On the fast and accurate computer solution of partial differential systems

ON THE FAST AND ACCURATE COMPUTER

SOLUTION OF PARTIAL DIFFERENTIAL

SYSTEMS

Michael T. Hill

A Thesis Submitted for the Degree of PhD

at the

University of St Andrews

1974

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D E C L A R A T I O N S

I H E R E B Y D E C L A R E T H A T T H I S T H E S I S H A S B E E N C O M P O S E D B Y M Y S E L F ,

T H A T T H E W O R K O F W H I C H I T I S A R E C O R D H A S B E E N D O N E B Y M Y S E L F , A N D

T H A T I T H A S N O T B E E N A C C E P T E D I N A N Y P R E V I O U S A P P L I C A T I O N F O R

A H I G H E R D E G R E E .

M I CHAEL T. HILL

T H E R E S E A R C H W A S U N D E R T A K E N F U L L T I M E A T T H E V O N K A R M A N I N S T I T U T E

D U R I N G T H E P E R I O D O C T O B E R 1 9 7 3 T O D E C E M B E R 1 9 7 6 , A N D T H E S T U D E N T

W A S M A T R I C U L A T E D A S A F U L L T I M E R E S E A R C H S f U D E N T A T T H E

U N I V E R S I T Y O F S T . A N D R E W S F O R T H E P E R I O D O C T O B E R 1 9 7 4 - A U G U S T 1 9 7 9 .

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ON T H E F A S T A N D A C C U R A T E C O M P U T E R S O L U T I O N

OF P A R T I A L D I F F E R E N T I A L S Y S T E M S

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M i c h a e l T. Hill

B. S c . ( H o n s . ) A p p l i e d M a t h s , U n i v e r s i t y of St. A n d r e w s (1973)

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1. G E N E R A L I N T R O D U C T I O N ... 1

1.1 I n t r o d u c t i o n and h i s t o r i c a l s u r v e y ... 1

1.2 I n t r o d u c t i o n to h i g h e r o r d e r m e t h o d s ... 5

2. D E S C R I P T I O N O F T H E M E T H O D ... 8

2 . 0 I n t r o d u c t i o n ... 8

2 . 1 A p p l i c a t i o n to o r d i n a r y d i f f e r e n t i a l e q u a t i o n s . . 11

2. 2 S o m e r e s u l t s w i t h t h e m e t h o d ... .. . 25

3. A P P L I C A T I O N TO P A R T I A L D I F F E R E N T I A L E Q U A T I O N S . . . , . 27

3. 0 I n t r o d u c t i o n ... 27

3 . 1 P a r a b o l i c e q u a t i o n s . . . ...27

3 . 2 E l l i p t i c e q u a t i o n s ... 37

3 . 3 H y p e r b o l i c e q u a t i o n s ... 39

3 . 4 S u m m a r y ...45

4. F A S T M E T H O D S ...48

4 . 0 D i s c u s s i o n ... 48

► 5. A R E L A X A T I O N M E T H O D F O R H Y P E R B O L I C E Q U A T I O N S ... 51

5. 0 I n t r o d u c t i o n ... 51

5 . 1 R e l a x a t i o n a p p l i e d to d i f f e r e n t i a l e q u a t i o n s . . . 53

5 . 2 R e l a x a t i o n a p p l i e d to d i f f e r e n c e s c h e m e s ...68

5 . 2 . 1 F i r s t o r d e r d i f f e r e n c i n g ... 69

^ 5 . 2 . 2 L a x ' s m e t h o d ... . 74

# 5 . 2 . 3 T h e L a x - W e n d r o f f m e t h o d ... 79

5 . 2 . 4 “ E u l e r " d i f f e r e n c i n g ... 83

. 5 . 2 . 5 A semi - i m p l i c i t s c h e m e ... 85

%

1

* 6. C O N C L U S I O N S ... 89

R E F E R E N C E S ... 90

TABLE OF CONTENTS

A B S T R A C T

T w o m e t h o d s are p r e s e n t e d for use on an e l e c t r o n i c c o m p u t e r for t h e s o l u t i o n of p a r t i a l d i f f e r e n t i a l s y s t e m s .

T h e f i r s t is c o n c e r n e d w i t h a c c u r a t e s o l u t i o n s Of d i f f e r e n t i a l e q u a t i o n s . It is e q u a l l y a p p l i c a b l e to o r d i n a r y d i f f e r e n t i a l e q u a t i o n s and p a r t i a l d i f f e r e n t i a l e q u a t i o n s , and c a n be u s e d for p a r a b o l i c , h y p e r b o l i c or e l l i p t i c s y s t e m s , and a l s o for n o n - l i n e a r a n d m i x e d s y s t e m s . It can be u s e d in c o n j u n c t i o n w i t h e x i s t i n g s c h e m e s . C o n v e r s e l y , t h e m e t h o d can be u s e d as a v e r y f a s t m e t h o d of o b t a i n i n g a r o u g h s o l u t i o n of t h e s y s t e m . It has an a d d i t i o n a l a d v a n t a g e o v e r t r a d i t i o n a l h i g h e r o r d e r m e t h o d s in t h a t it d o e s n o t r e q u i r e e x t r a b o u n d a r y cond i ti o n s .

T h e s e c o n d m e t h o d is c o n c e r n e d w i t h th e a c c e l e r a t i o n of t h e c o n v e r g e n c e r a t e in t h e s o l u t i o n of h y p e r b o l i c s y s t e m s . T h e n u m b e r of i t e r a t i o n s has b e e n r e d u c e d f r o m t e n s of t h o u s a n d s w i t h t h e t r a d i t i o n a l L a x - W e n d r o f f m e t h o d s to t h e o r d e r of t w e n t y i t e r a t i o n s .

A n a l y s e s f o r b o t h t h e d i f f e r e n t i a l and the d i f f e r e n c e s y s t e m s a r e p r e s e n t e d . A g a i n t h e m e t h o d is e a s i l y a d d e d to e x i s t i n g p r o g r a m s .

ACKNOWLEDGEMENTS

T h e a u t h o r m u s t t h a n k :

the S c i e n c e R e s e a r c h C o u n c i l and t h e von K a r m a n I n s t i t u t e f o r f u n d i n g p a r t s of t h i s r e s e a r c h , P r o f e s s o r H-J W i r z f o r s u p e r v i s i n g t h e w o r k , and M m e T o u b e a u

1. G E N E R A L I N T R O D U C T I O N

1.1 I n t r o d u c t i o n and h i s t o r i c a l s u r v e y

1

-T h e d i s c i p l i n e n o w k n o w n as " c o m p u t a t i o n a l f l u i d d y n a ­ m i c s " a p p e a r s to h a v e as its d a t e o f b i r t h the y e a r 1910 * w h e n L.F. R i c h a r d s o n p r e s e n t e d his h i s t o r i c a l p a p e r (1) to the Royal S o c i e t y . R i c h a r d s o n u s e d as his e x a m p l e s the i t e r a t i v e s o l u t i o n of L a p l a c e ' s e q u a t i o n , t h e b i h a r m o n i c e q u a t i o n and o t h e r s . His

" c o m p u t e r s " w e r e b o y s , b e i n g p a i d at a r a t e p r o p o r t i o n a l to the n u m b e r o f c o r r e c t c a l c u l a t i o n s c a r r i e d out. He c o m b i n e d all of his p r o p o s e d m e t h o d s i n t o a l a r g e s c a l e p r a c t i c a l e x a m p l e , but, b e c a u s e of t h e sm al l n u m b e r o f t i m e s t e p c a l c u l a t i o n s w h i c h w e r e p e r f o r m e d , the i n s t a b i l i t y p r e s e n t in o n e of his p r o c e d u r e s was n o t d i s c o v e r e d at t h e ti me . T h i s f a c t s e r v e s to h i g h l i g h t the

s t a t e of a f f a i r s in t h a t n u m e r i c a l m e t h o d s and t h e i r c o r r e s p o n d i n g s t a b i l i t y a n a l y s e s m u s t p r o c e e d t o g e t h e r .

In 1 9 2 8 C o u r a n t , F r i e d r i c h s and L e w y p u b l i s h e d t h e i r c l a s s i c a l p a p e r (2) in w h i c h t h e y e s t a b l i s h e d c e r t a i n e x i s t e n c e t h e o r e m s and u n i q u e n e s s t h e o r e m s f o r p a r t i a l d i f f e r e n t i a l s y s t e m s . A l t h o u g h t h e a u t h o r s w e r e p r i m a r i l y i n t e r e s t e d in u s i n g f i n i t e d i f f e r e n c e f o r m u l a t i o n s as a tool f o r p u r e m a t h e m a t i c s , t h e i r w o r k has s i n c e b e c o m e t h e " c o r n e r s t o n e " for m o d e r n p r a c t i c a l f i n i t e d i f f e r e n c e s o l u t i o n s .

-- 2 ~

t y pe " e l l i p t i c a l p r o b l e m s ( w h e r e t h e p r o b l e m is s o l v e d o v e r the w h o l e f i e l d s i m u l t a n e o u s l y ) . T h e f i r s t s t u d y of the n u m e r i c a l s o l u t i o n of a v i s c o u s f l u i d d y n a m i c s p r o b l e m w a s p r e s e n t e d by T h o m (3) in 1933. He s t u d i e d the v i s c o u s f l o w a r a u n d - c i r c u 1 ar c y l i n d e r , f o r w h i c h s o m e a n a l y t i c a l s o l u t i o n s are p o s s i b l e for c o m p a r i s o n .

In 1 9 3 8 S h o r t l e y and W e l l e r (4) p r e s e n t e d w h a t w a s b a s i c a l l y an i m p r o v e d v e r s i o n of R i c h a r d s o n ' s m e t h o d , u s i n g o v e r r e l a x a t i o n . T h e y a l s o i n c l u d e d , f o r t h e f i r s t t i m e , an i d e n t i ­ f i c a t i o n and a n a l y s i s of t h e c o n v e r g e n c e r a t e s .

D u r i n g t h e S e c o n d W o r l d W a r , J. v o n N e u m a n n and o t h e r s at the Los A l a m o s S c i e n t i f i c L a b o r a t o r y in the U S A had d o n e m u c h w o r k on t h e d e v e l o p m e n t of t h e f i r s t e l e c t r o n i c c o m p u t e r s , and t h e i r a p p l i c a t i o n i n i t i a l l y to b a l l i s t i c p r o b l e m s . A c o n s i d e r a b l e a m o u n t of e f f o r t w a s g i v e n to t h e c o n s i d e r a t i o n of c o n v e r g e n c e , n u m e r i c a l s t a b i l i t y and t h e u n i q u e n e s s of the s o l u t i o n s . M u c h of the w o r k c a r r i e d o u t d u r i n g t h e w a r w a s c l a s s i f i e d as s e c r e t , and, in 1 9 46 , S o u t h w e l l (5) p r e s e n t e d a r e l a x a t i o n m e t h o d w h i c h c l e a r l y f u l f i l l e d two a i m s . F i r s t l y , it o b t a i n e d a b e t t e r r a t e of c o n v e r g e n c e , and s e c o n d l y , it s u c c e e d e d in m a k i n g the w o r k m o r e i n t e r e s t i n g for the h u m a n c o m p u t e r s . T h i s l a t t e r w a s b e c a u s e t h e y h a d to s c a n t h e c o m p u t a t i o n a l m e s h fo r the l a r g e s t r e s i d u a l ( s ) and u p d a t e t h e s o l u t i o n a c c o r d i n g l y . In f a c t , t h i s a d v a n t a g e

b e c o m e s a d i s a d v a n t a g e w h e n t h e m e t h o d is a p p l i e d on e l e c t r o n i c c o m p u t e r s , b e c a u s e s c a n n i n g t h e g r i d of m e s h - p o i n t s w o u l d t a k e l o n g e r t h e n t h e a r i t h m e t i c i n v o l v e d . T h u s , e l e c t r o n i c d i g i t a l

. of t h e L i e b m a n n v a r i a t i o n o f R i c h a r d s o n ' s m e t h o d . In 1 9 5 0 F r a n k e l (6) p r e s e n t e d the m e t h o d n o w k n o w n as " s u c c e s s i v e o v e r r e l a x a t i o n " w h i c h is st il l w i d e l y u s e d f o r e l l i p t i c p r o b l e m s .

As t h e e l e c t r o n i c c o m p u t e r b e g u n to b e c o m e a v a i l a b l e so t h e c e n t r e of i n t e r e s t m o v e d f r o m e l l i p t i c ( u s u a l l y s t e a d y -s t a t e ) p r o b l e m -s to p a r a b o l i c (time d e p e n d e n t ) p r o b l e m -s . T h i -s w a -s b e c a u s e it b e c a m e f e a s i b l e to a t t e m p t t i m e d e v e l o p m e n t p r o b l e m s . T h e m o s t we ll k n o w n o f t h e m a n y p a r a b o l i c m e t h o d s to be p u b l i s h e d in t h e 1 9 4 0 s w a s in t h e C r a n k - N i c o l son p a p e r (7) p u b l i s h e d in 1947. A l t h o u g h t h i s (C r a n k - N i c o l son m e t h o d ) is still us ed in

m a n y m e t h o d s p u b l i s h e d at t h e p r e s e n t t i m e , it us es a v e r y s i m p l e f o r m u l a fo r c a l c u l a t i n g t h e n e x t tirne-like s t e p , and is p e r h a p s b e t t e r s u i t e d to h u m a n c o m p u t e r s t h a n to e l e c t r o n i c d i g i t a l c o m p u t e r s . T h i s s i t u a t i o n m a y b e c o m e reversed, w h e n p a r a l l e l p r o ­ c e s s i n g b e c o m e s w i d e l y a v a i l a b l e .

In the e a r l y 1 9 5 0 s , t h e w a r t i m e w o r k c a r r i e d o u t at Los A l a m o s b e g a n to be p u b l i s h e d , i n c l u d i n g von N e u m a n n ' s (8) f a m o u s c r i t e r i o n for s t a b i l i t y of p a r a b o l i c f i n i t e d i f f e r e n c e e q u a t i o n s , and a m e t h o d of a n a l y s i n g a l i n e a r i s e d s y s t e m .

4

-A l t h o u g h the f u n d a m e n t a l p a p e r d e a l i n g w i t h h y p e r b o l i c s y s t e m s h a d b e e n p r e s e n t e d as e a r l y as 1 9 2 8 (by C o u r a n t , F r i e d r i c h s and L e w y ) , l i t t l e w o r k a p p e a r s to h a v e b e e n d o n e on t h e m un ti l

t h e m i d 1 9 50 s. T h e 1 9 2 8 p a p e r p r e s e n t e d a n e c e s s a r y s t a b i l i t y c r i t e r i o n , t h a t the f i n i t e d i f f e r e n c e d o m a i n of d e p e n d e n c e m u s t i n c l u d e th e c o n t i n u o u s ( i . e . , d i f f e r e n t i a l ) d o m a i n of d e p e n d e n c e T h e s i m p l e s t and e a r l i e s t m e t h o d f o r h y p e r b o l i c e q u a t i o n s is t h a t of L a x (11) in 1954. T h i s w a s i m p r o v e d in t h e L a x - W e n d r o f f (12) m e t h o d ( 1 9 6 0 ) , i m p r o v e d a g a i n in tw o s t e p m e t h o d s s u c h as

R i c h t m y e r ' s (13) (1 9 6 3 ) a n d M a c C o r m a c k 1s (14) ( 1 9 6 9 ) . L a x ' s 1 9 5 4 p a p e r a l s o p r e s e n t e d an a r g u m e n t f o r w r i t i n g the s y s t e m in c o n s e r v a t i o n or d i v e r g e n c e f o r m - t h a t is r e t u r n i n g to t h e p h y s i c s o f the p r o b l e m and w r i t i n g t h e e q u a t i o n s by a n a l o g y to N e w t o n ' s l a w o f c o n s e r v a t i o n of m a s s , m o m e n t u m and e n e r g y r a t h e r t h a n e x p a n d i n g th e t e r m s to a " n e a t'1 m a t h e m a t i c a l f o r m u l a t i o n . T h i s f o r m is e s p e c i a l l y i m p o r t a n t in c a l c u l a t i o n s i n v o l v i n g s h o c k s .

T a y l o r (15) r e m a r k s t h a t the m a j o r i t y o f the c u r r e n t

w o r k d o n e on h y p e r b o l i c s y s t e m s is b e i n g c a r r i e d o u t at Los A l a m o s , w h e r e " a l m o s t u n l i m i t e d r e s o u r c e s are a v a i l a b l e " . T h e s e i n c l u d e t h e £ a r t i c l e £ n £ e l l ( P I C ) m e t h o d of H a r l o w and o t h e r s (16) and th e j i x p lo si ve

l

_n £ e l l ( E I C ) m e t h o d of M a d e r ( 17) ( 1964 ). T h e s e arid o t h e r m e t h o d s i n c l u d e t r e a t m e n t of f l u i d / f l u i d b o u n d a r i e s as w e l l as d i s c o n t i n u i t i e s .

5

-p r e s e n t e d h e r e i n use t h e c o m -p u t e r to a g r e a t e r b e n e f i t d u r i n g the s o l u t i o n of t h e s y s t e m .

T h e m e t h o d p r e s e n t e d in t h e f i r s t p a r t of th is p a p e r a l l o w s t h e c o m p u t e r to s o l v e a g i v e n p a r t i a l d i f f e r e n t i a l s y s t e m ( w h e r e s y s t e m is u n d e r s t o o d to i n c l u d e a set o f p a r t i a l d i f f e r e n ­ tial e q u a t i o n s and t h e n e c e s s a r y a n d s u f f i c i e n t b o u n d a r y c o n d i t i o n s ) to a n y r e q u i r e d d e g r e e of a c c u r a c y . P a r t of t h i s d e s c r i p t i o n has b e e n p u b l i s h e d p r e v i o u s l y by t h e c u r r e n t a u t h o r (18). In the s e c o n d p a r t of t h i s p a p e r a m e t h o d is p r o p o s e d w h i c h will s o l v e h y p e r b o l i c partial. d*if t e r e n t i al s.ystems up to two o r d e r s of m a g n i t u d e f a s t e r t h a n e x i s t i n g m e t h o d s . T h e o b v i o u s c o m b i n a t i o n of t h e t w o m e t h o d s t h e r e f o r e p r e s e n t s a v e r y f a s t and v e r y a c ­ c u r a t e s o l u t i o n of p a r t i a l d i f f e r e n t i a l s y s t e m s .

1.2 I n t r o d u c t i o n to h i g h e r o r d e r m e t h o d s

H i g h e r o r d e r m e t h o d s ( H O M s ) a r e of i n t e r e s t n o t o n l y as s u c h , b u t a l s o b e c a u s e of th e f a c t t h a t an i m p r o v e m e n t in a c ­ c u r a c y w i l l l e a d to a s i t u a t i o n w h e r e l a r g e r g r i d s t e p s can p r o d u c e t h e s a m e ( l o w e r ) a c c u r a c y . T h e r e f o r e a b a l a n c e b e t w e e n a c c u r a c y and g r i d s i z e w i l l l e a d to an o p t i m i s e d m e t h o d .

F o r e l l i p t i c e q u a t i o n s , c l a s s i c a l l y s e c o n d o r d e r d i f f e r e n c i n g is c o n s i d e r e d a d e q u a t e , g i v i n g ( i n t w o d i m e n s i o n s ) a s i m p l e f i v e p o i n t d i f f e r e n c e f o r m u l a . (The “ M e h r s t e l 1 en"

of 1 0~ 6 , a g r i d of 1000 x 1 00 0 p o i n t s is r e q u i r e d , t h a t is to say, a m e s h s i z e of 1 0 - 3 . If e a c h i t e r a t i o n on e a c h g r i d p o i n t r e q u i r e s o n l y 3 m u l t i p l i c a t i o n s ( i g n o r i n g a d d i t i o n s ) , t h a t g i v e s a to ta l of

o f 3 x 1 06 p e r i t e r a t i o n , and so 33 i t e r a t i o n s g i v e an u n r e a s o n a ­ ble 1 08 m u l t i p l i c a t i o n s . A f o u r t h o r d e r m e t h o d w o u l d r e q u i r e a gr i d o f 30 x 30 p o i n t s , a g r i d s i z e of .03 g i v i n g 900 g r i d p o i n t s and 8 9 . 0 0 0 m u l t i p l i c a t i o n s for 33 i t e r a t i o n s . C l e a r l y a s i x t h o r d e r m e t h o d n e e d s o n l y a 10 x 10 g r i d , 3 0 0 m u l t i p l i c a t i o n s p e r i t e r a t i o n

or 1 0 . 0 0 0 for 33 i t e r a t i o n s . C l e a r l y t h i s w o u l d r e d u c e the c o m ­ p u t a t i o n t i m e f r o m h o u r s to s e c o n d s .

Th e s o l u t i o n is u n f o r t u n a t e l y n o t as s i m p l e as th is . One of the p r o b l e m s e n c o u n t e r e d in u s i n g h i g h e r o r d e r m e t h o d s is t h a t t h e m a j o r i t y of t h e m n e e d a d d i t i o n a l b o u n d a r y c o n d i t i o n s . T h i s f a c t o f t e n p r o v e s to be a m a j o r h u r d l e , as t h e s e a d d i t i o n a l

b o u n d a r y c o n d i t i o n s m u s t be c o m p a t i b l e w i t h t h e (as y e t u n k n o w n ) s o l u t i o n s . T h i s is a n u m e r i c a l e f f e c t and u s u a l l y has l i t t l e to do w i t h t h e p h y s i c s of t h e p r o b l e m - s a v e in t h e c o n s e r v a t i o n of m a t t e r , m o m e n t u m and e n e r g y . T h i s e f f e c t can e a s i l y be d e m o n s t r a t e d by c o n s i d e r i n g a s i m p l e e x a m p l e . T h e e q u a t i o n

6

-h a s a s o l u t i o n : t

u = u0 e .

7

-V l ’ uj = At uj

i s

u j + l = u j + A t >

and o n c e a g a i n if u0 is g i v e n t h e n t h e p r o b l e m is c o m p l e t e l y s o l v e d . W h e n s o l v e d n u m e r i c a l l y w i t h a s e c o n d o r d e r m e t h o d s u c h as a c e n t r e d d i f f e r e n c e , t h e n t h e d i f f e r e n c e e q u a t i o n is

Uj+ 1 ~ Uj “ l = 2At UJ

2. D E S C R I P T I O N OF T H E M E T H O D

2 . 0 I n t r o d u c t i on

In t h i s s e c t i o n a m e t h o d is d e s c r i b e d for a c h i e v i n g a n u m e r i c a l s o l u t i o n o f a d i f f e r e n t i a l e q u a t i o n to a n y o r d e r of a c c u r a c y . It is c a p a b l e of g i v i n g a d e g r e e of a c c u r a c y b o u n d e d o n l y by t h e a c c u r a c y of t h e c o m p u t e r u s e d , if n e c e s s a r y , b u t is m o r e l i k e l y to be u s e d as a m e t h o d of o b t a i n i n g f o u r t h or s i x t h o r d e r a c c u r a c y in p l a c e of f i r s t or s e c o n d o r d e r p r e v i o u s l y .

A c o r o l l a r y o f a h i g h e r o r d e r m e t h o d is a m e t h o d y i e l d i n g the s a m e ( l o w ) a c c u r a c y w i t h f e w e r nodal p o i n t s - t h a t is w i t h less c o m p u t a t i o n a l e f f o r t .

T h e m e t h o d is e a s i l y a p p l i e d to all t y p e s of p a r t i a l d i f f e r e n t i a l e q u a t i o n s , b u t in o r d e r to m a k e its u t i l i s a t i o n c l e a r , f i r s t its u s e is d e m o n s t r a t e d by s o l v i n g an o r d i n a r y d i f f e r e n t i a l e q u a t i o n .

To a v o i d t h e u s e of a c o m p u t e r i n i t i a l l y , a s i m p l e l i n e a r s e c o n d o r d e r o r d i n a r y d i f f e r e n t i a l e q u a t i o n is t a k e n , w h i c h has a w e l l k n o w n a l g e b r a i c s o l u t i o n . T h i s e q u a t i o n can be r e d u c e d in d i s c r e t e v a r i a b l e s to a s i m p l e l i n e a r s e c o n d o r d e r d i f f e r e n c e e q u a t i o n , a l s o w i t h k n o w n s o l u t i o n .

9

-( 1 )

d x2

t o g e t h e r w i t h t h e b o u n d a r y c o n d i t i o n s

f.‘(x = 0) = 0

f (x = 4 ) = i

(

2

)

T h i s e q u a t i o n can e a s i l y be s h o w n to p o s s e s s t h e e x a c t s o l u t i o n

a d i f f e r e n c e r e p r e s e n t a t i o n m u s t be f o u n d f o r the s e c o n d d e r i v a

-t d ^ f •

t i v e , ---- . T h e m o s t w i d e l y u s e d d i s c r e t e f o r m is o b t a i n e d by d x2 '

e x p a n d i n g f in a T a y l o r s e r i e s a b o u t s o m e p o i n t x at i n t e r v a l s h, as f o l l o w s

f ( x + h ) = f (x) + h f ' ( x ) + — f " ( x ) + — f ' " ( x ) + 0( h M (4)

2

6

f ( x - h ) = f (x ) - h f ' ( x ) + — f " ( x ) - — f ‘ " ( x ) = 0( ) (5)

2

6

w h e r e p r i m e s d e n o t e d i f f e r e n t i a t i o n w i t h r e s p e c t to x. A d d i n g t h e s e t w o e x p r e s s i o n s and s u b t r a c t i n g 2 f ( x ) y i e l d s

f (x + h ) - 2f(x) + f ( x - h ) = h2f " ( x ) +

O( h^ )

(6)

We n o w i n t r o d u c e s o m e n o t a t i o n f o r the d i s c r e t e v a r i a b l e s . Let

f (x ) = A sin-rrx + B C O SttX

( 3a)

w h i c h t h e b o u n d a r y c o n d i t i o n s t h e n r e d u c e to

f ( x ) - sin-rrx

(3b)

10

-x

j

=

J

h

j

= 0,1

...N

(7)

f j = f ^ xd^

( 8 )

sxf j = f ( xj +h) -

2

f (Xj) +

(

9

)

U s i n g t h i s n o t a t i o n , o u r o r i g i n a l d i f f e r e n t i a l e q u a t i o n (1) c a n n o w be w r i t t e n as a s e c o n d o r d e r d i f f e r e n c e e q u a t i o n , n a m e l y

s V . + h 2 i r 2 f , = 0 ( 1 0 )

X J J

c o r r e c t to o r d e r h 2 .

T h e d i f f e r e n c e e q u a t i o n (10) is a l s o l i n e a r , and by com p a r i s o n w i t h (3a) can be s h o w n to h a v e the s o l u t i o n

fj = A s i n ir A j + BcosirAj (11)

and on a p p l y i n g t h e b o u n d a r y c o n d i t i o n s

f0 = 0

f N = 1

(1 2)

t h e s o l u t i o n b e c o m e s

f j = A s i n

ttA

j

( 13)

A =

1

si n ( — )

2h

11

-E x p a n d i n g

X

in a p o w e r s e r i e s in h g i v e s

rs 1

IT

= h

i t + l i t i + i ! t i + 0 ( h 7)

2 2 4 4 8 0

1 + rc2h 2 +

12 2 4 0

+ 0 ( h 6 ) j

(14)

(15)

It can i m m e d i a t l e y be s e e n t h a t t h e s o l u t i o n (13) is

f

j

=

1

si n — (1+0(h 2 )) 2

s i n tt h j

1 + 0 ( h 2 ) j

(1 3a )

T h a t is to s a y , t h e s o l u t i o n is f o r m a l l y a c c u r a t e to s e c o n d o r d e r in h. T h e m o s t usual w a y of express! ng thi s f a c t is to w r i t e

f (x ) = f • + 0( h 2 )

b u t i n s t e a d of an a d d i t i v e o r d e r f u n c t i o n a m u l t i p l i c a t i v e f u n c t i o n c a n be d e f i n e d by w r i t i n g f ( x) = a f . B w h e r e a = 1 + 0( h 2 ).

vi

T h i s i d e a f o r m s a b a s i s f o r t h e h i g h e r o r d e r m e t h o d .

2. 1 A p p l i c a t i o n to o r d i n a r y d i f f e r e n t i a l e q u a t i o n s

12

-C o n s i d e r a n e w a p p r o x i m a t i o n to --- g i v e n by d x2

d 2 f V

S - I = a — ( 1 6 )

d x2 h2

w h e r e t h e c o e f f i c i e n t a has y e t to be d e t e r m i n e d and wi ll p r o ­ b a b l y be a f u n c t i o n of x.

R e w r i t i n g t h e d i f f e r e n c e e q u a t i o n (10) by u s i n g t h e d e f i n i t i o n ( 1 6 ) t h e n l e a d s to t h e f o l l o w i n g e q u a t i o n :

2 2 t t 2

5x f j + —

f j ■ 0

<1 7 >

a

T h i s d i f f e r e n c e e q u a t i o n a l s o p o s s e s s e s an e x a c t s o l u t i o n , n a m e l y

f. = a sinttyj + b c o s7ryj (18)

J

and t h e n u p o n a p p l y i n g th e b o u n d a r y c o n d i t i o n s (1 2), t h i s r e d u c e s

to :

fj “ a sin-iryj

a = --- ---- (19)

s i n

2h

- 2 . - 1 rf h x

y - — sin — ) .

tt 2 / a

T h i s s o l u t i o n is s i m i l a r to (13); in f a c t , it is the s a m e if a = 1. W i t h a d i f f e r e n t v a l u e for a we c a n , n a t u r a l l y , o b t a i n d i f f e r e n t s o l u t i o n s , and w i t h t h e " r i g h t " v a l u e t h i s can be t h e e x a c t s o l u t i o n o f t h e o r d i n a r y d i f f e r e n t i a l e q u a t i o n (1).

13

-T h e o u t s t a n d i n g p r o b l e m is h o w to e v a l u a t e a to g i v e th is e x a c t s o l u t i o n . By c h o o s i n g a we a r e a b l e to e n s u r e t h a t the s o l u t i o n (19) is t h e s a m e

a t t h e - p o i n t x .

as t h e s o l u t i o n (3b) of e q u a ­

te

t i o n (1). By c o m p a r i s o n it can be s e e n t h a t the c h o i c e s h o u l d be s u c h t h a t

a = 1

t h a t i s

w h i c h r e d u c e s to

= h

s i n (^-) = 1

2 h (

2 0)

(2 1)

ft

r

and h e n c e

u = - s i n" 1 ( - ^ - ) = h

¥

2 / a

(

2 2)

l e a d i n g to

irh 2

a

=

irh

(23)

It s h o u l d be n o t e d h e r e t h a t in t h e l i m i t as h 0, t h e n a ->• 1, and h e n c e th i s f o r m u l a t i o n is c o n s i s t e n t .

k

T r u n c a t i n g t h e p o w e r s e r i e s in h fo r a a f t e r a g i v e n n u m b e r o f t e r m s y i e l d s a d i f f e r e n t a c c u r a c y . For e x a m p l e , for a g i v e n v a l u e o f h, by u s i n g r e s p e c t i v e l y

= 1

a = 1 +

12

(24)

14

-„ = 1 + (26)

12 2 4 0

a s o l u t i o n c o r r e c t to o r d e r h 2 , h4 and h6 can be c a l c u l a t e d .

As an i l l u s t r a t i o n of t h i s f a c t , f i g u r e 1 is p r e s e n t e d . T h i s f i g u r e s h o w s a p l o t of log(k-n-h) ( h o r i z o n t a l l y ) v e r s u s

1 o g ( m a x | e r r o r | ), w h e r e k is t h e f r e q u e n c y . T h e f o u r c u r v e s r e ­ p r e s e n t t h e v a l u e s o b t a i n e d f o r s e c o n d , f o u r t h , s i x t h and e i g h t h o r d e r m e t h o d s . T h e f i g u r e c l e a r l y d e m o n s t r a t e s t h a t t h e e r r o r is, in f a c t , p r o p o r t i o n a l to h to t h e p o w e r 2, 4, 6 o r 8 r e s p e c t i v e l y , as is s h o w n b y the g r a d i e n t of t h e l i n e s . By m e a n s of t h i s f i g u r e t h e n e c e s s a r y g r i d m e s h - s i z e to g i v e s o m e p r e s c r i b e d e r r o r at a f r e q u e n c y k c a n be d e t e r m i n e d .

By c o n t r a s t , f i g u r e 2 s h o w s th e s a m e p l o t of log(kirh) v e r s u s 1o g ( m a x | e r r o r | ), b u t on t h i s o c c a s i o n t h e s o l u t i o n w a s e v a l u a t e d u s i n g a CII M i t r a 15 c o m p u t e r , w h e r e a s t h e d a t a f o r f i g u r e 1 w a s c a l c u l a t e d u s i n g a C o n t r o l D a t a 65 0 0 . (T he M i t r a has f o u r or f i v e d i g i t a c c u r a c y . By c o m p a r i s o n , t h e C D C c a r r i e s a b o u t 15 s i g n i f i c a n t d i g i t s ) . T h i s i l l u s t r a t e s c l e a r l y t h a t , of c o u r s e , t h e m e t h o d is l i m i t e d by the a c c u r a c y o f t h e c o m p u t e r us ed . S e c o n d , f o u r t h , s i x t h or e i g h t h o r d e r a c c u r a c y can be a c h i e v e d p r o v i d e d t h a t t h i s is w i t h i n the m a c h i n e a c c u r a c y . In t h i s c a s e , no s o l u t i o n can be o b t a i n e d w i t h an e r r o r b e t t e r

- 7 t h a n 10 .

15

-K e y to f i g u r e s 1 a n d 2

O r d e r 2

4

6

k = 1

A D G K

k = 5 B E H

L

k = 9 C

F

16

->>

►

N>

-1,200

i . 00

-7.00

L O G ( K P I H ) V E R S O S L0G< HftX E R R O R )

- . 9 0 0 - . 6 0 0

BOBO B B B B

EE E E E EE

B C C C C C C C ccc c

FF F F F E F F FF

F F F £

LH

H

C B C

B C B (

F E

£ F i

FF

F E 0

F H

J

H J K

H

J L

FIGURE 1

c c

17

-?>»■

r>

V'

fi

*~9

r

H

* 7 r

*-■>

*

*

k

-l .200

1.00

I---L 0 G ( K P I H> V E R S U S I---L O C I H A X E R R O R )

■ 0 . 9 0 0 - 0 . 6 0 0

K KKK

y. k

D K

K

K

DDD K

K

K K K

EE

E E E EE

L L H L H L

K L

C B C

B C B C

B C C C C C C C CCCC

E F FF

FF F F F E F F FF

hj k

J K

J H

J H

J HJ J J J H ) J HJ H

G H L HH

HL L H L

L L L L HHL H HH L K M K L

L

- 0 . 3 0 0 0 . 000 0. 3 0 0 0 . 6 0 0

0 . 00

-2 .00

- 3 0 0

5 . 0 0

-6.00

7 . 00

8 . 0 0

- 9 . 0 0

18

-e q u a t i o n poss-ess a s i m p l -e s o l u t i o n . W-e w o u l d t h -e n r -e s o r t to s o l v i n g t h e e q u a t i o n n u m e r i c a l l y by m e a n s of an e l e c t r o n i c d i g i t a l c o m p u t e r . T h a t t h i s m e t h o d is i d e a l l y s u i t e d to s o l u t i o n by c o m p u t e r will

b e c o m e o b v i o u s in w h a t f o l l o w s .

It wi l l be r e c o g n i s e d t h a t , so far, t h i s p r o c e d u r e o n l y s e e m s to be u s e f u l w h e n t h e s o l u t i o n is a l r e a d y k n o w n . T h i s is n o t t h e c a s e , a n d s o m e a l t e r n a t i v e m e t h o d s f o r c a l c u l a t i n g a are d e v e l o p e d b e l o w . If t h e s o l u t i o n is d e f i n e d on s o m e g i v e n i n t e r v a l , or is k n o w n to be a h a r m o n i c f u n c t i o n , F o u r i e r a n a l y s i s c a n be

h e l p f u l . T h e d i s c r e t e F o u r i e r t r a n s f o r m is d e f i n e d as f o l l o w s . G i v e n a f u n c t i o n f(x) w h i c h is p e r i o d i c on an i n t e r v a l 0 < x < X, w h e r e t h e i n t e r v a l is divided into N+l p o i n t s s u c h t h a t Nh = X, t h e n

N / 2

£

n = 1

N / 2

,

n

f ( x . ) = a0 + £ a cos2'irnx. + b n sin2irnx-j (27)

J n = 1 I J n J J

w h e r e x^ = ( j - l ) h j = 1, . . . , N+1

?

N

a n = — £ f (x .) cos 2ttn x • n = 1, . . . , ----1 (28b )

N j = 1 J 2

N/2

1 n N

— £ f ( x . ) COS 2i - X

N j = l J 2 ’

(28c)

„

Z

E f ( x i ) s i n 2irnx

N j = l

19

-No w, by r e w r i t i n g e q u a t i o n (16) as

<

j*

h2 d 2 f

a -

_'

_{"2 r}

d x2 6 X j

(29)

w i t h a F o u r i e r s e r i e s e x p a n s i o n f o r f, it is p o s s i b l e to e v a l u a t e t h e n u m e r a t o r o f t h i s e x p r e s s i o n . H e n c e , it is p o s s i b l e to e v a l u a t e

a. T h e s e c o n d d e r i v a t i v e c a n be a p p r o x i m a t e d as t h e l i m i t as h 0, n a m e 1y

<52 f . d 2 f

d x2

an-« lim x

j _

N/ 2 h->0 h2

( 2

it

)

s

n 2 (

a

c o s 2 i r n x

- + b

s

i

n2irnx , ) ( 3 0 )

P ^ 'I J n j

N/ 2

E (cos2'irnh~l)(a co s2irnx.+b sin2irnx.) (31)

n= l J n j

H e n c e ,

- ( 2 tt h )

N / 2

E n 2 (a cos2Trnx-+b sin2irnx_.)

n = 1 J J

N/2

E ( l “ C O s2irnh)(an c o s2nnx-H-bn s i n2TTnx.) n = l

(32)

If a n = 0 fo r all n e x c e p t ap, and b n

( 2irh ) 2 p2 a cos 2irpx ■ _______________r___________ vJ_

( l c o s 2 i r p h ) a c o s 2 T T p x

-r

J

= 0 for all n

a

=

( 2irhp) l - c o s2Trph

' ( trhp)

s i n tt h p ^

( 3 3 )

a

2 0

-If we h a v e m o r e t h a n o n e of the c o e f f i c i e n t s a , b n n o n z e r o , t h e n a will be d e p e n d e n t on t h e s e c o e f f i c i e n t s and it is t h e r e f o r e n e c e s s a r y to f i n d a w a y to c o m p u t e t h e s e . O n e s u c h a w a y is to s o l v e th e d i f f e r e n c e e q u a t i o n s e t t i n g a = 1 to g e t a f i r s t a p p r o x i m a t i o n to f of o r d e r h 2 . T h e n use t h i s f. to c a l

-J J

c u l a t e a b e t t e r a p p r o x i m a t i o n to

a

in e q u a t i o n (32) r e p e a t i n g t h i s p r o c e s s if n e c e s s a r y .

In t h e c a s e w h e r e t h e s o l u t i o n s o u g h t is k n o w n n o t to be p e r i o d i c , or n o t h i n g is k n o w n a b o u t t h e h a r m o n i c p r o p e r t i e s o f t h e s o l u t i o n , t h e n t h e a b o v e F o u r i e r a n a l y s i s m a y n o t be h e l p f u l , and t h e n s o m e o t h e r m e a n s f o r e v a l u a t i n g a s h o u l d be u s e d .

R e t u r n i n g to e q u a t i o n (29) f o r

a

s u g g e s t s t h a t a can be d e t e r m i n e d if the s e c o n d d e r i v a t i v e in the n u m e r a t o r can be e v a l u a t e d . U s i n g a p r o c e s s s i m i l a r to t h a t d e s c r i b e d a b o v e , if we f i r s t s o l v e the d i f f e r e n c e e q u a t i o n o b t a i n e d by s e t t i n g a = 1

we wi ll h a v e s o m e k n o w l e d g e of t h e f o r m of the s o l u t i o n , it is d 2 f

q u i t e f e a s i b l e to use a h i g h e r o r d e r e x p r e s s i o n f o r d x2

e x a m p l e

, for

j

d 2 f

d x2 12 h 2 ’- ( f .1 + 2+ f .i - 2 ) + l 6 ( f J + l + f l - l ) - 3 0 f 1l + O ( h ^ ) ( 3 4 )

T h e n s u b s t i t u t i n g t h i s e x p r e s s i o n i n t o e q u a t i o n (29) p r o d u c e s a s e c o n d a p p r o x i m a t i o n to a w h i c h can t h e n be u s e d to p r o d u c e a s e c o n d a p p r o x i m a t i o n to f*. A g a i n , t h i s p r o c e d u r e can

J

21

-v a l u e s o f f.. It a p p e a r s , in t h e -v a r i o u s c a s e s t r i e d to d a t e , t h a t t h r e e or f o u r i t e r a t i o n s p r o v e a d e q u a t e . A n o n c e n t r e d 5 p o i n t d i f f e r e n c e w o u l d be u s e d at j = 1 and j = N - 1.

U s i n g m a t r i x - o p e r a t o r n o t a t i o n , and s u p e r s c r i p t s in p a r e n t h e s e s t o d e n o t e t h e i t e r a t i o n l e v e l , th i s p r o c e d u r e can be w r i t t e n b r i e f l y as f o l l o w s . L e t

(T— )i ' Sx f i (35)

e.g. as g i v e n by e q u a t i o n (34) (36)

- J x j

( Pf )

=

3

d x2

E q u a t i o n (5) c a n be w r i t t e n as

Tf + h27r2 I f = b

(37)

w h e r e I is t h e u n i t m a t r i x , a n d jb c o n t a i n s the b o u n d a r y condi t i o n s . If

Tl - 1 +

h2tt2 1 (38)

t h e n e q u a t i o n (37) b e c o m e s

T il =

k

(39)

T h e i t e r a t i v e s c h e m e f o r £ wi l l t h e n be :

( 1)

T if = b (40)

( ! ) ■ ( 1 ) ( 1 ) ( 4 1 )

D T x f = Pf

(1) (2)

22

-(k) (k) . (k)

D T xf = Pf (43)

(k) ( k +1)

D T xf = b (44)

w h e r e e q u a t i o n s (40), (42) a n d (44) a r e u s e d to s o l v e f o r f and t h e n e q u a t i o n s (41) and (43) are u s e d to e v a l u a t e D, w h i c h is the d i a g o n a l m a t r i x w i t h e l e m e n t s a j .

If, in t h i s s e q u e n c e { f ^ ) } and c o r r e s p o n d i n g l y { D ^ ^ > k is a n u m b e r l a r g e r t h a n 4, t h e n s o m e a c c e l e r a t i o n t e c h n i q u e s h o u l d be a p p l i e d . For i n s t a n c e , an o v e r r e l a x a t i o n e x p r e s s i o n of the f o r m

(k) (k) (k)

D — ( 1 “* co) D + o )d

f k )

w h e r e d v

'

is t h e r e s u l t o f s o l v i n g an e q u a t i o n of the f o r m (43) f o r D .

It is s t r a i g h t f o r w a r d to s h o w t h a t , if t h e s e q u e n c e of i t e r a t e s { f ^ ) } t e n d s to a l i m i t f" say, t h e n t h i s l i m i t is a

h i g h e r o r d e r s o l u t i o n . A s s u m i n g t h a t a l i m i t e x i s t s , t h e n e q u a t i o n s (43) and (44) b e c o m e

d"

J 1i A =

Pf" (45)

D "

Ti I X = £

( 4(5)

a n d h e n c e

P f " = b (47)

23

-of e q u a t i o n (1) u s i n g a f o u r t h o r d e r a p p r o x i m a t i o n for t h e s e c o n d d e r i v a t i v e . For e x a m p l e , if P is c h o s e n to us e all t h e N = J L

Z h

p o i n t s of t h e s o l u t i o n , t h e n t h e s o l u t i o n f will be Nth o r d e r c o r r e c t in h. If t h e s o l u t i o n is a p o l y n o m i a l of o r d e r m, w h e r e m < N, t h e n t h e s o l u t i o n w i l l be e x a c t .

T h e e x t e n s i o n to i n c l u d e f i r s t o r d e r d e r i v a t i v e s is s i m i l a r . To s o l v e t h e e q u a t i o n

d

2

f d f

5 - 1 +

21 2 1

+ k f = 0 (48)

d x2 dx '

a s e c o n d o r d e r d i f f e r e n c e f o r t h e f i r s t d e r i v a t i v e is i n t r o d u c e d w i t h an u n k n o w n c o e f f i c i e n t , 3,

df

dx - 3 2 h^ (49)

or if D2 is t h e d i a g o n a l m a t r i x w i t h e l e m e n t s 3, and T2 is the o p e r a t o r :

( T * ! ) j ■ fj + 1 - f j . i (50)

t h e n e q u a t i o n (48) b e c o m e s in d i s c r e t e f o r m

T i f + h £ T 2f + h 2 k I f = _b (51)

o r i f

Ti + h* T2 + h2 kl = T3 (52)

24

-T h e e q u a t i o n c o r r e s p o n d i n g to e q u a t i o n (41) is

t o

D2 T 2 f = P 2 f ( 5 4 )

l e a d i n g to the i t e r a t i v e s y s t e m

( 1)

T s l

=

k

(55)

( 1 )

( 1 )

(1 )

D T jf = Pf (56)

( 1 )

(1 )

( 1 )

D T 2 f = P 2 f (57)

(

1

)

D

Tj + h t D2

(

1

)

T + h2 kl 1 f

(

2

)

= b (58)

and so on. H e r e P2 is s o m e h i g h e r o r d e r d i f f e r e n c e o p e r a t o r for t h e f i r s t o r d e r d e r i v a t i v e .

T h e a d v a n t a g e s of u s i n g s u c h a s c h e m e are i m m e d i a t e l y c l e a r . To s o l v e e q u a t i o n (55) f o r jP r e q u i r e s the i n v e r s i o n of t h e m a t r i x T , w h i c h is a t r i - d i a g o n a l m a t r i x , i . e . , t h e o n l y non z e r o e l e m e n t s are the d i a g o n a l a n d o n e e l e m e n t to e a c h s i d e of t h e d i a g o n a l . T h e r e e x i s t s a w e l l k n o w n a l g o r i t h m for i n v e r t i n g s u c h a m a t r i x , w h i c h r e q u i r e s m u c h le ss w o r k t h a n i n v e r t i n g the m a t r i x P w h i c h has m o r e non z e r o e l e m e n t s , and m a y e v e n h a v e no

z e r o e n t r y .

t h a t , a l t h o u g h th e a c c u r a c y of t h e s o l u t i o n m a y be f o u r t h o r d e r or h i g h e r , th e m e t h o d o n l y r e q u i r e s t w o b o u n d a r y c o n d i t i o n s , as w o u l d be e x p e c t e d for a s e c o n d o r d e r d i f f e r e n t i a l e q u a t i o n . It

25

-is m o r e o f t e n t h e c a s e t h a t h i g h e r o r d e r m e t h o d s r e q u i r e m o r e b o u n d a r y c o n d i t i o n s t h a n th e d i f f e r e n t i a l e q u a t i o n , and d e t e r ­ m i n i n g t h e s e e x t r a b o u n d a r y c o n d i t i o n s s a t i s f a c t o r i l y can i n v o l v e a d i s p r o p o r t i o n a t e l y l a r g e a m o u n t of w o r k .

2.2 S o m e r e s u l t s w i t h t h e m e t h o d

To s h o w h o w q u i c k l y t h e i t e r a t i v e p r o c e d u r e o u t l i n e d a b o v e c o n v e r g e s to the h i g h o r d e r s o l u t i o n , t h e f o l l o w i n g e x a m p l e s w e r e u s e d :

1) w i t h

a -

1, h = 0 . 0 5 , a f i v e p o i n t f o r m u l a s u c h as e q u a t i o n (45), g i v e s i m m e d i a t e l y ; a = 1 . 0 0 2 0 5 ,

w h i c h is t h e c o r r e c t v a l u e as g i v e n by e q u a t i o n (23); 2) w i t h

a

= 1, h = 0 . 0 1 , e q u a t i o n (45) g i v e s a v a l u e :

a = 1 . 0 0 0 0 8 2

w h i c h a g a i n is c o r r e c t and so t h e r e is no n e e d for i t e r a t i o n ; 3) t a k i n g a t o o s m a l l , n a m e l y , a = 0 . 1 and h = .05, o n l y two

i t e r a t i o n s are r e q u i r e d . T h e s e c o n d o r d e r s o l u t i o n g i v e s :

a = 1 . 0 2 0 5 6 2

f i r s t i t e r a t i o n y i e l d s a = 1 . 0 0 2 0 1 5

and a s e c o n d i t e r a t i o n r e s u l t s in

a = 1 . 0 0 2 0 5 2

as e x a m p l e (1) ;

4) t a k i n g a t o o l a r g e , a = 10, w i t h h = .05 as in e x a m p l e (3), g i v e s on s u c c e s s i v e i t e r a t i o n s :

26

-T h e s e f o u r e x a m p l e s s h o w c l e a r l y , p r o v i d e d t h e v a l u e of a is irh

2 /a

< 1, t h a t any s u c h t h a t y e x i s t s in e q u a t i o n (2 2), n a m e l y

v a l u e f o r a c a u s e s t h e p r o c e d u r e to c o n v e r g e r a p i d l y to a hi gh o r d e r s o l u t i o n .

3. A P P L I C A T I O N T O P A R T I A L D I F F E R E N T I A L E Q U A T I O N S

3.0 Introduction

T h e e x t e n s i o n of t h i s a p p r o a c h to t h e h i g h e r o r d e r s o l u ­ t i o n of p a r t i a l d i f f e r e n t i a l e q u a t i o n s is r e a s o n a b l y s i m p l e . I n ­ s t e a d of o u r m u l t i p l i c a t i v e v a r i a b l e s a etc b e i n g s i m p l e c o n s t a n t s or f u n c t i o n s of one v a r i a b l e , t h e y w i l l n o w be m o r e c o m p l i c a t e d f u n c t i o n s of t w o or m o r e v a r i a b l e s , and we will h a v e o n e s u c h c o e f f i c i e n t f o r e a c h d e r i v a t i v e .

For t h e s a k e o f c l a r i t y , t h e t h r e e b a s i c t y p e s of

p a r t i a l d i f f e r e n t i a l e q u a t i o n s w i l l e a c h be t r e a t e d s e p a r a t e l y to s h o w s o m e o f t h e f e a t u r e s p e c u l i a r to e a c h and h o w our h i g h e r o r d e r m e t h o d c o p e s w i t h them . T h e f i r s t t y p e t r e a t e d is p a r a b o l i c e q u a t i o n s .

3.1 P a r a b o l i c e q u a t i o n s

As an e x a m p l e of a p a r a b o l i c e q u a t i o n , c o n s i d e r t h e l i n e a r i z e d f o r m of t h e B u r g e r s e q u a t i o n :

u t + U u x = U xx (5 9 >

u i + 1 " u i-1

u

=

3

- + 0 ( h 2 )

(62)

x

2h

k k k

u i + i " 2u.- + u . ,

u = a - i--- j -i l i +

0(

h 2 )

(63)

XX h2

w h e r e a , 3 and y are u n k n o w n f u n c t i o n s of x and t, l e a d s to the

. k + 1

d i f f e r e n c e e q u a t i o n f o r u. : J

k+1 k o k k

uj

= uj - u ( Xj >t k) - — <u j + l - u j - l )

a k k k

+ ; ^ ( u j + i " 2 u j + uj - i ) ( 5 4 )

S e t t i n g the c o e f f i c i e n t s a, 3 and y to u n i t y will l e a d to a s o l u t i o n for u c o r r e c t to o r d e r x + h 2 . T h i s s o l u t i o n can t h e n be u s e d to d e t e r m i n e n e w v a l u e s of t h e s e c o e f f i c i e n t s a , 3 and y to o b t a i n a b e t t e r a p p r o x i m a t i o n to t h e s o l u t i o n of e q u a t i o n (5 9) f o l l o w i n g t h e p r o c e d u r e u s e d a b o v e in s e c t i o n 2.2.

29

-■*> If we n o w p r e s c r i b e t h e f o l l o w i n g i n i t i a l and b o u n d a r y c o n d i t i o n s

^ u ( x sO ) = x + sin2TT(jox

I

u ( 0 , t ) = 0 (6 6)

i*

u (1, t ) = 1

r

pi

t h e n t h e e x a c t s o l u t i o n of t h e differential e q u a t i o n

u t “ u xx (67)

i s

"4fr2w 2 t

u (x j t ) = X + e s i n2iro)X . (6 8)

T h e e x a c t s o l u t i o n o f t h e d i f f e r e n c e e q u a t i o n (65) s u b j e c t to t h e i n i t i a l and b o u n d a r y c o n d i t i o n s (6 6) is

k k

Uj = jh + c sin27Twjh (69)

w h e r e

£ = 1 - 4s s i n2n(uh (70)

and

s = — - . (71)

h2 Y

C o m p a r i n g e q u a t i o n s (6 8) and (69) it c a n be s e e n t h a t t h e e x a c t v a l u e f o r c is

•“4

tt

2

w

2

t

C ex = e ■ (72)

To g i v e t h e s e v a r i a b l e s s o m e n u m b e r s , let

30

-C = 1 - 0 . 4 s i n2 O . l u = 0 . 8 4 7 2 1 3 5 9

w h i c h s h o u l d be c o m p a r e d w i t h

C = 0 . 8 5 3 9 2 3 5 0 .

t. X

U s i n g a t h r e e p o i n t f o r m u l a f o r t h e n u m e r a t o r of y ( c o r r e c t to o r d e r c2 = 0 . 0 0 0 0 1 6 ) and a f i v e p o i n t f o r m u l a fo r t h e n u m e r a t o r of a (as in t h e d e f i n i t i o n of a , e q u a t i o n ( 2 9 ) ) , y = 1 . 0 9 0 1 7 0 0

and a = 1 . 0 3 1 8 3 0 5 , g i v i n g

C _{a 3 y}v = 0 . 8 5 5 3 8 9 8 2

w h i c h is a b e t t e r v a l u e .

R e p e a t i n g t h i s p r o c e d u r e g i v e s

C

_{U 3 y}

v = 0 . 8 5 4 6 3 7 6 4

w h i c h is a f u r t h e r i m p r o v e m e n t , a n d a l r e a d y c o r r e c t to o r d e r

h

+

t

2 .

At t h e end of t h e s e c t i o n a b o v e on o r d i n a r y d i f f e r e n t i a l e q u a t i o n s it w a s p o i n t e d o u t t h a t , w h e r e a s m o s t o t h e r h i g h e r o r d e r m e t h o d s n e e d e x t r a b o u n d a r y c o n d i t i o n s , t h i s m e t h o d r e q u i r e s no a u x i l i a r y c o n d i t i o n s . T h e s c h e m e u s e s o n l y t h o s e c o n d i t i o n s w h i c h ar e g i v e n as i n p u t to t h e p r o b l e m . In g e n e r a l , w i t h an n^*1 o r d e r

. t h .

d i f f e r e n c e f o r an m o r d e r d e r i v a t i v e , an e x t r a (n-m) c o n d i t i o n m u s t be a r t i f i c i a l l y d e f i n e d . In m a n y c a s e s the s a t i s f a c t o r y d e f i n i t i o n o f t h e s e n o n - p h y s i c a l b o u n d a r y c o n d i t i o n s can r e q u i r e

as m u c h w o r k as the s o l u t i o n o f t h e r e s t of t h e p r o b l e m . An a r ­ b i t r a r y c h o i c e can o f t e n r e n d e r a w e l l p o s e d p r o b l e m ill p o s e d .

31

-As an e x a m p l e of t h i s , c o n s i d e r a g a i n e q u a t i o n (59)

u t + U u x = \ x ( 5 9 >

We h a v e a f i r s t d e r i v a t i v e w i t h r e s p e c t to t, a f i r s t and a s e c o n d w i t h r e s p e c t to x. T h u s we n e e d o n e c o n d i t i o n (in t h i s c a s e it is an i n i t i a l c o n d i t i o n ) in t and t w o in x. Th e d i f f e r e n c e f o r m u l a ­ t i o n (65) has a f i r s t d i f f e r e n c e w i t h r e s p e c t to k, and a s e c o n d d i f f e r e n c e w i t h r e s p e c t to j. T h e r e f o r e , it r e q u i r e s one i n i t i a l c o n d i t i o n and t w o b o u n d a r y c o n d i t i o n s , the s a m e as th e o r i g i n a l d i f f e r e n t i a l f o r m u l a t i o n . If, on t h e o t h e r h a n d , f o r a m o r e a c c u ­ r a t e s o l u t i o n , s e c o n d o r d e r d i f f e r e n c i n g f o r t and f o u r t h o r d e r f o r x a r e u s e d , t h e n t w o i n i t i a l c o n d i t i o n s and f o u r s p a t i a l b o u n ­ d a r y c o n d i t i o n s a r e r e q u i r e d . H o w e v e r , t h e s t a t e m e n t of the p r o ­ b l e m p r o v i d e s o n l y o n e i n i t i a l and t w o b o u n d a r y c o n d i t i o n s , so the r e m a i n i n g c o n d i t i o n s m u s t be i n v e n t e d . W i t h t h e p r e s e n t h i g h e r o r d e r s c h e m e , t h e e q u a t i o n is w r i t t e n as in e q u a t i o n (65) e x a c t l y as if t h e f o r m u l a t i o n w e r e f i r s t o r d e r in t i m e and s e c o n d o r d e r in s p a c e . T h e e f f e c t of t h i s is t h a t p r e c i s e l y t h e s a m e n u m b e r of c o n d i t i o n s is r e q u i r e d f o r t h e d i f f e r e n c e s y s t e m as for t h e d i f ­ f e r e n t i a l s y s t e m . T h e s a m e is t r u e for e q u a t i o n s of e l l i p t i c and h y p e r b o l i c t y p e . T h i s is o b v i o u s l y a v e r y v a l u a b l e s i d e e f f e c t of t h e m e t h o d . W h e n e x t r a ( a u x i l i a r y ) b o u n d a r y c o n d i t i o n s a r e a d d e d i n d i s c r i m i n a t e l y t h e n t h e p r o b l e m c a n s o o n b e c o m e ill p o s e d , w i t h the r e s u l t t h a t the s o l u t i o n ' b l o w s u p1 or d i v e r g e s , e v e n w h e n t h e c o n t i n u o u s s o l u t i o n is b o u n d e d .

-t i o n (29) is e v a l u a -t e d a-t -t i m e l e ve l k + 1, -t h e n -th e r e s u l -t i n g s y s t e m is ' i m p l i c i t 1 . All c a l c u l a t i o n s are d o n e u s i n g an e x p l i c i t s c h e m e , b u t th e c o n v e r g e d s o l u t i o n will be as if t h e s c h e m e w e r e i m p l i c i t . T h u s t h e s c h e m e b e c o m e s i n d e p e n d e n t o f t i m e - l i k e s t e p - size. An a d d i t i o n a l a d v a n t a g e o f an i m p l i c i t s c h e m e is t h a t it a d d s in t h e s p a c e - l i k e c o o r d i n a t e c o n d i t i o n s at t h e n e w level i m m e d i a t e l y r a t h e r t h a n w o r k i n g o u t t h e i n t e r i o r p o i n t s i n d e p e n ­ d e n t l y o f the b o u n d a r y c o n d i t i o n s and i n c l u d i n g t h e m a f t e r w a r d s .

S o m e r e s u l t s o f all t h e s e e f f e c t s are s h o w n in s o m e

e x a m p l e s . T a b l e s 1, 2 and 3 i l l u s t r a t e the r a p i d i n c r e a s e o b t a i n e d by t h e a p p l i c a t i o n of t h i s m e t h o d to t h e d i f f u s i o n in d i f f e r e n t f o r m s . T h e f i v e ro ws of t h e s e t a b l e s are :

1

. t h e c o m p u t e d v a l u e s of t h e s o l u t i o n o b t a i n e d by a p p l y i n g t h e s t a n d a r d s e c o n d o r d e r in s p a c e , f i r s t o r d e r in t i m e , i m p l i c i t d i f f e r e n c e and p r o c e e d i n g to a t i m e t = 0.4;

2

. t h e e x a c t s o l u t i o n at t i m e t = 0. 4;

3. t h e a b s o l u t e v a l u e o f the e r r o r , t h a t is the d i f f e r e n c e b e t w e e n l i n e s 1 and 2;

4. th e v a l u e s g i v e n by c a l c u l a t i n g the c o e f f i c i e n t s and c o r r e c t i n g t h e s o l u t i o n at t i m e t = 0 . 4 ;

5. the r e m a i n i n g e r r o r , t h e d i f f e r e n c e b e t w e e n l i n e 2 and l i n e 4.

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