andrew benjamin white, jr. numerical solution of two-point boundary-value problems

7/28/2019 Andrew Benjamin White, Jr. Numerical Solution of Two-point Boundary-Value Problems

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NUMERICAL SOLUTION OF TWO-POINTBOUNDARY-VALUE PROBLEMS

Thesis ByAndrew Benjamin White, Jr.

In Partial Fulfillment of the Requirements

For the Degree ofDoctor of Philosophy

California Institute of Technology

Pasadena, California 911091974

(Submitted March 13, 1974)


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Acknowledgements

The au thor wishes t o thank Professo r Herber t B , K e l l e r f o rh i s p a t i e n t g ui de nc e and i n v a l ua b l e a s s i s t a n c e i n t h e p r e p a r a t io nof t h i s t h e s i s . H i s encouragement and enthusiasm were e s se nt ia lt o th e comple tion of t h i s work .

The o p p o r t u n i t y t o c a r ry o u t t h i s r e s e a rch was p ro v id ed byCal i fo rn i a I n s t i t u t e of T echn ol og y Gradu a te T each in g A s s i s t an t s h i p sfo r wh ich t h e au t h o r i s g r a t e f u l . H e a l s o wi sh es t o e x p r e ss h i sapp rec ia t i on t o th e en t i r e Depar tment of Appl ied Mathematics, bo thf a c u l t y a nd s t u d e n t s , f o r p r o vi d in g a n i n v i g o r a t i n g c l i m a t e i nwhich t o pursue t h i s work.

The autho r wishes to thank M r s . Janet Smith and Mrs. MaryBe th Br i gg s fo r t h e i r e x p e r t t y p i n g of a d e t a i l ed and t ed i o u smanuscript .


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ABSTRACT

The app rox im ati on of two-point boundary-v alue problenls byg e n e r a l f i n i t e d i f f e r e n c e schemes i s t r e a t e d . A necessary ands u f f i c i e n t c on d it io n f o r t h e s t a b i l i t y of t h e l i n e a r d i s c r e t eboundary-value problem i s d e r iv e d i n terms o f t h e a s s o c i a t e dd i s c r e t e i n i t i a l - v a l u e p rob le m. P a r a l l e l sh o o t i ng m eth ods a r eshown to be e qu iva l en t t o t he d i s c r e t e boundary-va lue p roblem.O ne-s tep d i f f e r e n ce s chemes a r e co n s i d e red i n d e t a i l and a c l a s so f c o m p u t a ti o n al l y e f f i c i e n t s chem es o f a r b i t r a r i l y h i g h o r d e r ofaccuracy i s e x h i b it e d . S u f f i c i e n t c o n d i ti o n s a r e found t o i n s u r et h e c o nv er ge nc e of d i s c r e t e f i n i t e d i f f e r e n c e a p p ro x im a ti on s t onon l in ear boundary-value p rob lems wi t h i s o l a t ed s o l u t i on s . ~ e w t o n ' s

method i s c o n si d er e d a s a p r oc e du r e f o r s o l v i n g t h e r e s u l t i n gn o n l i n ea r a l g e b ra i c equ a t i o n s. A new, e f f i c i e n t f a c t o r i z a t i o nscheme fo r b l o ck t r i d i ag o n a l ma t r i ce s i s d e r i v ed . The theorydeveloped i s ap p l i ed t o t h e n umer i ca l s o l u t i o n o f p l an e Co u e t t eflow.


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Table of Contents

TITLE PAGEIntroductionChapter 1. Linear Two-Point Boundary-ValueProblems

1. Existence Theory2. Numerical Methods3. Convergence for Linear Boundary-Value ProblemsChapter 2. Difference Schemes

1. Taylor Series2. Quadrature3. Gap Schemes4. Other Schemes5. Stability of Triangular Schemes6. Asymptotic Error Expansions7. Increased Accuracy

Chapter 3. Parallel Shooting1. Parallel Shooting2. Method of Complementary Functions3. Method of Adjoints

Chapter 4. Nonlinear Two-Point Boundary-ValueProblems1. Finite Difference Schemes2. Existence of Solutions3. Newton's Method4. Equivalence of Shooting and ImplicitSchemes

Chapter 5. Block Tridiagonal Matrices1. Block LU Decomposition2. A Split Decomposition


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Chapter 6 . A Numerical Example: Pl an e Co ue tt eFlow

1. I s o l a t e d S o l u t i o n2. Numerical Procedure3. Numerical Resul ts

Appendix AReferences


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

T hi s t h e s i s d e a l s w it h t h e a p p li c a t io n of f i n i t e d i f f e r e n c eschemes t o two-point boundary-value p roblems. The assu mptio n i s madeth roughou t t ha t th ese boundary-va lue p rob lems have i so la te d so lu t ion s ;t h a t i s , t h e homogeneous, l in ea r i z ed p rob lem has on ly th e t r i v i a ls o l u t i o n , The g en e ra l t h eory d eve lo p ed p l ace s no r e s t r i c t i o n s o n t h eform o f t h e d i f f e r e n ce eq u a t i o n s ,

In Chapter 1, t h e a p p l i c a t io n of a n a r b i t r a r y , c o n s i s t e n td i f fe re nc e scheme t o a l in e a r boundary-value p rob lem i s t r e a t e d . I nt h e main t heorem o f t h i s ch ap t e r , Theorem 1 .1 6, t h e s t a b i l i t y of t h ed is c r e te boundary-value problem i s shown t o b e eq u i v a l e n t t o t h es t a b i l i t y of t h e a s s o c ia t e d d i s c r e t e i n i t i a l - v a l u e problem . T h isa s s o c i a t ed i n i t i a l -v a l u e p ro bl em emp loy s t h e same d i f f e r en ce eq u a t i o n st o a pp ro xi ma te t h e d i f f e r e n t i a l e q u a t io n , b u t i n i t i a l c o nd i t i on sr ep l ac e t h e b ou nd ary co n d i t i o n s . From t h i s r e s u l t , i t i s c l e a r t h a ta s imple shoot ing method i s , i n f a c t , a s p e c i f i c pr o ce d ur e f o r s o l v i n gth e d i sc r e te boundary-value problem.

Spec ia l emphasi s i s p laced on one-s t ep d i f f e r en ce schemes i nChapter 2 , H i g h o rd e r accu ra t e d i f f e r en ce ap p ro x i ma t i o n s a r e developedusing both Taylor ser ies and t h e i n t e g r a l f orm of t h e d i f f e r e n t i a le q u at i on , I n p a r t i c u l a r , o n e- st ep schem es of a r b i t r a r y o r d e r a r eder ive d which re qu ir e t h e ev al ua ti on of a minimum number of newf u n c t i o n s ( e. g d e r i v a t i v e s of A ( t ) , f ( t ) ) . The eq u i v a l en ce shown i nChapter 1 i s u se d t o exam ine t h e s t a b i l i t y of t r i a n g u l a r d i f f e r e n c e


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schemes.In Chapter 3 , th e equivalence re s u l t o f Theorem 1 .16 i s genera l -

i zed t o i nc lude a l l pa ra l l e l shoo t ing m ethods . Theorem 3 . 2 2 showsth a t t hese m e thods a r e e ac h a p a r t i c u l a r p r oc ed ur e f o r s o l v in g t h eequ at i ons der ive d from approximating l i n e a r boundary-value problems.The Method of Complementary Functions i s exam ined i n d e t a i l a s a nexample of methods f o r s olv ing problems with sep ara ted boundaryco nd iti on s. The Method of Ad jo in ts i s a l so cons ide red and i t i s shownt h a t t h i s method i s n o t i n g e n e r a l e q u iv a le n t t o t h e d i s c r e t eboundary-value problem,

Nonlinear boundary-value problems ar e de al t wi th i n Chapter 4.The di ff er en ce schemes examined i n Chapter 2 a r e g en e r al i ze d t o bea p p l i c a b l e t o n o n l i ne a r d i f f e r e n t i a l e qu a t io n s . F ol lo wi ng Keller [ 6 1,ex i s t e nce and un iquenes s o f t he se d i s c re t e app rox im at ions i s shown.W e n o te t h a t ~ e w t o n ' smethod converges quadrat ica l ly .

Chapter 5 i s concerned wi th t h e p ra c t i c a l p robl em of so lv ingth e sys tem s o f a l g eb r a i c equa t i ons a r i s i n g f rom the app roxim ation ofboundary-value problems wi th sep ara ted boundary con di t i ons . Thesee q u at i on s a r e w r i t t e n i n b l oc k t r i d i a g o n a l f or m, Mx = b. The specia lz e r o s t r u c t u r e of t h i s sy st em i s exp lo i t ed t o show tha t , w i th anappro p r i a t e row swi t ch ing s t r a t e gy , such a m a t r i x pos ses ses a s imp leblock LU decomposi t ion i f and only i f M i s nons ingular .

A numerical example i s presen t ed i n C hap te r 6. The equat ionscons idere d model pla ne Couet te f low. The Gap4 scheme, a s der ive d i nChapter 4 , i s used t o d i s c re t i ze t he non l i nea r boundary-va lueproblem and ~ewton ' smethod i s employed t o so lv e t he r e s u l t i ng s e t of


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n o n l i n ea r eq u a t i o n s .A c o n s i st e n t e f f o r t i s made t o u s e o t o r e p re s e n t z e ro o r

a z e ro v e c t o r and O ' f o r z e r o m a t r i c e s , e x c ep t i n t a b l e s o r e q u a t i o nnumbers . The numbering of theorems, equat ion s , o r t a b le s i s doneco n s ecu t i v e l y t h ro u g ho u t each ch ap t e r .


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Chapter I

Li ne ar Two-Point Boundary-Value P roblems

1. Exis tence Theory

W e co n s i d e r t h e s ys tem of n f i r s t -o rd e r , l i n e a r o r d i n a r yd i f f e r e n t i a l e q u a t i o n s :

where u, f, 6 a r e n -v ec t o rs and A , B , B a r e n x n ma t r i ce s . Be fo re0 1proceed ing to th e numer ica l approx imation o f ( l . l a , b ) , we p re sen tan ex i s t e nce and uniqueness re s u l t conven ien t fo r our purposes .

Theorem 1 . 2 . Let A( t ) E cm[o , l ] f o r s o me m 5 o . Def ine th efundamental m a tr ix ~ ( t ) s t h e s o l u t i o n o f

X' ( t ) - A(t ) X( t ) = o X(O) = I .

Then f o r e a c h f ( t ) E cm[o , l ] an d 6 E", problem ( l . l a ,b ) has au n i q u e s o l u t i o n y ( t ) E cm+l0, ] i f f [BO+BIX (1) ] i s nons ingu lar .Pro o f: The s o l u t i o n t o t h e i n i t i a l - v a l u e p ro bl em

i s i n g e ne r al


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Uniqueness fo r t he i n i t i a l - va l ue p roblem ins u re s t h a t X( t ) i snons ingu la r on [o , l ] . The boundary-value problem ( l . l a ,b ) has aso l u t i on i f and on ly i f we c a n de f in e a n n-ve ctor r s uc h t h a t y ( t )s a t i s f i e s t h e b oundary c o n d i t i on

Thi s r e qu i re s

Thus, ( l . l a ,b ) has a unique so lu t i on i f and only i f [B + B X(1) 1 i s0 1nons ingu lar . That y ( t ) E cm+'[o,l] i s an observa t ion f rom the formo f d i f f e r e n t i a l e q u at i on ( 1 .l a ) .

2 . Numerical Methods

Here w e dis cus s some s tandard concepts of numerica l ana l ys i sand develop some no ta t io n. In approximating th e so lu t i on of

i = J( l . l a , b ) , w e w i l l e m p l o y a n e t o f p o i n t s I t , } on [ o , l ] a n d a n e ti= Jfunc t ion {v , 1 d e f in e d on t h i s n e t .

Each v i s an n-vector and we d e f i n e V suc h tha ti


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where V i s an n ( J + l ) -v ec t o r , W e de f i ne th e mesh wid thsmax

hi = t i - t i = l , , . . , J , a n d h = l l i l J h \Je f u r t h e r r e q u i r ei-1' o i 't h a t f o r ea ch hi t h e r e e x i s t s a A E [ E , ~ ] , > 0 , s u ch t h a ti

T h i s c o n d i ti o n m er el y s t i p u l a t e s t h a t

E s min h./max hi 5 11

and t hu s t he mesh becomes dense i n [o , l ] a s ho + o.The norm we w i l l employ i s 1 I V 1 1 max0 S i l J I l v i l l and fo r

b lock ma t r i c es , th e usua l induced normmax

l l ~ l l I lv l l = 1 I Iw II n t he n= l case , t h i s induced norm equ als t he maximum abs ol ute rowsum, however t h i s r e s u l t d o es n o t g en e ra l i ze t o n > 1. We maye a si ly produce t he upper bound

max JI I"I I ' si


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The boundary condi t io ns ( l . l b) a r e approximated by

We d e f i n e t h e t r u n c a t i on e r r o r T t o bei

where y( t ) i s any s o l u t i o n of t h e d i f f e r e n t i a l e qu a ti o n ( l o l a ) .S im i l a r l y , we de f ine a t runc a t i on e r ro r T a s s o c i a t e d w i t h t h e0boundary condi t ions

where y ( t ) i s any so lu t ion of (1 . lb ) . Note that ro w i l l always b eze ro . In cons ide ri ng the accuracy of approximations (1.5a), (1 .5b),w e a r e concerned wi th y ( t ) a so lu t i on of ( l . l a ,b ) and we de f ine

Combining th e d is c r e te approximations (1.5a) and (1.5b), th ed is c r e te boundary-value problem may be w ri t t en a s


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More briefly, we will write

where B is an n(J+l) x n(J+l) matrix and r is an n(J+l)-vectorhwith ro = f3. Employing Euler's method to approximate the differentialequation (lala), this formulation of the discrete problem becomes

This example will be used throughout to illustrate various points ofinterest.

Consistency. The difference approximation (1.5a) is saidto be consistent with the differential equation (l.la) iff

where y(t) is a solution of (l.la,b). ~uler's ethod provides anexample. If y(t) E cP[o,l], p 2 2, then


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Thus, ~ u l e r ' smethod i s con s i s t en t and we say tha t i t i s f i r s t - o r d e racc u ra t e b ecau se t h e l e ad i n g t e rm i n t h e t r u n ca t i o n e r r o r ex pan si oni s l i n ea r i n h .i

Order of acc uracy . A numerical scheme has an order ofaccuracy p > o i f p i s t h e l a r g e s t i n t e g e r s uc h t h a t

f o r a l l n e t s wi th h o 5 H.S t a b i l i t y . L e t V be a s o l u t i o n t o (1 .8) . The d i f f e r en ce

scheme (1.8) i s s a i d t o be s t a b l e i f f t h e r e e x i s t c o n st a nt sK 2 o, and H > o , such th a t0

f o r a l l n e ts w it h ho 5 H. For t h e l i n e a r c a s e , t h i s c o n d i ti o n i sequ iva le n t t o showing th a t a KO 2 0 and H > o e x i s t s uc h t h a t

f o r a l l n e t s w it h ho 5 H.Convergence. The nu me ri ca l scheme (1.8) i s covergent iff

maxoSi


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A s tand ard r e s u l t may now be s ta te d ,

Lemma 1.10 Let th e boundary-value problem ( l . la , b ) have t h e ex ac tso lu t i on y ( t ) . Le t t h e d i s c re t e boundary-va lue p robl em

be con s i s t en t w i th ( l . l a ,b ) and s t a b l e . Then t h e method i s convergent ,t h a t i s ,

P ro of : The d e f i n i t i o n of t h e t r u n c at i o n e r r o r ~ [ y ( t ) ] ombined with(1 .5a , b ) g ive s

By hypoth es i s , t he d i f fe re nc e scheme i s s t a b l e , t h us t h e r e e x i s tc o n s t a n t s KO 2 0 , H > o s u c h t h a t

f o r a l l n e t s wi th ho 5 Ha Also by hypoth esi s , t h e method i sc o n s i s t e n t , t h a t i s


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Hence,

maxl y - 'I l = .,i


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S i m i l a r l y , we d e f i n e t h e d i s c r e t e i n i t i a l - v a l u e m a t r i x T h a s s o c i a t e dwi th any % (1 .7) to be

To form I w e r e p l a c e Bo , B i n t h e f i r s t b lo ck row of i3 wi thh' 1 h1 , 0 r e s p e c t i v e l y .

3 . Convergence f o r li n e a r boundary-value problems--T h e main re su l t of t h i s c hap te r i s that a numerical scheme

ap pl ie d to a boundary-value problem wit h a un ique so lu t i on i s s t a b l eand c ons i s t e n t i f a nd on ly i f t he a s soc i a t e d i n i t i a l - va lu e p roblemi s s t a b l e and c o n s i s t e n t . I t w i l l be use f ul t o prove sev er a l l emmasf i r s t .

Lemma 1.12 (F ac to ri za ti on ) Let 7 b e t h e i n i t i a l - v a l u e m a t ri xha s s o c i a t e d w i t h . Then'I,


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where L is a (J+l)n x n+l matrix

N is an (n+l) x (J+l)n matrix

and 6 is an (n+l)-vector

Proof: We take (1.7) to be the most general form of the boundary-valuematrix Bh.

The associated initial-value matrix is


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It may be d i re c t l y ve r i f i e d t h a t L,N,C a r e t h e p roper f a c t o r s ,bu t t h e se quence of ope ra t i ons below may c l a r i f y t he de r iva t i o n .

The ide n t i t y ro = 6 i s used here .

' Lemma 1.13 (Reducing) Let L ,N be pxq and qxp ma tr ice s resp ec t iv el y.Let x and b be a p-vector and a q-vector re sp ec t iv ely . F u r th e r , l e tL have r ank q 5 p. Then th e mat r ix equa t ion

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

i f and o nl y i f

Proof : The suf f ic i enc y i s t r i v i a l on s u b s t i t u t i o n of Lw i n t o ( 1 . 1 4 ) .Since L i s a pxq matr ix with rank q 5 p, th e columns of L


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a r e l i n e a r l y i nd e pe n de n t. T h e re f o re , t h e r e e x i s t s a p x ( P - ~ )m a t r i x L s u c h t h a t

W e may write an y v ec t o r x E EP u n i q u e l y a s

U sin g t h i s r e p r e s e n t a t i o n i n ( 1. 14 ),

The vector x i s a s o l u t i o n of (1.14 ) i f and o n l y i f

and

The n e c e s s i t y f o l l o w s .m

N ote t h a t t h i s lemma r ed u ces t h e s o l u t i o n o f p s imul taneouseq u a t i o n s

t o s o l u t i o n of q e q u a t i on s , q 5 p ,

Now w e s t a t e and pr ov e t h e m ain r e s u l t o f t h i s c h a p t e r.


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Theorem 1.16 Let th e boundary-value problem ( l , l a , b ) have a1u ni qu e s o l u t i o n y ( t ) E c [0 ,1 ] . Then t h e fo l l ow i n g a r e eq u i v a l en t :

a ) The d is c r e t e boundary-value problem i s s t a b l e , c o n s i s t e n t(and convergent) .

b ) The a s s o c i a t ed i n i t i a l -v a l u e p ro bl em i s s t a b l e , c o n s i s t e n t(and convergent) .

Proof: (b => a ) The Fa c to r i z a t ion Lemma (1 .12) s t a t e s th a t fo r th eboundary-value problem (1.8)

The in i t i a l -v a l ue p roblem i s s t a b l e by h y p ot h e s is , t h e r e f o r e Ih i sn on si ng u la r f o r a l l ho 5 H, say. Lef t mul t ip l y ing (1 .17) by-1Ih w e o b t a i n

Define Z and z by

where Z i s an n (J+ l )xn mat r ix and z i s an n ( ~ + l ) -v ec t o r , From (1 .1 8 ),we no te th a t Z s a t i s f i e s

and thus i s an approx imat ion t o t he fundamenta l mat r ix so lu t ion (1-,3),By Lemma 1. 10, t h e d i s c r e t e i n i t i a l - v a l u e pr ob le m i s convergent


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maxo r i l J I I z i - x ( t i ) I I +- o a s h -+ o0and i n p a r t i c u l a r

I Iz J - x ( ~ ) I I + o a s h o -+ o .By v i r t u e of (1, 18) , we may w r it e (1.17) a s

v + [z : ,z ] rw - 5 1 = 0 .1The odumns of th e n(J +l) xn+ l m at rix [Z , z ] a r e l i n e a r l y i nd ep en de nt

s tprov ided tha t r, # o, 1 5 i 5 J . I f th e (n+ l)-- column of L i s t h eI

ze ro v ec t o r w e remove i t , r e p l a c e = k ] y J = [ e ] and thedegeneracy i s removed, We complete th e proof assuming r # o , f o rjsome j .

By t h e Redu cing Lemma, (1.20) has a so lu t io n i f and only i f- 1 a t i s f i e s ,

and th e so lu t i on must be of th e form

W e r e c a l l t h a t Zo = I and take A = 1, s o t h a t ( 1.21 ) i s eq u i v a l en t t o

Compare t h i s w i t h the condition de.ri.vc?(l i n Theorem ( 1 . 2 ) f o r the.


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so l u t i on o f th e boundary-va lue p roblem ( l . l a , b ) . Ex i s t e nce an du n iq u enes s o f t h e s o l u t i o n , V, of t h e d i s c r e t e bo un da ry -v al ueproblem now hinge on th e non -s in gul ar i ty of (B + BIZJ). We w r i t e

0

By hypo thesis y ( t ) i s unique , and Theorem 1.2 s t a t e s th a t[Bo + BIX(l)] must be non-s ingul ar. I t has al re ad y been shown th at

t h u s , by t h e Banach Lemma, B + B Z i s n on -s in gu la r f o r a l l n e t s0 1 Jwith ho 5 H, provided H i s s o s m a l l t h a t f o r s o m e p E ( o , l )

It i s c l ea r t h a t t h e d i s c r e t e b ou nd ary-va lu e p ro bl em i s s t a b l e s i n c e

and Z and z a r e s o l u t i o n s of d i s c r e t e i n i t i a l - v a l u e p ro bl em s, whichwere assumed s table .

( a => b) The p roof o f t h i s pa r t i s e s s e n t i a l l y t h e s a m e .C on si de r t h e d i s c r e t e i n i t i a l - v a l u e pr oble m

I h V - r = o .

Following t h e Fa ct or iz at io n Lermna (1.1 2), w e may w r i t e

Th v - r = Bhv - L{NV + 6 1 .By hypoth esis , th e d is c r e t e boundary-value problem i s s t a b l e , t h u s


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A8 i s non -sin gula r and we may d e f i n e [ Z ! 21 byh

The Reducing Lemma (1.13) now im pl ie s t h a t t h e so lu ti o n V of (1.23)must be of th e form

and e x i s t s i f and o n ly i f

AThere fore , w e wish to show that Z i s n on -s in gu la r f o r a l l n e t s w it h0h 5 H, f o r H su f f i c i e n t ly sma l l . Theorem 1 . 2 de r iv e s t h e so l u t i on0of t h e boundary-value problem to be

where r m u s t s a t i s f y

Thus, t h e so lu t i on to t he boundary-value problemA2' ( t ) - A ( t ) ~ ( t ) o

BoX(o) + BIX(l) - I = 0i s given by


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where R i s a n n x n matr ix and must s a t i s f y

By hypot hes is y ( t ) i s unique , t h u s t h e m a t r i x R i s non-s ingula r ,A At h a t i s , X(o) i s non-s ingula r . W e n o t e t h a t Z a s d e f i ne d i n ( 1. 24 )

i s a convergent approximat ion t o th e equa t i ons (1 .26) , hence

Aa nd , a s b e fo r e , Z i s n on -s in gu la r f o r a l l n e t s w it h h 5 H , where0 0H i s su f f i c i e n t ly sm a l l . Now,

and t h e d i s c r e t e i n i t i a l - v a l u e pro blem h a s a s t a b l e s o l u t i o n .m

W e r e c a l l t h a t t h e o r de r o f a c c u r a c y o f a scheme i s determinedby t h e l a r g e s t i n t e g e r p s uc h t h a t

I l ~ [ ~ ( t ) ]1 5 K h f o r a l l h 5 H.0Coro l la r y 1 .27. Le t th e boundary-value problem ( l . la ,b ) have a uniques o l u t i o n y ( t ) E C P + ~ [ O , ~ ] ,o r some in t e g e r p 2 1. L et t h e d i s c r e t eapproximat ions (1 .5a ,b) be p-order ac cu ra t e , Fur t her , l e t t h e d i s c r e t ei n i t i a l - v a l u e p ro bl em a s s o c i a t e d w i t h ( 1. 5 a, b ) b e s t a b l e . Then f o ra l l n e t s w i th ho 5 H,

where V i s t h e n ( J+ l ) - ve c to r so lu t i on o f ( 1 .5a ,b ) and k i s a constant


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independe nt of hoe

Proo f. The proof fol lo ws t h a t of Theorem 1.16.

These r e s u l t s and t he man ipu la t ions l e ad ing to them. sugges ts e v e r a l f u r t h e r t o p i c s , C ha pt er 11 w i l l be concerned wi th theo p e r a t i o n a l c h a r a c t e r s i c s of v a r i o u s s p e c i f i c n umer i ca l schemes,Theorem 1 .1 6 w i l l b e e x p l o i te d t o d e te r mi ne t h e s t a b i l i t y p r o p e r t i e sof a p a r t i c u l a r c l a s s o f s ch emes. W e w i l l examine asymptoticer ro r expansions and d i s cus s methods o f inc rea s in g th e accuracy o fd i s c r e t e ap pro xi mat io n s. I n Chapt er 111, we w i l l en l arg e uponTheorem 1 .16 and the equ iva lence of d i sc re te in i t i a l - va lu e andboundary-value problems.

D r . H , 0 , Kreiss h as r ec en t l y p u b li s h ed a p ap er [ 7 ]d e a l i n g w i t h t h e s t a b i l i t y of d i s c r e t e ap pr ox im at io ns t o a r b i t r a r yo r d e r s ys te ms o f l i n e a r o r d i n a r y d i f f e r e n t i a l e qu a t io n s . Thus, i ta p pe a rs t h a t t h i s c h a p t er i s p e rh aps a s p ec i a l c a s e of Kreisst r e s u l t s .However, i n Appendix A, we g i v e a n i n d i c a t i o n t h a t K r e i s s t work i scomplementary t o our own and no t inc lu si ve . Fu rt he r, Theorem 1.16g i v e s a n e ce s sa r y and s u f f i c i e n t c o n d i ti o n f o r s t a b i l i t y of t h emost general schemes, whereas D r . Krei s s de a l s wi th k -s t ep methods.


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Chapter I1

Difference Schemes

I n t h i s c h a pt e r, w e w i l l examine some spe c i f i c d i f f e r en ceschemes ; of par t i cu la r i n t e re s t a r e one-s tep ( two-po in t ) methods.One-s tep methods f o r l i n e a r p rob lems have the ch ar ac te r i s t i c fo rm

T hese metho ds a r e o f s p e c i a l i n t e r e s t f o r s e v e ra l r ea s o n s . They ad mitnonun ifo rm ne t s and , a s Ke l l e r 1 5 1 has shown, piecewise smooths o l u t i o n s . I n a d d i t i o n , t h e c a l c u l a t i o n s in v o lv e d i n s o l v i n g t h em a t r i x e q u a t io n

f o r s e p a r a t e d b oun dar y c o n d i ti o n s a r e p a r t i c u l a r l y s im p le . I n t h i sc a s e, t h e m a t r i x 8 may be pu t i n t o a b lock t r id ia go na l form whichhpos ses ses a s imp le LU-decomposition.

One of two methods i s u s u a l l y employed t o g en e ra t e d i f f e r en ceap pro xi mat io n s t o ( l o l a ) and ev a l u a t e t h e i r o rd e r of a ccuracy . Thef i r s t a pp ro ach i s v i a T ay lo r s e r i e s and n e c e s s i t a t e s e v a l u a ti n g t h et r u n c a t i o n e r r o r f u n c t i o n s a t some r e f e r e n c e p o i n t t ( t ) whereR i

Quadra tu re fo rmulae may a l s o be used t o gen era te us ef u l approx imat ions .I f we w r i t e (1. a ) i n t h e e q u i v a l e n t form


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t h e a p p l i c a t i o n of q u a d r a t ur e i s immediately obvious.

1. Tav lo r S e r i e s

The centered-Euler o r Box-scheme (Ke l ler 1 :5]) i l l u s t r a t e st h e u s e of T a y lo r s e r i e s . A pp li ed t o ( l o l a ) t h e c e n t e re d E u l e r scheme

where t - - - ii-112 - ti 2

I f we assume th a t y( t) E C2nrfl [o , l ] and l e t tR( t i ) = t i-1l2,

t h e t r u n c a t i o n e r r o r i s shown t o b e

2and the l ead ing term, 0 (hi) , i s

Here the choice of tR -- t i-1/2 i s impor tant due t o th e symmetry of

(2 .2 ) . However, t h e f i r s t t e rm o f t he t runca t i on e r r o r expans ion ,(2.3b), i s in va ri an t and hence the or de r of acc uracy of any scheme


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must b e i n v a r i a n t t o t h e r e f e r e n c e p o i n t t I f we were t o useR.-tR - ti-l' th e t r u n c a t i o n e r r o r i s shown t o be

and t he l ead in g term i s

However, r e ca l l i ng t ha t y ( t ) i s an exac t so lu t i on o f (1 . l a ) , t h i sexp res s ion i s shown t o be equ iv a l en t t o t h e fo l l owing

Thus, th e le ad in g term of (2.3a) i s exa ct ly t he same a s (2 .4a) wi th= tt~ i-1/2 and t =R 5-1 r e spe c t i ve ly . Note t h a t t he nex t t erms

i n each expansion a r e no t t he same.The centered-Euler scheme may be gene ra l iz ed t o an a r b i t r a r i l y

h igh ord er by employing the exact form of th e t runc at i on er ro r . In -2c o r p o r a ti n g t h e f i r s t t er m, O (h i) , o f t h e t r u n c a t i o n e r r o r e xp a ns io n


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Act) and f (t ) , we must a l s o ev al ua te A") ( t ) , A(') ( t ) , f ( t ) ,f ( l ) ( t ) and var ious p roducts of thes e funct ions .

Since the funct ions C( t) and g( t ) w i l l co n t in u e t o ap p ea r,we need t o d e f in e th ese q u an t i t i e s i n more g en e ra l i t y . The usualprocedure w i l l b e t o e v a l u a t e y ( p ) ( t ) i n terms of d e r iv a t iv es an dcombinat ions o f A( t ) , f ( t ) and y( t ) . For s i m p l i c i t y , w e d e f in e

where C ( t ) i s an n x n matr ix and g ( t ) i s an n-vector.v VThus,

Y ( 0 ) ( t ) = y ( t ) => Co = I , go = 0 .The te rms of p a r t i cu la r in t e r e s t a r e co nta in ed in th e fo l lo win g t ab le .

Table 2.7

Employing t h i s no t a t i on , the four th -order genera l ize d cen tered-Eulerscheme (2.5) i s


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I n t h e n e x t s e c t i o n , t h i s m ethod w i l l be improved upon i n t he sen set h a t t h e f u n c t i o ns Co, C1, C 2 , and C w i l l b e u se d t o d e f i n e a s i x t h -3order-method.

2. Qu adr atu re Employing (2.1) we g e t

where y ( t ) i s t h e e x a c t s o l u t i o n t o ( l . l a , b ) . I f we a p p ly t h et r a p e z o i d a l r u l e t o ( 2. 9) , t h e r e s u l t i s

Thus a se co nd o r d e r a c c u r a t e d i f f e r e n c e scheme ( ~ r a p e z o i d a l u l e )maybe def ined by


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This may be e q u i v a le n t l y w r i t t e n a s

u s i n g t h e f u n c t i o n s i n Tabl e 2.7. This one-step method and ot he rs ofa r b i t r a r i l y h i g h o r d e r may b e d e r iv e d i n a more g e n e r a l s e t t i n g bymeans of (2.9) .Lemma 2.11. L e t b ( t ) E cdl[a,b]. Also l e t polynomia l s p i ( t ) bed e f i n ed by

wh.er e 5, E [arb]. Then

+ (- l)& l (s) b(&') ( s) ds ,

Pro of : Rep ea ted i n t eg ra t i o n by p a r t s y i e l d s ( a ) i mmed ia t el y.Induc t ion on v y i e l d s ( b ) . The indu c t ion hypo thes i s i s s t a r t e d by


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(2.12a) and, for ma lly , i s

v-1IP,-,(t)l 5 Ib-al

Thus,

Innumerable schemes pr es en t themse lves by way of t h i sLemma. The Euler-Maclarin sum formula may be derived using polynomialswhich look l i k e

where the s c a l e f o r e ach p o ly no mia l i s n e c e s sa r i l y d i f f e r e n t . Thus,th e Eu ler -Maclarin fo rmula may be used t o de f in e a d i f f e r en ce scheme


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of a r b i t r a r i l y high o rd e r . A f i f t h -o rd e r method d e f i n ed i n t h i smanner i s

One und esi rab le as pe ct of (2.13) i s e v a l u a ti n g c 4 ( t ) and g ( t ) . I n4or de r t o employ (2.13) w e must eva lua te ( t ) , ( t ) , A(') ( t ) ,A(t) , f (3) (t) , f C 2 ) t ) , f ( I ) ( t ) , f ( t ) and a ss or te d products and sums oft h e s e fu n c t i o n s .

3. Gap schemes Thus fa r , w e have cons idered se ve ra l one-s tep schemes.The centered-Euler scheme w a s g e n e r a l iz e d i n S e c t i o n 1 t o a f o u rt h -o rder accura te scheme. T h i s i n c r ea s e i n accu racy i s b o ug h t a t t h ep r i c e of ev a l u a t i n g A C2) ( t ) , A(') ( t ) , f (2 ) ( t ) , a nd f ( I ) ( t ) i n a d d i t i o nt o A( t) and f ( t ) . I n Se ct io n 2, th e Euler-Maclarin Sum Formula wasu sed t o d e f i n e a f i f t h -o rd e r meth od, b u t t h i s d i f f e r en c e schemereq u i r ed ev a l u a t i o n o f ( t ) , f (3) ( t ) and a l l l ow er o rd e r d e r i v a t i v e s .S ince each new func t i on t o be eva lu a ted inc rea ses bo th p rogrammingcomplexity and computation t ime, we want t o examine hig h-o rde r, one-s t e p methods wi th a minimum of de ri va t i ve s r equ ire d.

The cente red- Euler scheme and th e Tra pez oid al Rule (2.10)possess ano th er us ef u l p roper ty . Each o f these d i f fe re nc e schemeshas a t ru nca t ion e r r o r expansion which con ta ins on ly even powers of 11i s


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For reasons tha t w i l l become c le ar , se e Sec t ion 6 on asympto t i c e r ro rexpansions , w e want t o p reserve th i s p ro p e r t y i n co n s i d e r i n g h ig h-order methods.

R eca ll in g Lemma 2.1 1, we ha ve the r e s u l t

b+ ( - l ) & l f P&l (s ) b(&') ( s ) ds .

aV=mI f th e sequence of polynomials (pV(t) lvmocould be d e f i ne d i n s u ch a

way t h a t13y(a) = pV(b) = 0 v=n+l,. , 2n

then a d i f fe re nce method o f o rd er 2n can be def ined which r equ i re s theev a l u a t i o n o f A - ) 1 , f("-') ( t ) , and a l l l ow er-orde r d e r i v a t i v e s .For i n s t an ce , a f o u r th -o rd er method s o d e f i n ed r eq u i r e s ev a l u a t i n g~ ( " ( t ) and f(') ( t ) where a s the genera l i zed cen te red-Euler schemeneeds ~ ( " ( t ) , f ( 2 ) ( t ) , e t c .

1 n nLenrma 2.14 Let pZ n( t) = ( t -a) ( t -b) . Define the sequence ofpolynomials {py ( t ) 3 , V 2 0 , s u c h t h a t


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Thena) (p v( t) ) s a t i s f y conclus ion a) of Lemma 2.11.

Proof: We can re pl ac e (.2.15a) w it h

and the sequence s a t i s f i e s conclu sion a) of Lemma 2.11.

b. 5 Ib-alZn and the re s u l t i s immediate byinduct on .

c. Immediate from (2.1 5a).

d. To prove t h i s p a r t of th e Lemma, i t i s s u f f i c i e n t t oshow th a t p2r( t) , r 2 n , i s an even function about t =-+a Without2 -b+alo s s of g en er al i t y , we assume th a t-2 - 0 , thus tZr 0 . BYhypothesis , p ( t ) i s even about F, ence we assume th a t p2n 2 (v-1) ( t>i s an even function.

Therefore , p ( t ) i s an odd function.2v-1


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3nt Iw


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-(L)R e c a l l i n g t h e d e f i n i t i o n o f P ( t ) , f o r L < n, P ( t i ) = 0 .-

Thus,

- m!2n! m-n) ! (2n-m)! ! (hi)2n-m*

Define 8 (-1)k (2n-k) ! "I and (2.16) becomes2n!(n-k)! k!

Table 2.17

A s examples, w e wr i te ou t the four th-order and s ix th-o rder schemes.


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Fourth-order scheme:

Sixth-order scheme:

The matrices C and the ve c to r s gv a r e def ined i n Tab le 2 .7 .V

D r . Ke ll er noted t ha t the se gap schemes might b e d e r i v edu s i ng He rm it e i n t e r p o l a t i o n t o e s t i m a t e t h e f u n c t i o n

b ( t ) = A ( t )y ( t ) + f ( t ) .For the fou rth- ord er gap scheme (2. 18) , we w r i t e

b ( t ) = H(t) + ( b ( t ) - H ( t ) )

where B(t) i s the lowes t o rder po lynomia l th a t matches b (ti) , b(') ( t i ) ,b ( t i - and b0(tiel). I n t h i s c a s e ,


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Re c a l li ng t he i n t e g r a l e qua tion (2 .1 ) , we have

f f ( t ) may be i n t e g r a t e d e x a c t ly t o g e t

S inc e b ( t ) = J1\t) , t h i s formulat ion w i l l y i e l d a di f fe rence schemee xa c t ly t h e same a s (2 .18) wi th t h e t runc a t ion e r ro r


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4 . Other Schemes We a r e p a r t i c u l a r l y i n t e r e s t e d i n o ne -s te pdi f fe re nc e schemes of th e form (2.2) , (2.5) , and (2.18) , thus on ly abr ie f ment ion w i l l be made of o th er methods . The i n t e g r a l e q u a ti o n(2.1) su gg es ts k-ste p methods of th e form

Note th a t (2.20) re pr es en ts (J-k+l)n equ at i ons i n (J +l ) n unknowns, andkn more eq uat ion must be found t o determine V uniquely. These ex t ra

equat ions are given by t he u sua l boundary condi t ions (1.5b) and byI1s t ar t i ng " m-step d i f f er en ce schemes , m < k , usu al ly of lower ord erthan (2.20). I n many case s the "s ta r t i ng " scheme i s a one-s te p scheme.Combining a l l of t hese equa t i ons i n to t he form

t h e i n i t i a l - v a l u e matrix, Ih, i s l ower t r i a ngu l a r . I f t he one- st epI1s t a r t i ng" s chem e i s of lower orde r than the k-s tep method, the meshs i ze , h ( l ) , a s soc i a t ed wi th t h e one- st ep method shou ld be sm al l e r t hant h a t of th e k-s te p method, h(k) . I f th e methods ar e of orde r p and qr e spec t i ve ly (p < q) , t hen C oro l l a ry 1.27 i n d i c a t e s t ha t h (1) andh (k ) should be r e l a t e d by

5. St a b i l i ty of Tr iangula r Di f ference Schemes A l l of th e methodsmentioned her e and most of tho se commonly used giv e r i s e t o blockt r i a n g u l a r i n i t i a l - v a l u e m a tr i ce s . That i s , they may b e w r i t t e nI h


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i n the form

The factor of J a pp ea ri ng w i t h a l l M ' s i s a normal iza t ion of t heiinve rse of the mesh s i ze . We would l i k e t o examine the s t a b i l i t y offami l ies of mat r ices 1 of t h i s form. The s t a b i l i t y t heory f o r d i s c r e tehini t ia l -va lue problems i s well-developed prov ided t h a t a k-s tepdifference scheme i s used and a l l Mij c a n be wr i t t e n a s M. . I , M a1 J i js c a l a r . We a r e i n t e r e s t e d i n a r e s u l t w hich do es no t r e q u i r e t h e s etwo r e s t r i c t i o n s on IheLemma (2 .22) . L e t Ih be a ma tri x of th e form of (2.21) . DefineDv, Nv, V=0 ,1 , by


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-1 1L e t Do b e n o n s in g u la r an d l e t I IDo I I = d , I D ~ I = dl, 1 1 ~ ~ 1= no,0

0and I IN1/ 1 = nl, indepen dent of J. Fu r th e r , l e t -- 1. Then0

Proof : T h e matrix 1 i s w r i t e n a s- ho r , e q u i v a l e n t l y

where

and

-1By hyp oth esi s and the Banach Lemma, D + J D i s n o n s i n g u l a r f o r0 IIJ > J o = - , and

do


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Since (Do + J-'D~)-' i s block dia gon al and (No + J - h l ) i s block

n i lp o te n t , the p roduc t (Do + J - ~ D ~ ) - ~N o + J m b l ) i s b lo c k n i l p o t e n t .

Thus,

where


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Combining Coro ll ar y 1.27 and Lemma ( 2.22) , we s t a t e the fo l lowingTheorem wit hou t pro of,

Theorem 2.24. Let the l i n e a r boundary-value problem (l . la ,b ) have a1unique so lu t i o n y ( t ) E C [O,l ]. Also , l e t the d i f f e ren ce scheme be

pth-order accurate. L e t Bh be of the form

-1 1where M i s nons ingula r f o r every i=O , ., and I 1 M 1 1 5 2 ,ii vv 0-I M ~ ~ ~5 dl, independent of J. F u r th e r, l e t

n0and - 5 1. Then f o r h < H , H s u f f i c i e n t l y s m a l l ,d 0 -0

This Theorem gives a precise est imate on the convergence of finitedi ff er en ce sol ut io ns . However, we now want t o look a t a more d et ai le daccounting of the e r r or incurred by approximat ing ( l . l a , b) wi th adif fe ren ce scheme.


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6. Asymptotic Error ExpansionsI n S e c t i o n 3, the gap schemes were der ived s o th at th e

tru nc at io n e r r o r expansion would have on ly even powers of h Thei*advantage of t h i s arrangement w i l l b e e v id e n t i n t h i s d i s c u s s i on .The genera l d i f ference scheme i s def ined as

The t run cat ion er ro r i s h e r e d e fi n ed t o b e

where z(t) e c ~ + ~ [ 0 , 1 ] .For example , Euler ' s method has the t run cat ion er r or , l e t t i n g

-tR - t im l '

I n general , the t runca t ion e r ro r w i l l be assumed t o be of th e form

= Lz(tR) - ctR)


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where each f (A,f) i s a l i n e a r d i f f e r e n t i a l o p e ra t o r of a t most o r d e rkk+ l which depends, in genera l non l ine ar ly , upon A ( ~ ) t R ), f (k ) ( tR )and a l l l ow er o r d e r d e r i v a t i v e s . For Euler 's method,

Theorem 2.27. Let the d i f f er e n t i a l equat ion ( l . la ,b ) have a uniques o l u t i o n y ( t ) E cq + l ~ ~ , l ]here A(t) E cq [O, l ] and f ( t ) E c q ~ 0 , 1 ~ .Let

b e a s t a b l e , con s is t en t d i f f ere nce approximation to (1. la ,b) . F ur the r ,l e t the t runca t ion e r ro r be g iven by

where z ( t ) E CP+l[O,l] and tR = t i + O(hi) . A lso l e tF k ( ~ ( t ) f ( t ) ) z ( t ) E cv[0, l] where v = midp-k,q-k} . Then fo r a l ln e t s w i t h h < H, H s u f f i c i e n t l y s m a l l ,0 -


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where yo(t) = y( t ) and yk( t ) , k 1, i s defined by

Proof: By induc t ion and the con t in u i ty of F (A,) , A(t) , a n d f ( t ) ,- ki t can be shown t h a t

Immediately upon substi tuting

in to th e t runcat ion e r ro r expansion (2 .28b), we get

Since yk(t) s a t i s f y (2.30a) , we have immediately

and hence

The boundary conditions and .ro[w(t) J c l e a r l y y i e l d


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By hypothesis , th e di ff ere nc e scheme i s s t a b l e , t h u s

O r equ iva len t ly ,

The exp res sio n (2.29) f o r vi - y ( t .) i s termed the asymptotic1er ro r expansion. In the following discu ssion we assume t h a ty ( t ) E cm[O,l], A(t) E cm[O,l] , and f ( t ) E cm[O,l]. Suppose that thetr un ca ti on e rr o r -ri[z(t)] has only even powers of h t h a t i s ,i'F2k+l = 0 , k=O,l, . The func t ion y l ( t ) i s defined by

S in c e t h e d i f f e r e n t i a l eq u at i on ( l . l a , b ) h a s a un ique s o lu t ion ,

Assume that y2k-l(t) Z 0 , k=1,2,..., , then y2vtl(t) i sdefined by


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nBr-l+dh)V

Uh* r-l b?" g ,4 &" 3 :z r t

3. " ! g& rlh k i

k r l m It+ a$ 8 kh *rl 0U ka &8 a a5 5 "k 4- U L )

(I]*I4d 3a (I]a, * -rlg*dUr3I]:


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2 4because the h. and h terms a r e nonvanishing i n (2.32). However,1 i

closer examinat ion of F2 and F4 y i e l d s t h e f ol lo w in g i d e n t i t i e s :

d- ~ ( A ' ~ ) - ~ A A ( ~ ' - ~ A ( ~ ) At [Lz tR) f ( tR) ]

+ ~(A(~)A+A~(~)+~A(~)A(~))Lz tR ) ( tR) ]1-Thus, y 2 ( t ) i s defined by

However, from (2.33) i t can b e s e e n t h a t

thus , y2( t ) = 0 . I n a similar fa sh io n we can show t h a t y ( t ) = o.4


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Theorem (2.27) an d the i d e n t i t i e s ( 2 .33 ) and ( 2 . 3 4 ) show that for t h eGap6 dif fe re nc e scheme

where y6(t) i s def ined by

7. Increased Accuracy W e w i l l consider two methods of increasingt h e accu racy of n u meri cal s o l u t i o n s t o t h e d i f f e r e n t i a l eq u at i on(1.a , ) . These methods ar e , by name, R ichards on h -t o e x t r a p o l a t i o nand FOX'S method of Def erred Cor rect ions (Fox [ 2 1, Pereyra [ 9 1) .Both methods rely on t h e e n t i r e t r u n c a ti o n e r r o r s e r i e s rather thanth e orde r of accuracy. Richardson ex t ra pol at i on employs a so lu t i onof th e d i s c r e t e p ro blem on two d i f f e r e n t n e t s t o e l i mi n a t e s u cce s s i v ee r r o r terms; the method of Deferred Cor rect i ons uses a s o l u t i o n t o t h ed i s c r e t e prob lem t o app ro xi ma te t h e f i r s t t r u n ca t i o n e r ro r t erm.

To i l l u s t r a t e th e method of Deferred Corre ct ion s we look a tth e center ed-E uler method (uniform mesh)


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U[I]krl'44 Q)k3a

Q)00ka[I]rl6Ucd5Q)U0E:

n.9,cddI-lu0U

gdU3rl0rnQ)Ucdk50cdkQ)ak0

rlakQ)3a[I]

3C20dUC)akk0Uaa,kkQ)w8w0a0588hurCVw

32rn

hl\dI=rlwInrlIdfdl *rl3UN1rlI

InrlIr ld?f-1 -rl3U

rlld


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disadvantages. At the boundary t=O, the quantity vo must be defined-1and thus we have to examine the solution outside the domain of interest.Also, the solution procedure for VO is significantly different than(2.38) where the approximation to the first truncation error term isincluded. Evaluation of higher-order derivatives of y(t) is at best adelicate procedure and must be done with the utmost care.

Richardson extrapolation avoids these particular problems.Once two aolutions are calculated, each on a different net, the powerstructure of the asymptotic error expansion is used to eliminate theleading error term at those points common to both nets. For example,the sixth-order gap scheme has an asymptotic error expansion of theform

If we solve the discrete problem twice for {vi (1)3 and {vi(2) 1 , then

Let ti(l) = t. (2), where t (1) is the ith point of the first net andJ it (2) is the j h point of the second net.J

Defh e


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then

This elimination procedure may, of course, be carried on as long asi s pract ical , tk at i s , u n t i l the continuity of y(t) or round-offerrors make further refinement unproductive.


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Chapter I11

P a r a l l e l S h o o t i n g

T h i s ch ap t e r exami nes t h e r e l a t i o n s h i p be tw een i m p l i c i tf i n i t e d i f f e r e n c e p ro ce du re s and d i s c r e t e i n i t i a l - v a l u e t ec h ni qu e s.The proof of Theorem 1.16 yi e ld s th e fo l lowing re s u l t : I f bo thth e boundary-value procedure

and t h e i n i t i a l - v a l u e p r oc e du r e

BVhave un' ique so lu t io ns , then v IV = Vi , i = o , . . J . That i s , t h e1procedure (3 .2a,b ,c) i s ac tu a l ly a method o f so lv ing th e equa t ions( 3 1 ) The main r e s u l t of t h i s c ha p te r s t a t e s t h a t any p a r a l l e lshoo t ing t echn ique i s a l s o eq u i v a l en t t o (3 .1 ) an d , t h u s , t o(3 .2a ,b ,c ) .

P a r a l l e l s h o o ti n g i n c l u d es s u ch p ro ced u re s a s s i mpl esh oo ti ng , Method of Complementary Fun ct io ns , and t h e Godunov-Conte


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or th ogo nal i za t ion p rocedure ( fo r example, see Sec t i o n 2 on theMethod of Complementary Functions). W e i n c l u d e a s e p a r a t e t r ea t m en tof th e Method o f Complementary Funct ions i n o rder t o i l l u s t r a t eth e impor tan t case o f sepa ra ted boundary cond i t ions . For comple tenesst he Method of A djo ints w i l l be d i scussed even though i t has no s imp ler e l a t i o n s h i p t o (3 .1 ) .1. P a r a l l e l S h o o t i n g

The boundary-value problem of i n t e r e s t i s

du- -d t A ( t )u - f ( t ) = o t E [0 ,11 a )(3. 3 )

The s imple shoot in g method (Theorem 1.2) use s t he s ol ut io n of (n+l)i n i t i a l -v a l u e p ro bl ems t o g en e r a t e t h e s o l u t i o n o f (3 .3 a ,b) . I np a r a l l e l s ho ot in g, t h e i n t e r v a l [ o , l ] i s d i v i d e d i n t o K n o n o v e r l a p p i n ~s u b i n t e rv a l s , [T y , Tv+l], and s ep a ra t e i n i t i a l -v a l u e p ro bl ems a r eso lved on each sub in te rva l . he so l u t i on o f (3 .3a ,b ) i s co n s t ru c t edby r e q u i r i n g c o n t i n u i t y a t Tv, v = 1,...,K-1,and s a t i s f a c t i o n ofth e boundary con di t i ons (3 .3b).

Dis cre te pa r a l l e l shoo t ing methods mimic t h i s p rocedure .we pl ac e a ne t ( t y ~ f : ~n each s u b i n t e rv a l [ Tv-l, ' fV]. On t h i s n e t ,a d i s c r e t e i n i t i a l - v a l u e p ro ble m i s s ol ve d w i t h i n i t i a l c o n d it i on sV V

Zo , zo. These n+ l i n i t i a l c o nd i ti o ns a r e a r b i t r a r y e x ce pt t h a tVwe requ i re the n x n mat r i x Z be nons ingu lar . T hi s i n i t i a l - v a l u e0

s o l u t i o n can b e r ep re s en t ed a s


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o r , eq u i v a l e n t l y we w r i t e

V ' Vz 1 z 1 = j rv ] .I I

I n p a r a l l e l s h o o ti n g, i t i s s ometi mes n eces sa ry t o s t a t e anin i t i a l - va lu e p roblem on some in te rv a l [TV ' Tvcl] backward byforma l ly g ivi ng n+l "boundary" cond i t io ns a t T r a t h e r t h a nv+l" i n i t i a l " co n d i t i o n s a t T However, Th eo ren 1 .1 6 s t a t e s t h a t f o rv *a l l n e t s w i th ho 5 H, H s u f f i c i e n t l y sm a ll , t h e s o l u t io n t o t h i sbackward problem i s eq u a l t o t h e s o l u t i o n of (3.4) w i t h zV and zV0 0app ropr ia t e ly chosen. Thus, i n o rde r to exani ine th e mos t genera lp a ra l l e l s h o o ti n g meth od s, i t i s s u f f i c i e n t t o c on s id e r d i s c r e t ei n i t i a l -v a l u e p ro bl ems o f t h e form (3.4).

VDefining Z , zv by (3 .4 ) , t h e s o l u t i o n of an y i n i t a l -v a l u ev Jproblem on th e ne t { t i j i i 0 i s given by

Thus, t h e c o n t i n u i t y r eq ui re m en ts f o r p a r a l l e l s h o o ti n g a r e

I n o rd e r t o meet t h e b ou nd ary co n d i t i o n s , t h e fo l l o w i ng r e l a t i o n s h i pmust b e s a t i s f i e d


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Combining ( 3 . 5 ) , ( 3 . 6 ) , (3.,7), and (3.8), t h e p a r a l l e l s h o ot i ngs o l u t i o n , vPS, i s given by

S t a b i l i t y . F or t h e p ur po se s of t h i s d i s c u s s i o n , t h e d i s c r e t ep a r a l l e l initial value problem is said to be stable if

I n t h e p r o of s t o f o ll o w, d e f i n e t h e nJ v x n ( J +1)m a t r i xv

VM v i a t h e r e l a t i o ns h i p


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The in i t i a l - va lu e ma t r ix I a s so c ia t ed w i t h a p a r a l l e l shoot in^hprocedure i s

For convenience, we w i l l w r i t e t h i s m a tr ix a s (K = 3 )

where the (1,o) block element of M i s t h e (SV+2, Sv+ l ) blockelement of 1h'

1Theorem 3.13. Let (3.3a,b) have a unique so lu ti on y ( t ) E c [ o , l ] .Let th e di ff er en ce scheme be co ns ist en t with (3.3a). Then th efo l lowing a re e qu iva le n t :

a) The d i s c r e t e i n i t i a l - va lu e p roblem is stable.b) The d i s c r e t e p a r a l l e l ini t ia l value problem is s table .

Proof: b) => a) The matrix lh c an be fa c to re d i n t h e fo l l owing


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v -1 -By hy po th es is I I( lh) I I 5 K, v = l , . , . , ~ ,hen-K1 1 1 1 5 max IK , 11-

a) => b) Without lo ss of g en er a l i ty , w e w i l l cons idert h e c a s e i n w hich t h e i n t e r v a l [ o , l ] i s d i vi d ed i n t o 3 s u b i n t e r v a l s ,K = 3. The f i r s t s t e p i s t o show th a t the ma t r ix rh, efined below,i s nons ingular ,

Mlo' ' '


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By us in g t h e F a c to r i z a t io n Lemma (1.1 2), t h e Reducing Lemma (1 .13 ),and the convergence o f t he in i t i a l - va lu e p roblem, it can b e shownt h a t

where K i s independent of ho.11 2 3L et V ,V ,V be nJ1, n( J +I ) , nJ v e c t o r s r e s p e c t i v e l y .2 3-

Consider th e n (J+ l ) equa t ions

where g2 i s any n(J2+1)-vector . S ince 7 i s nons ingu lar , f o r e a c hh2g2 t h e r e i s a V s u c h t h a t

Rec a l l ing (3 .14) , we f i nd th a t

Thus,

This argument i s v a l i d f o r any v = 1,..., , th us we havev -1I I (lh) I I ' 1 V = 1, . a , K .


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Before p roceed ing w i th th e theorem which w i l l t i e a l lt h r ee of t h e s e d i s c r e t e metho ds t o g e t h e r , i t i s conven ien t topresent two lemmas.

Lemma 3.17. Le t C1, C b e n o n s in g u l a r m x m mat r i ce s . L e t C1 and- 2C2 b e r e l a t e d b y

where L, N a r e m x q, q x m m a t r i ce s r e s p e c ti v e l y. F u r t he r , l e tL have rank q I m. Then th e q x q m a t r i x [ I + NL ] i s nons ingu lar .Pro of. By hy po th es is C1 i s n o n s in g u l a r, t h e r e f o r e f o r e a ch b E E?,t h e r e ex i s t s a u n iq ue m-v ec to r, x, s u c h t h a t

R e c a l l in g ( 3. 1 8) , t h i s e q u a t io n c an b e r e w r i t t e n a s

o r e q u i va l e n tl y a s

The Reducing Lemma (1.13) s t a t e s th a t (3.19) onl y ha s so lu ti on s oft h e form

where


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Since a un ique x, s o l u t i o n o f ( 3. 19 ), e x i s t s f o r e v e ry b , t h em a t ri x ( I + hZ) must be nonsingular .

Lemma 3.20. L e t t h e ~ ( J + K ) ~ ( J + K ) xpanded boundary-value ma tr ix-8 b e d e f i n e d byh

Let t he boundary-value mat r i x Bh b e n o n s i n gu l a r . Thennons ingu lar .

Proof . Suppose that B~ i s s i n g u l a r , t h e n t h e r e i s a ~ ( J + K ) - v e c t o rhEV s u c h t h a t

EHowever, i f we de le t e the K-1 l ements v i = SV+v, v = 1,...,K-1,i'Efrom V an d co l l ap s e VE t o form a n(J+l)-vector , V , then


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By hy po th es is Rh i s nons ingu lar , thus 1 [ v [ = O . N ote t h a t f o rEeach element vE dele ted f rom V t o fo rm Vi

and vE appea rs a s an element of V. Thus,i-1

Eand by con t ra d ic t ion , w e have 8 nons ingu lar .hm

Now, w e s t a t e and prove th e main theorem of t h i s Chap ter , whichw i l l i n c l u d e Theorem 1.16.

1Theorem 3.22. Let (3. 3a, b) have a u ni qu e s o l u t i o n y ( t ) E c [ o , l ] .Let the d i f fe re nc e scheme employed be con s i s t e n t wi th (3 .3a) . Denoteby B V , IV, and P S the following methods:

BV. Di sc re te boundary-value procedure, (3 .1) ,I V . D i s c re t e i n i t i a l -v a l u e p ro ced u re , ( 3.2a ,b ,c ),PS . D i s c r e t e p a r a l l e l s h o ot i ng pr o c ed u r e, ( 3 . 9 a , b, c ) .

Let one of the d i sc re te p rob lems (3 . I ) , (3 .2 a ) , and (3 .9a ) be

stable .Then, f o r a l l n e t s w it h ho 5 11 H s u f f i c i e n t l y s m al l,

i) Each method uniquely de fi ne s an n(J+ l) ve ct or as anap prox ima ti on t o y ( t ) o n t h a t n e t , and

ii) This approximat ion i s the same for each method.


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Proof: Note that stability of (3.1) is equivalent to (3.2a) or (3.9a).Re i). Theorem 1.16 proves that methods SV and IV uniquely

define approximations to y(t). To show that method PS uniquelydefines an approximation it is sufficient to show that the nK x nKmatrix in (3.9b) is nonsingular. Without loss of generality, weconsider the case where K = 3. Thus, we want to show that the3n x 3n matrix P, defined by

is nonsingular. From conclusion i), Bh is nonsingular, hence byELemma 3.20, 8 is nonsingular. Recall the n(J+l)-vector r inh

E(3.la), and define the ~(J+K)-vector r by

Consider the matrix equation

EDefine the expanded parallel shooting matrix Ih by


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Reca l l ing th e p ar a l l e l shoo t in3 method [3 ,9a ,b ,c ) , eq-ua tion (3.23)c an b e w r i t t e n a s

where

and

-I n z e n e r a l , L i s a n(J+K)x nK 1 n a t r i x , < i s a n ( n ~ 1) -ve c to r ,and N i s a nK + 1 x n(J+K) ma tr ix . Re ca ll in g Lemma 3.17 , w e n o t et h a t

i s nons ingula r , where

When t h e pr op er s u b s t i t u t i o n s a r e made, i t i s c l e a r t h a t (I - NL)i s nons ingu la r i f and on ly i f P i s nons ingula r .

R e i i ) . Theorem 1.16 i s s u f f i c i e n t t o p ro v e t h a t


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vBV = vIV* Reca lling (3.23), t h e Reducing Lemma 0. 1 3 ) s t a t e st h a t

where th e n-vectors , wi, mus t s a t i s fy

Now, we have t h e r e la t io n sh i p

Recal l ing (3.23) and th e de f i n i t i on of B:, we no te t ha t


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and thus it is clear thatIVvps = vBv = v ,

Unfortunately, Theorem (3.22) contains no information onthe numerical stability of any parallel shooting technique. Tosay that a unique solution to the difference equations (3.9a,b,c)exists is not to say that it can be accurately approximatednumerically. It is well-known that a simple shooting method mayproduce disastrous numerical solutions if the linearly independentsolutions of (3.3a) grow at different rates. For example, considerthe problem

This problem has a unique solution for all f(t) E c[o,l] and2f3 E E , because the matrix

1 + e o 1[Bo + BIX(l) 1 = 1 60 -..]+ e l + e

is nonsingular. However, if we try to invert this matrix numerically,on a machine with less than 85-bit accuracy, the computer willconsider (Bo + BIX(l)) singular, because 1 + eg60 will be representedas 1. Thus, no matter how close ZJ gets to X ( 1 ) , this matrix will


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be numerically singular due to round-off error. For this reason,methods which are equivalent in the sense of Theorem 3.22 may yieldgreatly different numerical solutions.

2. Method of Complementary Functions

The Method of Complementary Functions (Goodman and Lance [ 3 ] )is used to approximate the solution of (3.3a,b) when the boundaryconditions are separated. That is, the boundary conditions (3.3b)can be written as

where Bo is a p x n matrix of rank p, B is a q x n matrix of rank1qy p + q = n, and Bo, B1 are a p-vector, q-vector respectively. Inthis section, we use the convention that a single matrix in parentheses,(B,), has p rows and a single matrix in brackets, [B 1, has q rows.1The original equations (3.3a9b) can now be rewritten as

In what follows, we assume that the boundary-value problem (3.27ayb)

has a unique solution y(t). If the boundary-value problem hasa unique solution, then there exists a vector q such that0

(Do)qo = (Bo)where qo is orthogonal to the null space of (E ). Let the orthonormal0


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s e t { ~ ~ } z = ~p an th e n u l l sp ace of (Bo), t h en

(qi, ?'lo) = 0 1 s i S q .

With t h ese n o t i o n s i n hand, t h e so lu t i on of (3 ,27a ,b) can b ec h a r a c t e r i z e d i n th e fo l lowing way. So lv e t h e q homogeneousi n i t i a l - v a l u e problems

and the inhomogeneous problem

The unique sol ut io n of (3 .27a9b) i s given by

w here t h e co n s t a n t s cii, i = . . q , a r e determined f rom

I n t h e i n t r o du c t io n t o t h i s c h a p t e r, i t was s t a t e d t h a t

t h e Method of Complementary Fu nc tio ns i s a s p e c i a l c a s e of p a r a l l e lsh o o t in g . T h i s i s t h e b e s t p o i n t a t wh ic h t o make t h i s r e l a t i o n s h i pc l e a r . P a r a l l e l s h oo t in g n a t u r a l l y c o n t a i n s t h e c a s e o f s i mp lesh o o t in g . I n o r d e r t o ch a r a c t e r i z e y ( t ) by means of s imp le sh o o t in g ,so l ve th e n homogeneous in i t ia l - va lu e problems


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and the inhomogeneous problem

- -x(t) - A ( t ) x ( t ) + f ( t )d t

where C i s a nons ingu la r n x n n a t r i x and c i s an n-vec tor . Now,

where th e n-vector i s determined by

The r e s u l t o f s p e c i a l i z i n g t h e s i m pl e s h o o ti n g p r oc e du r e ( 3 , 3 2 a, b ) ,(3.33a,b) , (3.34) , (3.35) by ta ki ng

i s t h e Method of Complementary Fu nc ti on s.The d i s c r e t e Method of Complementary Functi ons (h er ea f t er

re fe re nc e t o t h e Method of Conzplementary Fun cti ons w i l l imply thed i s c r e t e p r oc ed u re ) i s de r ive d by so lv i ng e qua t ions ( 3 .28a Yb) ,


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(3.29a,b), (3.30), and (3.31) numerically. Solve the q homogeneousdiscrete initial-value problems

and the inhomogeneous discrete initial-value problem

-

For convenience, we write (3.32), (3.33) as

c. -'li0

0L r

iwhere Z is a (J+l)n x q matrix with its ith column being z . Thediscrete approximation to y ( t ) is given by

where


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The in i t i a l - va lu e equat ion (3.34) may be formal ly wr i t t e n as

The f i r s t n (q+l ) equa t i ons of (3.37 ) s t a t e t ha t t h e columns of Zo

-I ( G o )II

I [o lIIl oIIIIIIIIl o ,

a r e or thonormal and belong t o t h e nu l l -space of Boy and a l s o t h a t

.

0 0Bo Zo = Po and zo i s or thogona l t o t h e nu l l - space o f B . Now,0we want t o show t h a t V? ca lc u la ted v i a (3.37) , (3 .35) , and (3 .36)i s a so lu t i on of t he equa t i ons

The di s c re te boundary-value problem i s

R eca l li ng t he F ac to r i za t i o n Ler~una (1 .12 ) , t h i s m a t r ix can b e w r i t t e na s


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where T is the n(~+1) x n(J+l) matrix in (3.37). Left-multiplyinghby T~~~nd recalling (3.37), this equation is shown to be

The Reducing Lemma (1.13) states that

where w and A are determined from

When this is unraveled, we have

Recall (3.35) and (3.36) and compare with (3.30) and (3.40) toshow that


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3. Method of A di o in ts

The Method of A djoints completes th e s tu dy of so lu t i onprocedures fo r d i s c r e t e boundary-va lue p rob lems. This methodi n t ro d u ces t h e ad j o i n t eq u a t i o n wh ich can n o t be d e f in e d v i a l i n e a rt r an s fo rma t i o n s of t h e o r i g i n a l p rob lem. For t h i s r ea s o n t h eequiv alenc e ex hi bi ted between boundary-value techn iques and pa ra l l e l -shoo t ing p rocedures i s l a ck i n g h e re . A good exp lan at io n of t heMethod of Adjoints i s foun d i n Goodman and Lan ce [ 3 ] .

The problem cons idere d i n [ 3 ] i s a v e r y s p e c i a l o ne :du- -d t A ( t )u = f ( t ) a )

The a d j o i n t d i f f e r e n t i a l e q u a t io n i s

From (3 .41) , (3 .42) , th e fo l lowing re la t i on sh ip i s d e r i v ed

I n t e g r a t i n g t h i s e x pr es si on , w e o b t a i n

By s o l v i n g t h e ad j o i n t prob lem n-p t i mes w i t h i n i t i a l co n d i t i o n s-V Au (0) = e v = p+l , ..., , t h e e q ua t i o n (3 .44) r e s u l t s i n a s e t ofv'n-p (or q) e q u a t io n s f o r t h e q u a n t i t i e s u i( o ), i = p + l , ..., .


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Now, the initial conditions u(o) are explicitly known and the solutionof a single initial-value problem determines y(t).

If we use the sane discretization for (3.41a,bYc) and(3.42), (3.44), it is not in general true that the solution of thediscrete Method of Adjoints, ?, s equal to vBV. However, if weemploy the centered-Euler scheme this equivalence does hold. Thediscrete analogue to (3.44) is naturally

-* MA -* itv~ "J i-I f(ti_ l)0 - (3.45)0 0 i=l 2

From the difference equations

and

the following identity can be derived

Upon summing from i = 1, .., , this identity yields- - ik(vi + v -1

BV BV YAThus, V also satisfies (3.66) and vi = vi , i = 0,. . , J .


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Remember t h a t t h i s equiv a lenc e has been shown only i f th e cen te r ed-Euler scheme i s u se d t o d i s c r e t i z e t h e Method of A d j o in t s. I d e n t i t i e ss i m i l a r t o ( 3 . 4 7 ) can be de r ived f o r any number o f o t h e r d i f f e r e n c eschemes , bu t th e summation which yie ld ed (3.48) d oe s n ot i n g e n e r a lshow equivalence.


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Chapter IV

Non lin ear Two-Point Boundary-Value Prob lems

T h i s ch ap t e r d ea l s w i t h n u meri ca l ap p ro x i mat i on s t o t h eso lu t i on of no nl in ear two-point boundary-value problems of t he form

w here u , f , b a r e n -v ec t o rs . A f t e r Keller [GI , we a r e i n t e r e s t e d i ni s o l a t e d s o l u t i o n s o f (4 . l a ,b ) . The s o l u t i o n of (4 , l a , b ) i s i s o l a t e di f th e l in ea r i ze d boundary-value p roblem

h as o n ly t h e t r i v i a l s o lu t io n . Note tha t w e h av e us ed t h e n o t a t i o n

When con sideri ng th e d is c r e te problern , w e w i l l a l so examine~ e w t o n ' smethod a s a means of ca lc u l a t i ng an approx ina t ion t o y ( t ) .The n(J+l)-vector Y i s d e f i n ed t o b e


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where y ( t ) i s a n i s o l a t e d s o l u t i o n o f ( 4 .l a ,b ) . We a l s o d e f i n e

S,,[g(t)l = Ix I x E E", 1 lx - g ( t ) l 1 5 p } ,

1. F i n i t e Di f fe re nce Schemes

Each of t h e d i f f e r en ce schemes d i s cu s s ed i n Ch apt e r 2 canbe modified t o make i t a p p l i c a b l e t o n o n li n e ar d i f f e r e n t i a l e q u at i on sof th e form (4 .l a) . Thus, th e cent ered -Eule r scheme (2.2) becomes

fo r a g en e ra l n o n l i n ea r eq u a t i o n. I f eq u a t i o n (4 . la ) i s l i n e a r i nu , t h a t i s ,

t h en t h e d i f f e r e n ce eq u a ti o n (4 .3 ) r ed uces t o (2 .2) .The n o n l i n ea r an al og u e of t h e i n t eg ra l eq u a t i o n (2 .1)

i s c l e a r l y

A s i n Chapter 2, quadr a tu re fo rnlu lae and in te gr a t io n by p a r t s c anbe used to genera te co ns i s t en t numer ica l schemes . I n t h i s manner t he


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2. Exis ten ce of Solu t ion s

W e w i l l now cons ider th e ques t ion o f e x i s t en ce of so l u t i onst o systems o f non l in ear equa t ion s o f the fo rm (4 .8a ,b ) . A s a p re l u d e ,l e t u s ex amine t h e r e s u l t o f ap p l y i n g t h e s e n o n l i n ea r d i f f e r en c e

schemes t o th e l i ne ar iz ed problem.

Lermna 4.10, Let th e boundary-value problem (4. la, b) have an is o la te ds o l u t i o n , y ( t ) . L et

b e t h e n ( J + l) eq u a t i o n s w hi ch r e s u l t f ro m t h e ap p l i ca t i o n o f t h ed i f fe r enc e scheme (4 .8a ,b ) t o the l ine ar i ze d prob lem:

1where g , f3 a r e n -vec to rs , and g ( t ) E c [ 0 ,1 ] . L e t t h e d i f f e r e n c escheme be co ns is te nt and s t a b l e when app l ied to a l i n e a r two-point

2boundary-value problem wit h a unique sol ut io n, z ( t ) E c [ o , l ] . L e t1j=(y,t) r [ y ( t ) ] x [o , l l ] and b(x ,y) E c [S [Y(o) ] S p [ ~ ( 1 ) ] I sP P

Then f o r a l l n e t s w i t h ho 2 H, H s u f f i c i e n t l y s m a l l , L i s nons ingu larand


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w i t h ho 5 EL@) , Il(p) s u f f i c i e n t l y s ma ll , t h e n ( ~ + l ) q ua ti on s

rl(v) = 0

have a un ique so lu t ion W E S [y ( t ) ] .Proof: W e d e f i n e t h e v e c t o r f u n c t i o n

I f t h i s h as a s o l u ti o n , $(W) = W , t hen n(W) = o. The Contract ingMapping Theorem w i l l b e used t o show t h a t a u n i q u e so l u t i o n ex i s t s .That i s , we need t o show th a t f o r some h E ( o , l )

i) 1 I Y - $(Y) I I 5 (1 -i i ) II1/'(v1) - $(v2) I I 5 x I I v ~ - v21[ , where

Vl, V2 E S p [ y ( t ) l

By t he hypoth esis th e numerical scheme i s co n s i s t en t , h en ce

R e i i )

Now, we employ the mean value theorem


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Co rol lar y 4.17 (Convergence). Let t h e hypo thes is of Theorem 4.13hold . Then, i f q ( W ) = o , w E s p [ y ( t ) ] ,

Proof . It i s s u f f i c i e n t t o pr ov e t h a t f o r a ny E > o , t h e r e e x i s t san H s u ch t h a t

f o r any n e t w i t h ho 5 H. However, th i s i s the conc lus ion o fTheorem 4.13 with E r ep l ac i n g p .

rnCo rol lar y 4.18. Let th e hypo the sis of Theorem 4.13 hold. Letth e di ff er en ce scheme (4.8b) be pth o rd e r accu ra t e ,

Then, f o r a l l n e t s w i t h ho 5 H, li s u f f i c i e n t l y s m a l l ,

Pr oo f. From t h e proof of Theorem 4.13, w e know t h a t W E S [ y , t ]Pi f p 5 po, ho 5 H(po), and

where X E (0,1) and fi xe d. By hyp othe sis


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where th e r ight-hand s i d e of (4 .19b) i s w r i t t e n s i m p l y a s si because

i t i s n o t of p a r t i c u l a r i n t e r e s t h e r e . From (4 .19 a ,b) , we canco mp le t e ly ch a r ac t e r i ze t h e e l emen t s ( L . . ) of L.

1 J

L et t h e ( i , j l th n X n block element of nV(Y) be Ni j . Then

and Noo = Loo. S i m i l a r l y , N = Lo , j = l , . . . , J . From 4 . 6 , i t can0jbe shown th a t

S i m i l a r l y ,

N . . = Lii, i = 1, .., .11


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Thus, th e Gap4 scheme s a t i s f i e s con d i t i on (4.14) and i n fa c t

Centered-Euler . The cen te red-Euler scheme s a t i s f i e s equa t ion (4 .14) ,however L i s n o t eq u a l t o n V (y ) . Employing th e cent ered -Eule r scheme,( 4 3 ) , t h e J a c ob i an m a t r i x 17 (Y) h as th e fol lo wing nonzero n x nvblock components:

The matr ix L h a s t h e s e n x n block e lements

Thus, we have

1 1C l e ar l y f o r f ( u , t ) E c [ S p [ y ( t ) l [ o , l ] l and y ( t ) E c [ o , l ] ,


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p r o p e r ty (4.14) h o lds . N ote t h a t t h e q u a n t i t y

need-o t ap p r oach ze ro l i k e hP i n o r de r f o r a pth o r d e r a c c u r a t e0scheme t o s a t i s f y3 . So lu t i on of q(V) = o by Newton's Method

I n t h e p re v i ou s s e c t i o n i t was shown th a t under c e r t a inc o n d i t i o ns a s o l u t i o n e x i s t s t o t h e n (J +1 ) n o n l i n e a r e q u a t i o n s

W e would l i k e t o examin e a sp ec i f i c p r o ced u re f o r so lv in g t h e see q u a t i o n s, nam ely ~ e w t o n ' smethod. That i s , w e want t o show t h a te ac h i t e r a t e d e f in e d by

V 0e x i s t s u n i q ue l y an d t h a t t h e s e qu en ce {V ) converges when V i sjud ic ious ly chosen . The proof of th e fol lowin g theorem i s i d e n t i c a lt o t h a t g iv en i n K e l l e r [ 6 ] , w i t h t h e a p p r o p r i a t e g e n e r a l iz a t i o n s .

Theorem 4 .23 Let t h e hyp oth esi s of Theorem 4.13 hold and l e t Wbe th e n ( J+l ) -vector o f th a t theorem where


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L e t VO b e s uc h t h a t VO E S [y(t) and [ IvO - w [ I i s s u f f i c i e n t l yPs m al l. Then f o r a l l n e t s w i t h h 5 H, H s u f f i c i e n t l y s m a ll , a nd0f o r a l l p , o < p 5 po, p s u f f i c i e n t l y small, t h e N ewton i t e r a t i o n0 v(4.22) un ique ly def i ne s a sequence {V 1 and t h i s sequence convergesq u a d r a t i c a l l y t o W,Proof: A s noted befo re , see Keller [ b ] .

Chapters 1 and 3 examined t he e quiva lence between sho ot ingmethods and im pl ic i t t e chn iques f o r l in e a r boundary-va lue p rob lems .I n t h e ne x t s e c t i o n , we want t o d e f i n e a n o n l i n ea r s h o o t in gprocedure and s tudy i t s r e l a t i o n s h i p t o t h e method d es c r i b ed h e re .

4 . Equivalence of Shoot ing and Imp l i ci t Schemes f o r Nonl inear Problems

The nonl in ear shoo t ing scheme has t he same under ly ing i de a as i nChapter 1. The in i t i a l - va lu e p roblem

i s d i s c r e t i z e d , s o lv e d, and t he n t h e i n i t i a l c o n di t io n x (o ) = c i sw ig gl ed u n t i l t h e d i s c r e t e bo und ary c o n d i t i o n b ( c , u J ( c ) ) = o i ss a t i s f i e d .

For example , th e fo l lowing sys tem of eq ua t ion s re s u l t i fwe u se ~ u l e r ' smethod t o approxima te (4.24a):


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Note t ha t t h ese equa t i ons can be so lved ex p l i c i t l y , however ani t e ra t io n scheme must be employed to s a t i s f y th e nonl in ear boundarycond i t i on

The a pp li c at io n of Newton's Method t o so lv e (4.26) i s s t r a i g h tforward

'dum us t be eva lua t ed a t each s t e p . Th i sowever, t h e n x n m a t r i x-c

m a t r i x i s de te rmined by so lv ing t h e va r i a t i o na l equa t i ons

A d i sc re t e shoo t ing p rocedure r e s u l t s f rom com bining (4 ,25a ,b ),(4.28a,b), and (4.27):


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An a l t e r n a t i v e t o t h i s p ro ce du re i s ~ e w t o n ' smethod a sde sc r ibe d i n t he p re v ious se c t i on . Tha t i s , Newton's method i semployed t o s olv e th e system of equat ions

T h is r e s u l t s i n t h e f o l lo w in g i t e r a t i o n scheme

vI t has already been shown that {V 1 e x is t s and converges .I t i s c l e a r t h a t i n e ac h c a s e t h e method d e f i n e s a s o l u t i o n

of th e same s e t of equ at ion s:


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The implicLt method (4,31a,b) a pp li e s Newton's method t o each of t h en(J+l) equ at io ns of (4 ,30a, b) . The sh oot ing method (4.29a, b , c , d , e )f o rm a ll y s e p a r a t e s t h e f i r s t n e q u a t i o n s fr om t h e r em ai ni ng n J ,

I n l i e u of t h i s set of e qua t ions , t he shoo t ing method su bs t i tu te st h e eq u a t i o n s

which can be so lved e xac t ly when an ex p l i c i t d i f fe ren ce scheme i sused t o approximate (4 .23a) . Then t h e i n i t i a l v a l u e c i s v a r i e du n t i l an i n i t i a l v a lu e i s found such that

The p r i n c i p a l d i f f e r en ce i s t h a t t h e s h o o t i ng p rocedu res a t i s f i e s t h e l a s t n J e q u at i o ns e x a c t l y and t h e i m p l i c i t Newtonprocedure does not . That i s ,

w here i n g en e ra l I IV I I # 0 .Theorem 4.34 Le t t h e hy po the sis of Theorem 4.16 hold. Le t t h e

V vshoo t ing Newton i t e r a te s be {U 1 , U E S [ y ( t ) 1, and l e t t h e i m p l i c i tP


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Chapter V

Block Tr id iago nal Mat r ices

This chapter develops a computat ional method f o r solv ingt h e s ys te ms of l i n e a r e q u a t i o n s t h a t h av e a r i s e n i n pr e v i o u schapters . That i s , we w i l l deve lop a t echn ique fo r so lv ing t he

n ( J + l) l i n e a r e q u a t io n s

o r , when th e boundary-value problem i s n o n l i n e a r , t h e e q ua t i o n s

%(vv) (vW1 - vV) + T,(vV) = 0genera t ed by ~ewton ' smethod, More spec i f i ca l l y we a r e i n t e r e s te di n m a t r i c es w hich r e s u l t from th e appr oxima tion of two-pointboundary-value problems wi th sepa rat ed boundary co nd i t i on s. Thati s , problems which may be w r it te n a s

u ' ( t ) = f ( u , t ) t E [ 0 , 1 ] a )

bo(u(o)) = 0 b )

b l ( u ( l ) ) = 0 c )

where th e boundary con di t io ns (5 . l b ) a n d (5. l c ) r e p r e se n t p and qc o n d i t i o n s r e s p e c t i v e l y , p + q = n.

To i l l u s t r a t e t h e form t h a t t h e s e m a t r i c e s w i l l t a k e ,


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consider the approximation of

by ~uler's method. The resulting finite difference equations ( J = 2 )are

where we have used the convention that a single matrix in parentheses,(B ) , has p rows and a single matrix in brackets, [B1], has q rows.0The matrix of (5.3) can be rewritten in block tridiagonal form

where each Ai, Bi, C is an n x n matrix. Block matrices of thisitype have a very special zero structure: the first p rows of Biand the last q rows of Ci are zero rows.

The solution procedure of interest is a block LUdecomposition:


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I d e a l l y , w e would l ik ; t o show th a t t h i s decompos it ion ex is t si f t h e m a t r i x 8 i s nons ingular . U n f or t un a te l y t h e r e s u l t i s n o thth a t c l ea r . F or example , i f A( t ) i n (5 .2a) i s a 4 x 4 d iagona lma tr ix and (Bo) = (1 o o o ) , t h en

and A i s cl ea r l y s ing ula r . No mat ter how smal l h becomes, t h i s0 0matrix w i l l remain s in gu la r . However, i f t h e 5th row of Rh i s

ndi n te rchanged wi th t he 2 row, then

x o o o-Ao - [ ! x ! ! ] ,

A i s nons ingular , and the LU fa ct or iz a t io n can proceed . Thus,0a row swi tch ing s t ra te gy w i l l be included i n th e decomposi t ionprocedure.

1. Block LU Decompostion

The b lock t r i d i a gon a l m a t r i x M i s cons ide red i n the fo l lowing


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g e n e r a l i t y ;

where each Ai, Bi, C. i s a n n x n matrix. W e a l s o r e q u ir e t h a t t h e1f i r s t p rows of e ac h Bi be zero and th e l a s t q rows of Ci bezero . Such a matr ix M w i l l b e r e f e r r e d t o a s a b l oc k t r i d i a g o n a lmatrix w i t h p / q z e r o s t r u c t u r e .

The s t r a i g h t f o r w a rd b l o ck decomp os it io n i s

where t he m at r i ce s Li, Di are def ined by

Once th e LU decomposit ion i s completed, t h e s t an d a rd p ro ced u re i sadop ted to so l ve MV = r :

LW= r ,w = W.,


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The n(J+l) ve ct or s W and V a r e de f in e d by t h e r e c u r s i o n r e l a t i o n s :

and

I f i t should occur that some Di i s s i n g u l a r , t h en t h e s e r e c u r s i onformulae w i l l h av e to b e a l t e r e d to a l lo w fo r th e ap p ro p r i a t e rowin terchanges .

Lemma 5.10. Let M M denote n(m+l) x n(m+l) block tr id ia go na lma t r i ce s wi th p / q zero st ru ct ur e. Denote th e block elements of

0 0 0MO, M by Ai, Bi, Ci and Ai, Bi, Ci r e s p e c t i v e l y . Let MO be0nonsing ular . Then e i t he r A i s nonsingular o r by an in te rchange0

of rows (p + 1,.. p + n) of MO a mat r ix M can be formed such t h a tA i s nonsingular .0

0 0Proof . m = o. Immediate becaus e M = Ao.--> o. Let (S ) be the p x n mat r ix composed of th e f i r s t- P0

p rows of Ao. Let S b e th e n x n ma t r ix whose f i r s t q rows a r ent he l a s t q rows of A and whose l a s t p rows a r e t h e f i r s t p rows0

0of C1 W ith th ese d e f i n i t io n s i n hand, th e matr ix MO can bew r i t t e n a s


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where Q i s an (nm-tq) x nm ma tr ix . By hy po th es is MO i s nons ingular ,th us t h e rows of (S ) a r e l i n ea r l y i ndependent and t h e (n+p) x nPm at r i x S def ined by

has r ank n . These two cond i t i ons imp ly t ha t t he re a r e q rows ofSn which, tog et he r wit h th e rows of (S ) , form a set of n l i n e a r l yPindependent row vec tors . Thus, by in terch anging the ap pr opr ia te

0rows of M we can form a matrix M such t ha t A i s nons ingu l a r .0

Sim ila r rows. Defin e rows of a b lo ck t r i d i a g o n a l m a t r i x w i t hp /q z e r o s t r u c t u r e t o b e s i m i l a r i f by i n t e r c h an g i n g t

andrew benjamin white, jr. numerical solution of two-point boundary-value problems

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