r ex am p l e , t h e w e l l k n o w n Q R

m e t h o d .


11/17

T.E. Simos, P.S. Williams~Journal of Computational and Applied Mathematics 79 1997) 189-205 199

In T ab l e 1 w e p re sen t t h e ab so l u t e e r ro r s [Ecalculated- Eexact [ fo r t h e en e rg y v a l u e s E , , n = 1 , . . . , 4

fo r th e p o t en t ia l 2 ) w i t h 2 = 7 = 0 u s i n g t h e n ew m e t h o d d ev e l o p ed i n S ec t i o n 3 an d t h e ex t en d e d

N u m e r o v m e t h o d o f F a c k a n d V a n d e n B e r g h e [ 1 6] , f o r h = 0 .1 .

In T ab l e 2 w e p re sen t t h e ab so l u t e e r ro r s IEcalculated- E exaCt] for t he en erg y va lues E , , n = 1 , . . . , 4

f o r t h e p o t e n ti a l 2 ) w i t h ) , = 7 = 0 . 1 u s i n g th e n e w m e t h o d d e v e l o p e d i n S e c t io n 3 a n d th e e x t e n d e d

N u m e r o v m e t h o d o f F a c k a n d V a n d e n B e r g h e [ 16 ], f o r h = 0 .1 .

In Ta ble 3 w e pr es en t t he ab so lut e err ors [ E Calculated - E exact [ f o r t h e e n e r g y v a l u e s E f o r t h e

p o t e n ti a l 2 ) a n d f o r v a r io u s c h o i c e s o f 2 a n d ~ u s i n g th e n e w m e t h o d d e v e l o p e d i n S e c t i o n 3

w h i c h i s p r e se n t e d a s m e t h o d [ b ] ) a n d t h e e x t e n de d N u m e r o v m e t h o d o f F a c k a n d V a n d e n B e r g h e

[1 6] w h i ch is p r e se n t ed a s m e t h o d [ a ] ) . R = 1 0 an d h = 0.1 . B y j + ) w e p re sen t t h e j t h ev en

e i g e n v a l u e , w h i l e b y j - ) w e p r e s e n t t h e j t h o d d e i g en v a l u e .

In T ab l e 4 w e p re sen t t h e ab so l u t e e r r o r s [ E c a l c u la te d E e x a c t [ f o r t h e e n e r g y v a lu e s E f o r th e

p o t e n ti a l 3 ) a n d f o r v a r i o u s c h o i c e s o f / ~ u s i n g th e n e w m e t h o d d e v e l o p e d i n S e c ti o n 3 w h i c h

is p r e s e n t e d a s m e t h o d [ b ] ) a n d th e e x t e n d e d N u m e r o v m e t h o d o f F a c k a n d V a n d e n B e r g h e [ 16 ]

w h i ch i s p r e sen t ed a s m e t h o d [ a ] ) . R = 4 an d h = 0 . 0 4.

I n T a b l e 5 w e p r e s e n t t h e a b s o l u t e

e r r o r s I E c alc ul ate d E e x a e t [

f o r th e e n e r g y v a l u e s E f o r th e

p o t e nt ia l 4 ) a n d f o r v a r io u s c h o i c e s o f n a n d m u s in g t h e n e w m e t h o d d e v e l o p e d in S e c t i o n 3

T a b l e 1

C o m p a r i s o n o f a b s o l u t e e rr o r s [ Ecalculated -EeXaet [ f or the p oten t ia l 2) wi th 2 = 7 = 0 , p rod uc ed by

t h e e x t e n d e d N u m e r o v m e t h o d o f F a c k a n d V a n d e n B e r g h e [ 16 ] a n d th e p r e se n t m e t h o d f o r h = 0 . 1 .

W e no t e t ha t R = 10 s e e [ 15 ] ) . To t a l r e a l t i m e o f c om p u t a t i on i n s ) f o r h = 0 .1

E x a c t e i g e n v a l u e s E n E x t e n d e d N u m e r o v m e t h o d [ 1 6 ]) P r e s en t m e t h o d

1 3 . 4 1 0 9 1 . 3

10 - 1°

3 3 . 0 . 1 0 - 8 1 . 7- 1 0 - 9

5 1 . 4 . 1 0 - 7 4 . 4 - 1 0 - l °

7 4 . 3 . 1 0 - 7 1 . 7 . 1 0 - 8

T o t a l r e a l t i m e o f c o m p u t a t i o n s ) 3 . 2 2 3 . 2 3

T a b l e 2

C o m p a r i s o n o f a b s o l u t e e rr o r s [ Ecalculated - ECX~t I for t he po ten tial 2) w ith 2 = y = 10.0, p ro du ce d

b y t h e e x t e n d e d N u m e r o v m e t h o d o f F a c k a n d V a n d e n B e r g h e [ 1 6 ] a n d t h e p r e s en t m e t h o d

f o r h = 0 .1 . W e n o t e t h a t R = 1 0 s e e [ 1 5 ] ) . W e a s s u m e a s e x a c t e i g e n v a l u e s t h e e i g e n v a l u e s

o b t a in e d u s i n g t h e e x t e n d e d N u m e r o v m e t h o d o f F a c k a n d V a n d e n B e rg ,he [ 16 ] w i t h R = 10 a nd

h = 0 .0 5 p r e s e n t e d i n th e p a p e r o f F a c k a n d V a n d e n B e r g h e [ 1 6] . T o t a l r ea l t i m e o f c o m p u t a t i o n

in s ) fo r h = 0 .1

E x a c t e i g e n v a l u e s E n E x t e n d e d N u m e r o v m e t h o d [ 16 ] P r e s en t m e t h o d

1 . 58 0 0 2 23 2 7 3 . 9 . 1 0 - 8 7 . 6 . 1 0 - 9

3 . 8 7 9 0 3 6 8 3 0 4 .1 - 1 0 - 8 6 . 0 . 1 0 - 9

5 .8327 67530 1 .9 - 10 - 7 3 .1 • 10 - 8

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

T o t a l r e a l t i m e o f c o m p u t a t i o n s ) 3.3 1 3 . 3 2


12/17

200

T.E. Simos, P.S. Williams~Journal of Computational and Applied Mathematics 79 1997) 189-205

T a b l e 3

Co m par ison of ab solu te errors [E calculated - Eexact[ for the pote nt ia l (2 ) for va riou s cho ices o f 2 an d 7 , prod uce d b y the

e x t e n d e d N u m e r o v m e t h o d o f F a c k a n d V a n d e n B e r g h e [ 1 6 ] ( p r e se n t e d a s [ a ] ) a n d t h e p r e se n t m e t h o d ( p r e s e n t e d a s [ b ] ) .

R - - 1 0 ( s e e [ 1 5 ] ) a n d h = R/N = 0 .1 . To ta l r e a l t ime o f comp uta t ion ( in s ) fo r h = 0 .1

) ,( j -+-) ) , Exa ct e ige nv alue s [a] [b]

- 0 . 4 2 ( 1 + ) 0 .1 0 . 8 a 1 . 4 . 1 0 - 9

- 0 .6 7 + 0.1 3V3V3X/~.05(2+) 0.1 2.3 + ~ a 6. 8. 10 -8

-0 .4 6( 1 - ) 0 . l 2 .4 a 1 .4 • 10 -8

- 0 . 7 3 + 0 . 1 ~ ( 2 - ) 0.1 3 .7 + ~ a 2 . 3 10 - 7

T o t a l r e a l t i m e o f c o m p u t a t i o n ( s ) 3 . 43

8 . 7 . 1 0 - j l

2 . 5 . 1 0 - 9

6 . 3 . 1 0 - j °

7 . 2 . 1 0 - 9

3.45

aExac t e ige nva lu e s p re s en ted in [15 ].

T a b l e 4

C o m p a r i s o n o f a b s o l u t e e r r o r s I Ecalculated E e x a c t I fo r the po ten t i a l (3 ) fo r

v a r i o u s c h o ic e s o f f l, p r o d u c e d b y t h e e x t e n d e d N u m e r o v m e t h o d o f F a c k

a n d V a n d e n B e r g h e [ 1 6] ( p r e s e n t e d a s [ a ] ) a n d t h e p r e s e n t m e t h o d ( p r e s e n t e d

as [b ] ) . R = 4 ( s ee [15 ] ) an d h =

R/N

= 0 .04 . Fo r the exac t e igenva lue s , s ee

[16 ] . To ta l r e a l t ime o f comp uta t ion ( in s ) fo r h = 0 .04

f l Pa r i ty Exac t e igenv a lue s [a] [b ]

11 E v e n - 8

0

8

13 O d d - 1 1 . 3 1 3 7 0 8 5 0 0

0

11 .313708500

15 E v e n - 1 5 . 0 7 7 5 0 8 5 1 0

- 3 . 5 5 9 3 1 6 9 4 3

3 . 5 5 9 3 1 6 9 4 3

15 .077508510

1 7 O d d - 1 9 . 1 5 8 4 1 6 0 1 0

- 5 . 7 4 0 6 5 2 9 1 6

5 . 7 4 0 6 5 2 9 1 6

19 .158416010

T o t a l r e a l t i m e o f c o m p u t a t i o n ( s )

2 .3 10 -8

4 .7 10 -8

2.8 10 -7

3.3 10 -8

1.6 10 -7

7.5 10 -7

4 .7 10 -8

1.4 10 -7

3.4 1 0 7

1.8 10 -6

6.3 10 -8

2.8 10 -7

9.5 10 7

3.8 10 -6

12.49

2 .0 1 0 9

3.7 1 0 9

1.7 10 -8

4 .4 10 -9

1.2 10 8

4 .2 10 -8

2 .6 10 -9

1.5 10 -8

2.1 10 -8

9 .9 10 -8

2 .8 10 -9

2 .9 10 -9

6.3 10 -8

2.1 10 7

12.51

( w h i c h i s p r e s e n t e d a s m e t h o d [ b ] ) a n d t h e e x t e n d e d N u m e r o v m e t h o d o f F a c k a n d V a n d e n B e r g h e

[ 1 6 ] ( w h i c h i s p r e s e n t e d a s m e t h o d [ a ]) . I n t h e ta b l e , as in p u t , w e p r e s e n t t h e a p p r o p r i a t e v a l u e s o f

R a n d h = R/N.

I n T a b l e 6 w e p r e s e n t t h e a b s o l u t e e r r o r s I E calculated - E ex act I f o r t h e e n e r g y v a l u e s E f o r t h e

p o t e n t i a l ( 5 ) u s i n g t h e n e w m e t h o d d e v e l o p e d i n S e c t i o n 3 ( w h i c h i s p r e s e n t e d a s m e t h o d [ b ] ) a n d

t h e e x t e n d e d N u m e r o v m e t h o d o f F a c k a n d V a n d e n B e r g h e [ 1 6 ] ( w h i c h i s p r e s e n t e d a s m e t h o d [ a ] ) .

R = 1 0 . 0 a n d h = 0 . 0 6 2 5 .


13/17

T.E. Simos, P.S. Williams~Journal of Computational and Applied Mathematics 79 1997) 189~05

201

T a b l e 5

C o m p a r i s o n o f a b s o l u t e

e r r o r s I E c alc ula te d E e x a c t

I fo r the po ten t i a l 4 ) fo r va r ious cho ice s o f m

a n d n p r o d u c e d b y th e e x t e n d e d N u m e r o v m e t h o d o f F a c k a n d V a n d e n B e r g h e [ 1 6 ] p r e s e n t e d

a s [ a ] ) a n d t h e p r e s e n t m e t h o d p r e s e n t e d a s [ b ] ). T h e v a l u e s o f R a n d h = R/N a re p re s en ted

in the s econd co lum n . Fo r the exac t e igenva lue s , s ee [30 ]. To ta l r e a l t ime o f com puta t io n in s )

f o r h g i v e n i n s e c o n d c o l u m n

n m R h)

Exac t e igen va lue s [a] [b]

1 1 2 . 4 8 0 . 0 2 4 8 ) - 2 1 . 3 . 1 0 - 1 °

0 4 . 6 - 1 0 - l °

2 1 2 . 4 8 0 . 0 2 4 8 ) - 2 1 + x / 2 ) 6 . 2 . 1 0 - 1 °

- 4 1 .1 • 10 -9

2 x / 2 - 1 ) 2 . 3 . 1 0 - 9

1 1 0 1 . 3 2 0 . 0 1 3 2 ) - 1 1 6 . 6 • 1 0 - 9

9 8 . 5 . 1 0 - 8

2 1 0 1 . 3 2 0 . 0 1 3 2 ) - 2 1 + l x /T - ~ ) 1 . 8 . 1 0 - 8

- 4 2 . 4 . 1 0

7

2 l ~ - 1 ) 2 . 3 . 1 0

T o t a l r e a l t i m e o f c o m p u t a t i o n s ) 8 . 33

4.2

1.3

7.4

2.5

6.8

5.7

2.3

3.2

2.2

1.5

8 .34

] 0 1 1

1 0 1 1

10-~1

10-10

10-11

1 0 1 o

10 9

10

L

1 0 - 9

1 0 - 8

T a b l e 6

Co m pa ris on of ab solu te erro rs [E Calculated - Eexact[ in 10 -6 un its for

t h e p o t e n t i a l 5 ) p r o d u c e d b y th e e x t en d e d N u m e r o v m e t h o d o f

F a c k a n d V a n d e n B e r g h e [ 1 6] p r e s e n t e d as [ a ] ) a n d t h e p r e s e n t

m e t h o d p r e s e n t e d a s [ b ] ) f o r h = 0 . 0 6 2 5 ; R = - 1 0 s e e [3 1 ] ) . F o r t h e

e x a c t e ig e n v a l u e s , se e [ 1 , 2 8 ] . T o t a l r e a l ti m e o f c o m p u t a t i o n i n s )

for h = 0 .0625

Exac t e igen va lue s [a] [b ]

E o = - 1 7 8 . 7 9 8 5 3 8 3 5 3 0 0

E 4 = - 1 1 0 . 8 0 8 5 7 2 0 7 4 9 21 4 9

E 8 = - 5 9 . 0 0 6 5 6 4 7 5 8 2 5 3 6 0 2 25

E l 2 = - - 2 3 . 3 9 2 5 1 6 4 0 1 4 2 2 6 1 3 8 5

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

T o t a l r e a l t i m e o f c o m p u t a t i o n s ) 6 . 4 3 6 . 4 5

In T ab l e 7 w e p re sen t t h e ab so l u t e

e rr o rs ]E calculated E

. . . . t I f o r t h e en e rg y v a l u e s E fo r t h e

p o t e nt ia l 6 ) u s i n g t h e n e w m e t h o d d e v e l o p e d in S e c t io n 3 w h i c h i s p r e s e n t e d a s m e t h o d [ b ] ) a n d

t h e e x t e n d e d N u m e r o v m e t h o d o f F a c k a n d V a n d e n B e r g h e [ 1 6 ] w h i c h i s p r e s e n t e d a s m e t h o d [ a ]) .

R---- 15.0 and h = 0.125.

F r o m t h e a b o v e - m e n t i o n e d r e s u lt s it i s e a s y t o s e e t h a t t h e n e w m e t h o d i s m u c h m o r e a c c u r a te

c o m p a r e d w i t h t h e e x t e n d e d N u m e r o v m e t h o d o f F a c k a n d V a n d e n B e r g h e [ 1 6 ].


14/17

202 T.E. Simos, P.S . Wil liams~Journal o f Computational and Appl ied Mathem atics 79 1997) 189-205

T a b l e 7

C o m p a r i s o n o f a b s o l u t e

e r r o r s [ E c alc ula te d E e x a c t [

in

1 0 9

uni t s f or the

p o t e n ti a l 6 ) p r o d u c e d b y t h e e x te n d e d N u m e r o v m e t h o d o f F a c k a nd

V a n d e n B e r g h e [ 1 6 ] p r e s e n t e d a s [ a ]) a n d t h e p r e s e n t m e t h o d p r e s e n t e d

as [ b] ) f or h = 0 .125; R = 15 . s ee [ 31] ) . For the exac t e igenv a lues see

[ 1]. Tota l r ea l t ime o f com puta t ion in s ) f o r h = 0 .125

Exac t e igenv a lues [ a] [ b]

E 0 = - 4 9 . 4 5 7 7 8 8 7 2 8

E 4 = - 4 1 . 2 3 2 6 0 7 7 7 2

E 9 = - 2 2 . 5 8 8 6 0 2 2 5 7

E 1 3 = - 3 . 9 0 8 2 3 2 4 8 0 8

T o t a l r e a l t i m e o f c o m p u t ~ i o n s )

0 0

8315 307

737 060 47 876

5 1 5 5 1 8 0 5 6 3 2 4 1

9.67 9.69

5 . C o n c l u s i o n s

I n t h i s p a p e r a n e w a p p r o a c h t o m e t h o d s f o r t h e n u m e r i c a l s o l u t i o n o f s o m e s p e c i f i c S c h r r d i n g e r

eq u a t i o n s i s d ev e l o p ed . T h i s ap p ro ach i s b a sed o n t h e m i n i m i za t i o n o f t h e p h ase - l ag . U s i n g t h i s ,

w e h a v e g i v e n a d i r e c t f o r m u l a to f i n d th e p h a s e - l a g o f a s y m m e t r i c fo u r - s te p m e t h o d a n d f r o m

t h i s fo rm u l a w e h av e co n s t ru c t ed a s i m p l e fo u r - s t ep m e t h o d . T h e n u m er i ca l r e su l t s i n d i ca t e t h a t

t h e n e w m e t h o d i s m u c h m o r e a c c u r a t e t h a n t h e e x t e n d e d N u m e r o v m e t h o d o f F a c k a n d V a n d e n

B erg h e [1 6] fo r a v a r i e t y o f p o t en t i al s . T h e t h eo re t i c a l r e a so n fo r th i s i s t h a t w e h a v e a m e t h o d

w i t h p h a s e - la g o f o r d e r e i g h t w h i l e th e m e t h o d o f F a c k e t a l . is a m e t h o d w i t h p h a s e - l a g o f o r d e r

s ix . T h i s i s th e f i r st t i m e t h a t t h e p h ase - l ag i s ap p l i ed to t h e c o n s t ru c t i o n o f a m a t r i x m e t h o d . W e

n o t e h e r e th a t f o r t h e c o m p u t a t io n o f t h e e i g e n v a lu e s in t h e c a s e o f th e M o r s e p o t e n ti a l 5 ) a n d

t h e W o o d s - S a x o n p o t e n t ia l 6 ) t h e r e a re , a ls o , o th e r n u m e r i c a l m e t h o d s w h i c h g i v e v e r y a c c u r a t e

r e su l ts s e e [ 3 7 , 2 0 - 2 2 ] ) . T h e s e m e t h o d s a r e b a s e d o n t h e s h o o t in g t e c h n iq u e , w h i c h i s c o m p l e t e l y

d i f f e r e n t f r o m t h e t e c h n i q u e u s e d b y t h e p r o p o s e d m a t r i x m e t h o d .

H o w ev e r , i t i s c l e a r t h a t w e can o b t a i n m u c h m o re accu ra t e r e su l ts i f t h e p h ase - l ag can b e

m i n i m i s e d e v e n m o r e . T h i s is p o ss i b le i f w e i n c lu d e o t h e r la y e r s in t h e m e t h o d 2 7 ) .

A c k n o w l e d g e m e n t s

T h e a u t h o rs w i s h t o t h a n k a n a n o n y m o u s r e f e r e e f o r h i s c a r e f u l r e a d i n g o f th e m a n u s c r i p t a n d

h i s f ru i t fu l co m m en t s an d su g g es t i o n s .

A p p e n d i x A . P h a s e l a g a n a l y s i s o f t h e m e t h o d o f F a c k e t a l . [1 6 ]

A p p l y i n g t h e m e t h o d o f F ack e t a l. [ 1 6 ] t o th e s ca l a r t e s t eq u a t i o n 8 ) w e h av e t h e d i f f e ren ce

eq u a t i o n 9 ) fo r k = 2 ) w i t h t h e a s so c i a t ed ch a rac t e r i s ti c eq u a t i o n 1 0 ) w i t h

1 2

A 2 H ) = - I ~ - r H ,


15/17

T.E. Sim os, P .S . W illiams~Journal of Computational and Applied Ma thematics 79 199 7) 189-205 203

( A . 1 )

I H ) = 1 + ~ H 2,

A o H ) - - -2 + ~ 2 oH2.

I f w e a p p l y th e m e t h o d o f F a c k e t a l. [ 1 6] w i t h c h a r a c te r is t ic e q u a t io n g i v e n b y ( 1 0 ) - ( 1 3 ) , w h e r e

A i H ) , i

= 0 ( 1 ) 2 a r e g iv e n b y ( A . 1 ) a n d e x p a n d i n g c o s ( 2 H ) a n d c o s ( H ) v i a T a y l o r se ri es , w e h a v e

t h e fo l l o w i n g ex p re s s i o n fo r t h e p h ase - l ag :

3 1 H 7

t - - - - + O (H 9) . (A .2 )

120 960

S o , th e m e t h o d o f F ack e t al . [ 1 6] i s a s i x t h -o rd e r m e t h o d w i t h p h ase - l ag o f o rd e r s ix .

Appendix B. Interval of periodicity

C o n s i d e r a s y m m e t r i c f o u r - s te p m e t h o d f o r th e n u m e r i c a l s o lu t io n o f ( 7 ) . A p p l i c a t io n o f t h is

m e t h o d t o th e s ca l a r t e s t eq u a t i o n (8 ) l e ad s t o t h e d if f e r en ce eq u a t i o n (9 ) w i t h k - 2 w h i ch i s

a s so c i a t ed w i t h t h e ch a rac t e r i s t i c eq u a t i o n (1 0 ) .

S u b s t it u ti n g in ( 1 0 ) ( w i t h k = 2 ) s = ( 1 + z ) / ( 1 - z ) w e h a v e t h a t

( 2 A 2 ( H ) - 2 A f f H ) + A o H ) ) z 4 + 2 ( 6 A 2 ( H ) - A o H ) ) z 2 + 2 A 2 ( H ) + 2 A I ( H ) + Ao H) = O.

( B . 1 )

F r o m B .1 ) w e h a v e t h e p o l y n o m i a l s

P f f H )

= 2 A 2 ( H ) - 2 A I ( H ) +

A o H ) ,

P 3 ( H ) = 2 A 2 ( H ) + 2 A , ( H ) + A o H ) .

P f f H )

= 1 2 A f f H ) - 2 A 0 ( H ) ,

( B . 2 )

Theorem B.1 . A l l sym m etr i c fo u r - s t ep m e th o d s h a ve a n o n em p ty in t erva l o f p er io d ic it y O ,Hg) if,

f o r a l l H E [0, H0],

P ~ (H )> ~ 0 , i = 1 (1 )3 , a nd N H ) = P 2 H ) : - 4 P I H ) P 3 H ) > ~O . ( B . 3 )

P r o o L F r o m D e f i n it io n 1 a n d w i t h s = e x p ( i 0 ) w e o b t a i n

z = i t an (0 / 2 ) , z 2


16/17

204 T . E . S im o s , P . S . W i l l ia m s ~ J o u r n a l o f C o m p u t at io n a l a n d A p p l i e d M a t h e m a t i c s 7 9 ( 1 9 9 7 ) 1 8 9 ~ 0 5

R e f e r e n c e s

[ ]

[ 2 ]

[ 3 ]

[ 4 ]

[ 5 ]

[ 6 ]

[ 7 ]

[ 8 ]

[ 9 ]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[ 1 8 ]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

[ 2 8 ]

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L 9 7 - L I 0 1 .

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T.E. Simos, P.S. Will iams~Journal of Computational and Applied Mathematics 79 1997 ) 189-205 2 0 5

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v a l u e p r o b l e m s , B I T 3 1 1 9 9 1 ) 1 6 0 -1 6 8 .

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[3 9 ] R .R . Wh i t e h e a d , A . W a t t , G .P . F l e ssa s a n d M.A . N a g a ra j a n , Ex a c t so lu t i o n s o f t h e Sc h r6 d in g e r e q u a t io n - d 2 / d x 2 ) +

x 2 + )~x2/ 1 + gx 2) )y x )= E y x ) , J. Phys. A: Math. Gen. 1 5 1 9 8 2 ) 1 2 1 7 -1 2 2 6 .

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