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A study of numerical techniques for the initial value problem of general relativity Choptuik, Matthew William 1982

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A STUDY OF NUMERICAL TECHNIQUES FOR THE INITIAL VALUE PROBLEM OF GENERAL RELATIVITY by MATTHEW WILLIAM CHOPTUIK B . S c , Brandon U n i v e r s i t y , 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n , THE FACULTY OF GRADUATE STUDIES P h y s i c s Department We ac c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA November 1982 © Matthew W i l l i a m C h o p t u i k , 1982 In presenting t h i s thesis i n p a r t i a l fulfilment, of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of P H V S I C ;  The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date M,ov/cM,SEg i69"t 3E-G n/R11 i i A b s t r a c t N u m e r i c a l r e l a t i v i t y i s concerned w i t h the g e n e r a t i o n of s o l u t i o n s t o E i n s t e i n ' s e q u a t i o n s by n u m e r i c a l means. In g e n e r a l , the c o n s t r u c t i o n of such a spacetime i s a c c o m p l i s h e d i n two s t a g e s : 1) the d e t e r m i n a t i o n of i n i t i a l d a t a which i s s p e c i f i e d on a s i n g l e s p a c e l i k e h y p e r s u r f a c e and s a t i s f i e s f o u r i n i t i a l v a l u e e q u a t i o n s , and 2) the e v o l u t i o n of the i n i t i a l d a t a t o g e n e r a t e the spacetime or some p o r t i o n of i t . One of the key problems i s the development of e f f i c i e n t a l g o r i t h m s f o r the s o l u t i o n s of t h e s e e q u a t i o n s , as t h e y a r e s u f f i c i e n t l y complex t o t a x the f a s t e s t p r e s e n t computers. T h i s t h e s i s p r e s e n t s a comparison of v a r i o u s a l g o r i t h m s f o r the s o l u t i o n of the i n i t i a l v a l u e e q u a t i o n s , c o n c e n t r a t i n g on the r e c e n t l y d e v e l o p e d m u l t i - g r i d method. The s p e c i f i c problem examined has been p r e v i o u s l y s t u d i e d by Bowen, P i r a n and York. T h e i r i n i t i a l d a t a has been i n t e r p r e t e d as r e p r e s e n t i n g " s n a p s h o t s " of t h r e e new f a m i l i e s of b l a c k h o l e s . Three of the f o u r i n i t i a l v a l u e e q u a t i o n s p o s s e s s a n a l y t i c s o l u t i o n s . The r e m a i n i n g 2 - d i m e n s i o n a l non-l i n e a r 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 i s s o l v e d n u m e r i c a l l y i n t h i s t h e s i s u s i n g f i n i t e d i f f e r e n c e t e c h n i q u e s . The performance of the m u l t i - g r i d method, w i t h r e s p e c t t o t h r e e more well-known methods, i s e v a l u a t e d t h r o u g h n u m e r i c a l e x p e r i m e n t s . The speed and r e l i a b i l i t y of the m u l t i - g r i d a l g o r i t h m a r e found t o be v e r y good. In a d d i t i o n , the r e s u l t s which had been p r e v i o u s l y c a l c u l a t e d n u m e r i c a l l y by P i r a n a r e i i i e s s e n t i a l l y r e p r o d u c e d , w i t h the c o r r e c t i o n of some e r r o r s i n t h a t work. P o s s i b l e e x t e n s i o n s of the work t o more complex i n i t i a l v a l u e problems a r e a l s o d i s c u s s e d . T a b l e of C o n t e n t s A b s t r a c t i i L i s t of T a b l e s v L i s t of F i g u r e s I . v i Acknowledgement v i i 1. I n t r o d u c t i o n 1 2. The I n i t i a l V a l u e Problem f o r G e n e r a l R e l a t i v i t y 7 2.1 S p l i t t i n g spacetime i n t o space and time 8 2.2 I n t r i n s i c and e x t r i n s i c c u r v a t u r e 16 2.3 The i n i t i a l v a l u e e q u a t i o n s 24 2.4 P r e s e r v a t i o n of the c o n s t r a i n t s 30 2.5 Y o r k 's approach t o the i n i t i a l v a l u e problem ...32 2.6 P a s t work on i n i t i a l v a l u e problems 44 3. A S p e c i f i c I n i t i a l V a l u e Problem 47 3.1.1 Topology of the s l i c e s and s i m p l i f y i n g a s s umptions . 48 3.1.2 S o l u t i o n of the momentum c o n s t r a i n t ....49 3.2 The H a m i l t o n i a n c o n s t r a i n t and i n v e r s i o n t e c h n i q u e s 51 3.3 D i s c r e t i z a t i o n of the H a m i l t o n i a n c o n s t r a i n t ...55 4. The M u l t i - g r i d Method 65 4.1 B a s i c m u l t i - g r i d p r o c e s s e s ..65 4.2 The f u l l a p p r o x i m a t i o n s t o r a g e scheme 75 •4.3 S o l u t i o n on the c o a r s e s t g r i d .• 82 4.4 Treatment of boundary c o n d i t i o n s 83 4.5 I m p l e m e n t a t i o n of the FAS a l g o r i t h m 85 4.6 Work e s t i m a t e s 87 4.7 A d a p t i v e d i s c r e t i z a t i o n 90 5. N u m e r i c a l E x p e r i m e n t s and R e s u l t s 94 5.1 Comparison of m u l t i - g r i d method w i t h o t h e r methods 94 5.2 . T e s t i n g of the a d a p t i v e m u l t i - g r i d a l g o r i t h m ..100 5.3 N u m e r i c a l r e s u l t s f o r b o o s t e d b l a c k h o l e s 108 5.4 N u m e r i c a l r e s u l t s f o r s p i n n i n g b l a c k h o l e s ....114 6. C o n c l u s i o n s and P o s s i b l e F u t u r e A p p l i c a t i o n s 125 B i b l i o g r a p h y 1 30 Appendix A - B a s i c N u m e r i c a l Techniques f o r E l l i p t i c PDE's 134 A.1 D i s c r e t i z a t i o n 135 A.2 F i n i t e d i f f e r e n c e s 137 A.3 Methods f o r s o l v i n g l i n e a r systems ...144 A.3.1 D i r e c t methods 145 A.3.2 I t e r a t i v e methods 150 A. 4 Methods f o r s o l v i n g n o n - l i n e a r systems 158 Appendix B - I m p l ementation of M u l t i - g r i d A l g o r i t h m 162 B. 1 G r i d o r g a n i z a t i o n f o r a d a p t i v e d i s c r e t i z a t i o n .162 B.2 Summary of major m u l t i - g r i d r o u t i n e s 165 B.3 R e l a x a t i o n - r o u t i n e s 166 B.4 C o n t r o l parameters 167 B.5 L i s t i n g of m u l t i - g r i d program 169 V L i s t of T a b l e s I . M u l t i - g r i d r e s u l t s f o r m o d i f i e d t e s t problem 97 I I . Point-SOR' r e s u l t s f o r m o d i f i e d t e s t problem 98 I I I . Line-SOR r e s u l t s f o r m o d i f i e d t e s t problem 98 IV. Newton-ND r e s u l t s f o r m o d i f i e d t e s t problem 99 V. T e s t i n g of i n t e g r a t i o n r o u t i n e s ...101 V I . E f f e c t of v a r y i n g t r u n c a t i o n e r r o r parameter 102 V I I . M u l t i - g r i d t e s t r e s u l t s f o r v a r y i n g momentum 103 V I I I . T o t a l energy v e r s u s momentum f o r b o o s t e d h o l e s ....110 IX. V a r i o u s e n e r g i e s of s p i n n i n g b l a c k h o l e s 124 v i L i s t of F i g u r e s 1. Spacetime D i s p l a c e m e n t i n 3+1 Language 14 2. E x t r i n s i c C u r v a t u r e Example 21 3. Domain of Boundary V a l u e Problem 58 4. U n i f o r m G r i d f o r D i s c r e t i z a t i o n of Boundary V a l u e Problem 61 5. FAS A l g o r i t h m 86 6. Energy v s . Momentum f o r Model Problem 105 7. Pre-CGC Smoothing of R e s i d u a l s 106 8. Post-CGC Smoothing of R e s i d u a l s 107 9. Energy v s . Momentum f o r Boosted B l a c k H o l e s 109 10. S Dependence of Conformal F a c t o r of Boos t e d H o l e s .111 11. R Dependence of Conformal F a c t o r of Boos t e d Holes .112 12. A n g u l a r Dependence of Conformal F a c t o r of Boosted H o l e s 113 13. Energy v s . Momentum f o r S p i n n i n g B l a c k H o l e s .115 14. S Dependence of Conformal F a c t o r of S p i n n i n g Holes 116 15. R Dependence of Conformal F a c t o r of S p i n n i n g H o l e s 117 16. A n g u l a r Dependence of Conformal F a c t o r of S p i n n i n g H o l e s 118 17. Conformal F a c t o r of S p i n n i n g B l a c k Hole (J=1000) ..119 18. A n g u l a r Momentum and Energy Parameters f o r S p i n n i n g H o l e s 122 19. D i s c r e t i z a t i o n of Domain of Boundary V a l u e Problem 136 20. E x t r a G r i d P o i n t s - D i s c r e t i z a t i o n of Boundary C o n d i t i o n s 142 21. N a t u r a l O r d e r i n g of a 10x10 G r i d 147 22. N e s t e d D i s s e c t i o n O r d e r i n g of a 10x10 G r i d 148 23. Non-uniform D i s c r e t i z a t i o n and A s s o c i a t e d Tree S t r u c t u r e 163 v i i Acknowledgement I would l i k e t o thank Dr W.G. Unruh f o r s u g g e s t i n g the t o p i c of t h i s t h e s i s and f o r h i s s u b s t a n t i a l c o n t r i b u t i o n s t o a l l p a r t s of t h i s work. I would a l s o l i k e t o thank Dr U. Ascher f o r i n t r o d u c i n g me t o the m u l t i - g r i d method. F i n a l l y , I would l i k e t o acknowledge the f i n a n c i a l s u p port of the N a t u r a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l of Canada. 1 CHAPTER 1 I n t r o d u c t i o n In the s i x and a h a l f decades s i n c e E i n s t e i n p r e s e n t e d h i s f i e l d e q u a t i o n s , g e n e r a l r e l a t i v i t y has become w i d e l y a c c e p t e d as the " b e s t " t h e o r y of g r a v i t a t i o n . However, d e s p i t e the s u c c e s s e s of the t h e o r y and the v a s t amount of e f f o r t which has been devoted t o i t s s t u d y , r e l a t i v e l y l i t t l e i s known about the consequences of g e n e r a l r e l a t i v i t y i n s i t u a t i o n s i n v o l v i n g a h i g h degree of asymmetry or s t r o n g g r a v i t a t i o n a l f i e l d s . S i t u a t i o n s of t h i s t y p e , such as the c o l l a p s e of a s t a r t o form a b l a c k h o l e , the c o l l i s i o n of two b l a c k h o l e s , or a supernova e x p l o s i o n a r e of g r e a t i n t e r e s t t o a s t r o p h y s i c i s t s and g e n e r a l r e l a t i v i s t s a l i k e . Due t o the c o m p l e x i t y and n o n - l i n e a r i t y of the f i e l d e q u a t i o n s i n such s c e n a r i o s , c u r r e n t a n a l y t i c t e c h n i q u e s a r e g e n e r a l l y unable t o p r o v i d e a c c e p t a b l e s o l u t i o n s . T h i s has prompted the development of a new b r a n c h of g e n e r a l r e l a t i v i t y which i s c o n c e r n e d w i t h the s o l u t i o n of the f i e l d e q u a t i o n s by n u m e r i c a l means. N u m e r i c a l r e l a t i v i t y , as the f i e l d i s c a l l e d , shows c o n s i d e r a b l e promise i n p r o v i d i n g i n f o r m a t i o n about complex, p h y s i c a l l y r e a l i s t i c g r a v i t a t i o n a l p r o c e s s e s . The t h e o r e t i c a l framework on which most of the c u r r e n t work i n n u m e r i c a l r e l a t i v i t y r e l i e s was d e v e l o p e d i n the l a t e 2 1950's and e a r l y 1960's by A r n o w i t t , Deser and M i s n e r and i s summarized i n t h e i r 1962 work "The Dynamics of G e n e r a l R e l a t i v i t y " [ 1 ] . The ADM f o r m a l i s m i n v o l v e s a d e c o m p o s i t i o n of spacetime i n t o "space" and " t i m e " so t h a t the d y n a m i c a l n a t u r e of g e n e r a l r e l a t i v i t y can be examined. Roughly s p e a k i n g , i n t h i s f o r m a l i s m a 4 - d i m e n s i o n a l spacetime i s r e g a r d e d as an i n f i n i t e l y extended s t a c k of 3 - d i m e n s i o n a l s p a c e l i k e h y p e r s u r f a c e s . The c o n f i g u r a t i o n of any p a r t i c u l a r h y p e r s u r f a c e i s i n t e r p r e t e d as r e p r e s e n t i n g the d y n a m i c a l s t a t e of the spacetime a t a p a r t i c u l a r " i n s t a n t " . The complete spacetime i s then the "time h i s t o r y " of a s i n g l e s p a c e l i k e s l i c e . When w r i t t e n i n the ADM f o r m a l i s m , the f i e l d e q u a t i o n s s e p a r a t e i n t o two d i s t i n c t groups. Four of the t e n e q u a t i o n s i n v o l v e q u a n t i t i e s which may be d e f i n e d on a s i n g l e h y p e r s u r f a c e and thus r e p r e s e n t e q u a t i o n s of c o n s t r a i n t on the q u a n t i t i e s c h a r a c t e r i z i n g a s p a c e l i k e s l i c e . The o t h e r s i x e q u a t i o n s govern the "time e v o l u t i o n " of t h e s e q u a n t i t i e s . The problem of p r o d u c i n g a s o l u t i o n t o the f i e l d e q u a t i o n s can t h e r e f o r e be s e p a r a t e d i n t o two s t a g e s : 1) the d e t e r m i n a t i o n of i n i t i a l d a t a which s a t i s f i e s the c o n s t r a i n t , or i n i t i a l v a l u e , e q u a t i o n s 2) the e v o l u t i o n of the i n i t i a l d a t a t o produce the complete spacetime or some p o r t i o n of i t . At the p r e s e n t t i m e , the i n i t i a l v a l u e problem f o r g e n e r a l r e l a t i v i t y i s by f a r the b e t t e r u n d e r s t o o d of the above two sub-problems. York and v a r i o u s c o l l a b o r a t o r s have d e v e l o p e d a p r o c e d u r e which p r o v i d e s a " c o v a r i a n t " s e p a r a t i o n of the i n i t i a l d a t a i n t o f r e e l y s p e c i f i a b l e and d e t e r m i n e d - f r o m -3 c o n s t r a i n t p i e c e s . T h i s p r o c e s s , f o r a g e n e r a l i n i t i a l v a l u e problem, a l l o w s the c o n s t r a i n t e q u a t i o n s t o be w r i t t e n as a set of f o u r c o u p l e d n o n - l i n e a r e l l i p t i c 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 f o r f o u r " p o t e n t i a l s " which can be i n t e r p r e t e d as the r e l a t i v i s t i c g e n e r a l i z a t i o n of the s i n g l e Newtonian g r a v i t a t i o n a l p o t e n t i a l . Methods f o r a p p r o x i m a t e l y s o l v i n g n o n - l i n e a r e l l i p t i c systems e x i s t and i n p r i n c i p l e , t h e r e would seem t o be no reason why a g e n e r a l i n i t i a l v a l u e problem c o u l d not be s o l v e d by such means. The s i t u a t i o n i s not the same f o r the e v o l u t i o n problem where many q u e s t i o n s c o n c e r n i n g b o t h the t h e o r e t i c a l n a t u r e of t h e problem and p r a c t i c a l methods f o r i t s s o l u t i o n , remain unanswered. A l t h o u g h the n u m e r i c a l s o l u t i o n of e l l i p t i c e q u a t i o n s , such as those which a r i s e i n the i n i t i a l v a l u e problem, i s c o m p l e t e l y f e a s i b l e i n p r i n c i p l e , t h e r e a r e p r a c t i c a l l i m i t a t i o n s on the s i z e and type of problem which can be s a t i s f a c t o r i l y s o l v e d a t p r e s e n t . Even w i t h the r a p i d i n c r e a s e i n the speed and memory c a p a c i t y of computers, i t i s s t i l l an easy t a s k t o pose a n u m e r i c a l problem whose s o l u t i o n i s beyond the c a p a b i l i t y of any e x i s t i n g machine. For t h i s r e a s o n , a g r e a t d e a l of r e s e a r c h i s s t i l l b e i n g devoted t o the s e a r c h f o r e f f i c i e n t methods f o r s o l v i n g e l l i p t i c PDE's n u m e r i c a l l y . In a d d i t i o n , t h e r e a r e o t h e r f a c t o r s which must be c o n s i d e r e d here as i n the s o l u t i o n of any problem by n u m e r i c a l means. Among these a r e the r e l i a b i l i t y of the n u m e r i c a l r e s u l t s and the a b i l i t y t o ext e n d the s o l u t i o n method t o more g e n e r a l problems. In t h i s t h e s i s , a f a i r l y s i m p l e i n i t i a l v a l u e problem i s / 4 s t u d i e d . The problem was o r i g i n a l l y f o r m u l a t e d and p a r t i a l l y s o l v e d by Bowen and York [ 3 ] ; the s o l u t i o n was completed by P i r a n and York [ 4 1 ] . The i n i t i a l d a t a found has been i n t e r p r e t e d as r e p r e s e n t i n g " s n a p s h o t s " of t h r e e new f a m i l i e s of b l a c k h o l e s . I t i s hoped t h a t the t e c h n i q u e s used t o determine the d a t a w i l l be u s e f u l i n the s o l u t i o n of an i n i t i a l v a l u e problem f o r a non-head-on c o l l i s i o n of two or more b l a c k h o l e s . The problem was re-examined by the a u t h o r f o r t h r e e main r e a s o n s . Most i m p o r t a n t l y , the problem s e r v e d t o i n t r o d u c e the a u t h o r t o the f i e l d of n u m e r i c a l r e l a t i v i t y . S e c o n d l y , the study was i n t e n d e d t o p r o v i d e a check of the r e s u l t s which p r e v i o u s l y had been o b t a i n e d n u m e r i c a l l y . F i n a l l y , t h e problem s e r v e d as a " t e s t problem" f o r an i n v e s t i g a t i o n of a n u m e r i c a l t e c h n i q u e which may prove u s e f u l i n the s o l u t i o n of more c o m p l i c a t e d i n i t i a l v a l u e problems. The remainder of t h i s i n t r o d u c t o r y c h a p t e r i s d e v o t e d t o an o u t l i n e of the m a t e r i a l c o n t a i n e d i n t h i s t h e s i s . Chapter 2 b e g i n s w i t h a r e v i e w of the f o r m a l i s m which i s used t o study g e n e r a l r e l a t i v i t y from a d y n a m i c a l v i e w p o i n t . The 3+1 d e c o m p o s i t i o n of a spacetime i n t o a f a m i l y of s p a c e l i k e h y p e r s u r f a c e s i s d e s c r i b e d a l o n g w i t h the q u a n t i t i e s which c h a r a c t e r i z e the d e c o m p o s i t i o n . The c o n c e p t s of m e t r i c , p a r a l l e l t r a n s p o r t and c u r v a t u r e a r e examined i n the 3+1 language, f o l l o w e d by a d e r i v a t i o n of the i n i t i a l v a l u e e q u a t i o n s . York's approach t o the i n i t i a l v a l u e problem i s then p r e s e n t e d and the c h a p t e r c o n c l u d e s w i t h a r e v i e w of some of the i n i t i a l v a l u e problems which have been p r e v i o u s l y s t u d i e d . 5 In Chapter 3, the s p e c i f i c i n i t i a l value problem studied for t h i s thesis is discussed. The work previously done on the problem, which includes analytic solutions of three of the four constraint equations, i s reviewed. The remaining i n i t i a l value equation i s posed as a boundary value problem. Using f i n i t e difference techniques, t h i s boundary value problem i s then approximated by a discrete problem suitable for solution by numerical means. Chapter 4 is concerned with a description of the multi-grid method for the numerical solution of d i s c r e t i z e d boundary value problems. This r e l a t i v e l y new technique, developed over the past decade primarily by Brandt and co-workers, has been successfully applied to many problems from various f i e l d s of study, but has not been used previously in numerical r e l a t i v i t y . The method can be very e f f i c i e n t in comparison to some of the more well-known techniques such as SOR (successive over-relaxation) but requires a considerable amount of e f f o r t to implement. In Chapter 5, the results of some numerical experiments, as well as the results of the numerical solution of the problems described in Chapter 3, are presented. Using test problems of varying size, the performance of the multi-grid method i s compared to that of three other methods. The multi-grid algorithm is also evaluated using another test problem having an exact solution. F i n a l l y , the numerical results for the three new families of black holes are presented and compared with the results which had been previously calculated 6 by P i r a n . Chapter 6 contains some conclusions about the study and a d i s c u s s i o n of p o s s i b l e future extensions of the work. Fo l l o w i n g t h i s are two appendices: Appendix A reviews some of the techniques used i n the numerical s o l u t i o n of e l l i p t i c PDE'S. F i n i t e d i f f e r e n c e methods are b r i e f l y discussed as are some of the methods f o r s o l v i n g the systems of a l g e b r a i c equations which r e s u l t from the d i s c r e t i z a t i o n of boundary value problems. Appendix B contains a l i s t i n g of the m u l t i -g r i d program used to obtain the r e s u l t s discussed i n Chapter 5 as w e l l as some of the d e t a i l s of the implementation. 7 CHAPTER 2 The I n i t i a l V a l u e Problem f o r G e n e r a l R e l a t i v i t y The main purpose of t h i s c h a p t e r i s t o r e v i e w the f o r m a l i s m used t o study the i n i t i a l v a l u e problem f o r g e n e r a l r e l a t i v i t y from a s p a c e - p l u s - t i m e (3+1) v i e w p o i n t . I t i s assumed t h a t t h e r e a d e r i s f a m i l i a r w i t h the " b a s i c s " of g e n e r a l r e l a t i v i t y and d i f f e r e n t i a l geometry. T h i s i n c l u d e s the n o t i o n s of m e t r i c , p a r a l l e l t r a n s p o r t and c u r v a t u r e , as w e l l as some f a m i l i a r i t y w i t h the f i e l d e q u a t i o n s . Most of the m a t e r i a l i n s e c t i o n s 2.1 - 2.3 can a l s o be found i n Chapter 21 of G r a v i t a t i o n ( M i s n e r , Thorne, and Wheeler [ 2 7 ] ) , h e r e a f t e r r e f e r r e d t o as MTW. The major r e f e r e n c e f o r s e c t i o n s 2.4 - 2.5 i s the paper The I n i t i a l V a l u e Problem and Beyond ( P i r a n and York [ 4 1 ] ) . MTW c o n v e n t i o n s a r e used throughout t h i s t h e s i s . U n i t s a r e chosen such t h a t G=c='n=1, and the spacetime m e t r i c has s i g n a t u r e ( - + + + ) . Q u a n t i t i e s h a v i n g the s u p e r s c r i p t a r e d e f i n e d on the spacetime m a n i f o l d ; t h o s e denoted by ^ a r e d e f i n e d on a s p a c e l i k e h y p e r s u r f a c e of the s p a c e t i m e . Greek { L a t i n } i n d i c e s t a k e on t h e v a l u e s 0,1,2,3 {1,2,3}; summation c o n v e n t i o n s f o r both t y p e s of i n d i c e s a r e employed. The re a d e r may n o t i c e a s h i f t i n v i e w p o i n t as the c h a p t e r p r o g r e s s e s . In the f i r s t f o u r s e c t i o n s , the e x i s t e n c e of some spacetime which s a t i s f i e s E i n s t e i n ' s e q u a t i o n s i s assumed. A f t e r the 8 a p p r o p r i a t e i n i t i a l d a t a f o r t h i s h y p o t h e t i c a l spacetime has been i d e n t i f i e d , and the i n i t i a l v a l u e e q u a t i o n s have been p r e s e n t e d , a t t e n t i o n i s f o c u s s e d on the problem of d e t e r m i n i n g i n i t i a l d a t a which may be used t o c o n s t r u c t a g e n e r i c o s p a c e t i m e . 2.1 S p l i t t i n g spacetime i n t o space and time As mentioned i n the i n t r o d u c t o r y c h a p t e r , t h e 3+1 approach t o g e n e r a l r e l a t i v i t y i n v o l v e s a d e c o m p o s i t i o n of the 4 - d i m e n s i o n a l spacetime m a n i f o l d i n t o an i n f i n i t e f a m i l y of e d g e l e s s 3 - d i m e n s i o n a l s p a c e l i k e h y p e r s u r f a c e s . (A h y p e r s u r f a c e i s s p a c e l i k e i f and o n l y i f the d i s p l a c e m e n t between an a r b i t r a r y p a i r of d i s t i n c t p o i n t s of the s u r f a c e i s s p a c e l i k e . ) There i s a g r e a t d e a l of a r b i t r a r i n e s s i n the way one can s l i c e up a spacetime i n t o s p a c e l i k e h y p e r s u r f a c e s . S p e c i f i c a l l y , i t i s the same freedom one has i n c h o o s i n g a s e t of c o o r d i n a t e f u n c t i o n s xM on the spacetime m a n i f o l d such t h a t the tangent v e c t o r f i e l d t o one of the c o o r d i n a t e s , x°= t , i s everywhere t i m e l i k e . H aving chosen such a c o o r d i n a t e system, the s p a c e l i k e s l i c e s £(t) a r e j u s t t h e t = c o n s t a n t s u r f a c e s . W i t h t f i x e d , the r e m a i n i n g t h r e e c o o r d i n a t e s , x v s e r v e as a p a r t i c u l a r i n t e r n a l l a b e l l i n g of I L ( t ) . A c t u a l l y , a s i n g l e s e t o f s p a t i a l c o o r d i n a t e s x l may not be s u f f i c i e n t t o c o v e r a h y p e r s u r f a c e c o m p l e t e l y . T h i s w i l l not a f f e c t the f o l l o w i n g a n a l y s i s i n any way, s i n c e as w i l l be shown, a g i v e n h y p e r s u r f a c e may be c h a r a c t e r i z e d by q u a n t i t i e s which are independent of the i n t e r n a l l a b e l l i n g . On the o t h e r hand, the 9 x°=t c o o r d i n a t e must be g l o b a l i n the sense t h a t the s l i c e s i t d e t e r m i n e s c o m p l e t e l y f i l l t he s p a c e t i m e . (See r e f e r e n c e [10] f o r a more d e t a i l e d d i s c u s s i o n . ) Any h y p e r s u r f a c e X i ( t ) chosen i n the above manner i s assumed t o be a w e l l - d e f i n e d d i f f e r e n t i a b l e m a n i f o l d which may be a n a l y z e d u s i n g the same t e c h n i q u e s of d i f f e r e n t i a l geometry t h a t a r e a p p l i e d t o t h e s p a c e t i m e i t s e l f . The n o t i o n s of a p o i n t P of the h y p e r s u r f a c e and a c u r v e X ( P ) : P $ £(t) -> UR a r e taken as p r i m i t i v e . The l o c a l s p a t i a l c o o r d i n a t e s x 4 , of the 3+1 c o o r d i n a t e system a r e t h r e e f a m i l i e s of such c u r v e s . The tangent v e c t o r f i e l d s e t o the c o o r d i n a t e c u r v e s form a s p a t i a l b a s i s a t each p o i n t on the s l i c e which i s independent of the way the c o o r d i n a t e system v a r i e s away from the h y p e r s u r f a c e . Any o t h e r v e c t o r f i e l d , A which r e s i d e s i n the s l i c e , may be e x p r e s s e d i n terms of i t s components A 1 i n the b a s i s e i W X - A l " e i (2.1) In a s i m i l a r f a s h i o n , one can a l s o i n t r o d u c e f i e l d s of one-forms on the h y p e r s u r f a c e which a r e w r i t t e n i n terms of t h e i r components i n an a r b i t r a r y one-form b a s i s ^ d c o 1 . For example B i W i £ o l (2.2) The e x i s t e n c e of the n a t u r a l s c a l a r p r o d u c t , c 3 ) < , > between v e c t o r s and one-forms of the h y p e r s u r f a c e i s assumed. I f e\ i s a c o o r d i n a t e b a s i s of tangent v e c t o r s then the c o o r d i n a t e one-10 form b a s i s , dx j i s d u a l t o e- . That i s W < ° \ b \ e t > - 6\ (2.3) Then the c o n t r a v a r i a n t components AL i n (2.1) a r e g i v e n by A 1 = WC^4^ , t 3 )A > (2.4) S i m i l a r l y , the c o v a r i a n t components i n (2.2) a r e B; - 1»<< 3>B , ei > (2.5) The fundamental q u a n t i t y which c h a r a c t e r i z e s the h y p e r s u r f a c e , 57_,(t), i s a p o s i t i v e d e f i n i t e 3 - m e t r i c , f t > c j , which d e t e r m i n e s the l e n g t h s of t a n g e n t v e c t o r s and d e f i n e s n a t u r a l mappings from v e c t o r s t o one-forms and v i c e v e r s a . The c o v a r i a n t components < 5 )9;,: of the 3 - m e t r i c d e t e r m i n e th e l i n e element of t h e s l i c e - t h e d i s p l a c e m e n t between two p o i n t s P and Q i n £(t) h a v i n g spacetime c o o r d i n a t e s ( t , x l ) and ( t , x l + d x l ) r e s p e c t i v e l y i s g i v e n by diS 2" = C d i s + a o c e f r o m P to Q )* (2.6) The a s s o c i a t e d c o n t r a v a r i a n t c o m p o n e n t s , r a g ^ , of t h e 3 - m e t r i c s a t i s f y 13) i j 13) r 'l (o i \ 11 The q u a n t i t i e s w g t J a n d '^g^ j may be u s e d t o " r a i s e a n d l o w e r i n d i c e s " o f g e o m e t r i c a l o b j e c t s ( t e n s o r s ) d e f i n e d on t h e h y p e r s u r f a c e . B e c a u s e t h e h y p e r s u r f a c e H ( t ) i s e m b e d d e d i n a h i g h e r d i m e n s i o n a l m a n i f o l d , i t i s i n s t r u c t i v e t o e x a m i n e t h e r e l a t i o n s h i p b e t w e e n o b j e c t s d e f i n e d on t h e s u r f a c e a n d t h o s e w h i c h r e s i d e i n s p a c e t i m e i t s e l f . The q u a n t i t i e s o f g r e a t e s t i n t e r e s t a r e t h o s e w h i c h may u s e d t o c h a r a c t e r i z e t h e h y p e r s u r f a c e i t s e l f o r w h i c h d e s c r i b e how t h e s l i c e i s a c t u a l l y e m b e d d e d i n t h e s p a c e t i m e . M o r e o v e r , t o k e e p t h i n g s a s g e n e r a l a s p o s s i b l e , s u c h q u a n t i t i e s s h o u l d n o t d e p e n d on how t h e h y p e r s u r f a c e s a r e l a b e l l e d i n t e r n a l l y - t h a t i s , t h e y s h o u l d t r a n s f o r m a s t e n s o r s u n d e r a n y c o o r d i n a t e c h a n g e t h a t l e a v e s t h e t = c o n s t a n t s u r f a c e s u n c h a n g e d ( c h a n g e o f s p a t i a l b a s i s ) . F i r s t , c o n s i d e r a n a r b i t r a r y 4 - v e c t o r f i e l d ( 2 . 8 ) r < 4 1 A 1 e; • c 4 V e 0 C l e a r l y , t h i s o b j e c t c a n n o t be c o n s i d e r e d t o be d e f i n e d on a s i n g l e h y p e r s u r f a c e b e c a u s e o f t h e p r e s e n c e o f t h e c o m p o n e n t t 4 1A° e 0 . An a s s o c i a t e d 3 - v e c t o r f i e l d ( 3 ) A r e s i d i n g on a s l i c e <•*> •* c o u l d c o n c e i v a b l y be c o n s t r u c t e d f r o m A b y " k i l l i n g " t h e u n w a n t e d c o m p o n e n t , m a k i n g t h e i d e n t i f i c a t i o n f » A l = ( 4 ) A ' H o w e v e r , t h e c o m p o n e n t s A a r e d e t e r m i n e d i n t e r m s o f s c a l a r p r o d u c t s w i t h t h e s p a c e t i m e b a s i s o f o n e f o r m s ( A ) 6x"~ 12 C4> A1 ( A ) <4) C ^ A > (2.9) and t h i s b a s i s depends on the way i n which the s p a t i a l c o o r d i n a t e system v a r i e s away from the h y p e r s u r f a c e . Thus the components iA)kl do not form the components of an o b j e c t which t r a n s f o r m s as a 3 - v e c t o r under change of s p a t i a l c o o r d i n a t e s . However, th e s p a t i a l components A L of a spacetime one-form f i e l d CA)h ( 4 ) A. = u ) < ( 4>A , e t > (2.10) may be i d e n t i f i e d w i t h the t h r e e components of a one-form d e f i n e d on the h y p e r s u r f a c e s i n c e , as s t a t e d p r e v i o u s l y , the tangent v e c t o r s e\ do not depend on the way the c o o r d i n a t e system v a r i e s away from the s l i c e . Thus, under a change of s p a t i a l c o o r d i n a t e s , the components ( , ) A t = C 4 > A t w i l l t r a n s f o r m as d e s i r e d . T h i s statement may be extended t o the g e n e r a l case of a p u r e l y c o v a r i a n t t e n s o r f i e l d ^'T^a-..- on the spacetime m a n i f o l d - the s p a t i a l components ( 4 >T l j U... of such a t e n s o r a t any p o i n t P h a v i n g c o o r d i n a t e s ( t , x " ) form the components of a t e n s o r f , ' T - j k . . . which i s i n t r i n s i c t o the h y p e r s u r f a c e £ ( t ) . I n p a r t i c u l a r , t he s p a t i a l components W g c j o r t n e spacetime m e t r i c may be i d e n t i f i e d w i t h the components '°g;j as w i l l now be shown. To d e t e r m i n e the f u l l r e l a t i o n s h i p between the components ( 4 > 9 , ^ °f the spacetime m e t r i c and thos e of a 3-met r i c i n a 3+1 c o o r d i n a t e system, c o n s i d e r two p o i n t s P and Q h a v i n g c o o r d i n a t e s and xM+dxM r e s p e c t i v e l y . These p o i n t s then 13 r e s i d e on d i s t i n c t but "nearby" h y p e r s u r f a c e s . The spacetime d i s p l a c e m e n t , o r - p r o p e r t i m e , between the two p o i n t s i s g i v e n by ( A W « d x ^ d x ' (2.11) There i s an o t h e r way of c a l c u l a t i n g t h i s d i s t a n c e as i s sugg e s t e d by F i g u r e 1. On the s u r f a c e £(t) l o c a t e the p o i n t P' which l i e s d i r e c t l y "beneath" Q i n the sense t h a t a s m a l l d i s p l a c e m e n t from P' i n the d i r e c t i o n normal t o the h y p e r s u r f a c e £(t) t a k e s one t o the p o i n t Q. D e f i n e the f u n c t i o n N ( t , x L ) such t h a t N ( t , x l ) d t i s the pr o p e r d i s t a n c e between P' and Q. In g e n e r a l t h e c o o r d i n a t e s of P 1 a r e not ( t , x L + d x L ) . One must i n t r o d u c e t h r e e more f u n c t i o n s N L ( t , x L ) such t h a t ( t , x l + d x l + N L d t ) a r e the c o o r d i n a t e s of P'. Then, u s i n g a 4 - d i m e n s i o n a l g e n e r a l i z a t i o n of the Pythagorean theorem, the prop e r d i s t a n c e from P t o Q i s l 4 W = ^ ^ ( d x v N M t K d x ^ NJat ) - ( M d t ) z (2.12) The f u n c t i o n s N and N l a r e c a l l e d t he l a p s e and s h i f t f u n c t i o n s r e s p e c t i v e l y s i n c e N d e t e r m i n e s the l a p s e i n prop e r time between s u c c e s s i v e l y l a b e l l e d h y p e r s u r f a c e s , w h i l e N l d e s c r i b e s the s h i f t i n the s p a t i a l c o o r d i n a t e s between the two s l i c e s . Together the f o u r f u n c t i o n s a r e s i m p l y a m a n i f e s t a t i o n of t h e f a c t t h a t one has chosen a p a r t i c u l a r c o o r d i n a t e system t o s l i c e t h e spacetime and l a b e l t he r e s u l t i n g h y p e r s u r f a c e s . I f a d i f f e r e n t s l i c i n g or l a b e l l i n g i s chosen, then N 1 and N 14 Figure 1 Spacetime Displacement i n 3+1 Language 15 must adjust so as to keep (2.12) v a l i d . The 3 functions N l(t,x 1') may be regarded as the components of a 3-vector f i e l d residing in the hypersurface "£(t). The associated covariant components are given by N j = M ' ( 2' 1 3 ) Using t h i s r e l a t i o n in (2 .12) and comparing the result with (2.11) leads to the i d e n t i f i c a t i o n f 4 ^ 0 0 - ( N * N" - N T ) ^ ^ (2.14) This i s the ADM expression for the spacetime metric in terms of the 3-metric of a spacelike hypersurface and the lapse and s h i f t functions. The components (,A)g^*,v of the reciprocal 4-metric may be expressed in t h i s language v i a the relationship <3 <3 /o (2.15) with the result (4) oo 1 9 • - 7T* °s * g - — , (2.16) Another quantity which i s useful for describing the embedding of a spacelike hypersurface in a spacetime i s the 16 unit timelike vector f i e l d n which i s normal to the hypersurface ( A ) - A - - \ ^ ' (2.17) n* A,* o where A* are the components of any one-form which can be considered to reside in the hypersurface, that i s with Ao=0. The components n* may be ea s i l y determined by observing that the components n* of the associated one-form n, ( <n,n> = -1 ) in the basis dx*1 = (dt,dx'' ) are just A o -- N (2.18) r» ; = 0 Then, using (2.16) ( 2 . 1 9 ) i N n 1 - N A p a r t i c u l a r coordinate system which i s very useful for the derivation of the i n i t i a l value equations results from choosing the unit normal as the timelike basis vector, which then implies that the lapse function i s equal to unity everywhere on the hypersurface. 2.2 I n t r i n s i c and e x t r i n s i c curvature To complete the description of the relat i o n s h i p between a hypersurface and the spacetime in which i t i s embedded, the concepts of p a r a l l e l transport and curvature must be examined 17 i n the 3+1 f o r m a l i s m . There a r e a c t u a l l y t h r e e i n t e r r e l a t e d n o t i o n s of c u r v a t u r e t o be d e a l t w i t h : 1) the c u r v a t u r e of the spacetime i t s e l f , c h a r a c t e r i z e d by the spacetime Riemann t e n s o r R^yS , 2) the c u r v a t u r e i n t r i n s i c t o the h y p e r s u r f a c e , d e s c r i b e d by the 3 - d i m e n s i o n a l c u r v a t u r e t e n s o r (^Rt^w:^ , and 3) the c u r v a t u r e which d e s c r i b e s how the h y p e r s u r f a c e i s embedded i n the s p a c e t i m e . R e c a l l t h a t the c u r v a t u r e a t a p a r t i c u l a r event of spacetime may be d e t e r m i n e d by examining the change i n d i r e c t i o n of an a r b i t r a r y v e c t o r which i s p a r a l l e l t r a n s p o r t e d around a c l o s e d l o o p i n a neighborhood of the e v e n t . The e x i s t e n c e of a w e l l - d e f i n e d o p e r a t i o n of p a r a l l e l t r a n s p o r t i s e q u i v a l e n t t o the e x i s t e n c e of a spacetime c o v a r i a n t d e r i v a t i v e o p e r a t o r which i s c o m p a t i b l e w i t h the 4 - m e t r i c 3*/s - ^ " 0 = ^ (2.20) The f a c t t h a t a v e c t o r r o t a t e s when c a r r i e d a l o n g a s m a l l c l o s e d l o o p i s r e f l e c t e d i n the n o n - c o m m u t a t i v i t y of c o v a r i a n t d i f f e r e n t i a t i o n . Thus, f o r an a r b i t r a r y v e c t o r (4)W = W^e*, (A) x h a v i n g an a s s o c i a t e d one-form W = W«dx , the components of the Riemann t e n s o r s a t i s f y the f o l l o w i n g r e l a t i o n ( 4 ) R " / 5 * * W « = V ^ V ^ W / s - V , V ^ W ^ (2.21) The c u r v a t u r e i n t r i n s i c t o a s p a c e l i k e h y p e r s u r f a c e Y L ( t ) may be a n a l y z e d i n the same way, by i n t r o d u c i n g c o n c e p t s of 18 p a r a l l e l t r a n s p o r t and c o v a r i a n t d e r i v a t i v e d e f i n e d w i t h r e s p e c t t o the 3-geometry of the s l i c e . The c o v a r i a n t d e r i v a t i v e a l o n g the b a s i s v e c t o r e;, taken w i t h r e s p e c t t o the geometry of the h y p e r s u r f a c e i s denoted by D;. I t i s c o m p a t i b l e w i t h the 3 - m e t r i c ~- D- ih\>* - 0 (2.22) The r e s u l t of a p p l y i n g t h i s o p e r a t o r t o an a r b i t r a r y v e c t o r — 1 i - * A=A e c tangent t o the h y p e r s u r f a c e i s D i C A ^ O - A \ i £ i * A J " Y f t c k (2.23) w h e r e T *t a r e the c o n n e c t i o n c o e f f i c i e n t s o f the 3-geometry which may be e x p r e s s e d , i n any c o o r d i n a t e system, i n terms of the 3 - m e t r i c and i t s f i r s t d e r i v a t i v e s WO p " C?.1 p . (2.24) P „ • - - ± / <*> . • fi> v Note, t h a t because ( ° g = c < 0g ; j ^r,-> » r c V * (2.25) S i m i l a r l y , f o r a one form •* )B=Bjdx J 19 Now c o n s i d e r the a p p l i c a t i o n of t o the v e c t o r i41A=A' e; = < 5 )A ^ ( A 1 ^ ) - A V u e , * A ^ P ^ e ^ (2.27) The r e s u l t c o n t a i n s a component, < G • A I i i e a / w h i c h i s normal t o t h e h y p e r s u r f a c e . I f t h i s component i s " p r o j e c t e d o u t " , then what remains i s p r e c i s e l y the c o v a r i a n t d e r i v a t i v e of A taken w i t h r e s p e c t t o the h y p e r s u r f a c e . That i s D;(A.J = V- (A'c i j ) ~ A i ( 4 ' Fy. , e . > S (2.28) Having d e f i n e d a n o t i o n of c o v a r i a n t d i f f e r e n t i a t i o n which i s i n t r i n s i c t o the h y p e r s u r f a c e , the components of the 3- d i m e n s i o n a l Riemann c u r v a t u r e t e n s o r ^ R ' j * * . may be det e r m i n e d from the f o l l o w i n g J (2.29) where V L a r e the components of an a r b i t r a r y v e c t o r tangent t o the s l i c e . In a d d i t i o n t o the i n t r i n s i c c u r v a t u r e '"R' J^ of a h y p e r s u r f a c e YL(t) , a n o t h e r measure of c u r v a t u r e may be d e f i n e d on the h y p e r s u r f a c e which c o m p l e t e l y d e s c r i b e s the way i n which H(t) i s embedded i n the sp a c e t i m e . A s i m p l e example 20 b e s t i l l u s t r a t e s the f e a t u r e s of t h i s c u r v a t u r e . C o n s i d e r two s h e e t s of paper, one l a y i n g f l a t , and the o t h e r r o l l e d up i n t o a c y l i n d e r ( F i g u r e 2 a ) . Both s h e e t s have the same i n t r i n s i c E u c l i d e a n geometry as c o u l d be d e t e r m i n e d by h y p o t h e t i c a l o b s e r v e r s c o n f i n e d t o the s h e e t s , by the l a c k of r o t a t i o n of t a n g e n t v e c t o r s p a r a l l e l t r a n s p o r t e d , w i t h r e s p e c t t o t h e i r 2-g e o m e t r i e s , around c l o s e d l o o p s ( F i g u r e 2 b ) . A 3 - d i m e n s i o n a l o b s e r v e r , however, sees a d e f i n i t e d i s t i n c t i o n between the two c a s e s , c l a i m i n g t h a t the c y l i n d e r i s " c u r v e d " by v i r t u e of the manner i n which i t i s embedded i n the h i g h e r d i m e n s i o n a l space. T h i s e x t r i n s i c c u r v a t u r e (or l a c k of i t ) may be measured by examining a v e c t o r f i e l d c o n s t r u c t e d n o r m a l l y t o the 2 - d i m e n s i o n a l s u r f a c e . The u n i t normal v e c t o r a t the p o i n t a t which the e x t r i n s i c c u r v a t u r e i s t o be measured i s p a r a l l e l t r a n s p o r t e d w i t h r e s p e c t t o the embedding space a s h o r t d i s t a n c e a l o n g the s u r f a c e . The p a r a l l e l - t r a n s p o r t e d normal i s then compared t o the normal which a l r e a d y r e s i d e s a t the new p o s i t i o n . I n the a p p r o p r i a t e l i m i t s , t h e d i f f e r e n c e between th e s e two v e c t o r s i s a n o t h e r v e c t o r t a n g e n t t o the s u r f a c e which p r o v i d e s a measure of p a r t of the e x t r i n s i c c u r v a t u r e ( F i g u r e 2 c ) . To complete the measurement i n 2-dimensions would r e q u i r e t h a t the p r o c e s s be r e p e a t e d by t r a n s p o r t i n g the v e c t o r i n a d i f f e r e n t d i r e c t i o n . In g e n e r a l , the e x t r i n s i c c u r v a t u r e d e s c r i b i n g the embedding of a s u r f a c e i n a "one-h i g h e r " d i m e n s i o n a l space i s a second rank t e n s o r h a v i n g no components i n the d i r e c t i o n of the normal t o the s u r f a c e and can t h u s be c o n s i d e r e d t o be d e f i n e d on t h e s u r f a c e . R e t u r n i n g t o the case of a s p a c e l i k e h y p e r s u r f a c e 21 F i g u r e 2 E x t r i n s i c C u r v a t u r e Example 22 embedded i n s p a c e t i m e , the c o v a r i a n t d e r i v a t i v e of t h e one-form n has components *^>/!> - ~ n ^ ^ ^ r ^ a (2.30) The s p a t i a l components of t h e above e x p r e s s i o n d e f i n e the c o v a r i a n t components K V j of the e x t r i n s i c c u r v a t u r e t e n s o r d e f i n e d on the h y p e r s u r f a c e K,-j - ( n;. j - n ^ M ' r ^ • ) (2.31) (t h e s i g n above has been chosen by c o n v e n t i o n and has no p h y s i c a l s i g n i f i c a n c e ) . U s i n g e q u a t i o n ( 2 . 1 8 ) , t h i s becomes Ki] = - N C 4 ) P • 3 (2.32) I t f o l l o w s from the symmetry of the P^v i n any c o o r d i n a t e system t h a t K i j . K j t (2.33) E q u a t i o n (2.28) may now be r e w r i t t e n w i t h the h e l p of (2.32) t o d i s p l a y the r e l a t i o n between the c o v a r i a n t d e r i v a t i v e s of the spacetime and h y p e r s u r f a c e , and the e x t r i n s i c c u r v a t u r e V - C A ' ^ J ) = D . C A 4 ^ ) - K i j A ' * (2.34) For the s p e c i a l case when the a r b i t r a r y v e c t o r A above i s a 23 s p a t i a l b a s i s v e c t o r , t h i s becomes A c l o s e r examination of equation (2.32) r e v e a l s an i n t e r e s t i n g r e l a t i o n s h i p between the e x t r i n s i c curvature and the 3-metric • — [T>. KJj - p j N i - ^ ] ( 2- 3 6 ) In the s p e c i a l case where the 3+1 coordinate system i s chosen such that N=1 and N'=0 (Gaussian normal coordinates) the above becomes I e> ^ = - t (2.37) Thus the e x t r i n s i c curvature may be v i s u a l i z e d as a " v e l o c i t y " of the 3-metric i n the d i r e c t i o n normal t o the hypersurface. I t must be emphasized, however, that the e x t r i n s i c curvature i s d e f i n e d on a s i n g l e hypersurface as opposed to a "true" v e l o c i t y of the 3-metric whose determination i n v o l v e s , i n p r i n c i p l e , s p e c i f y i n g the 3-metrics on two nearby s l i c e s and then c a l c u l a t i n g , the rate of change v i a some so r t of l i m i t i n g procedure i n which the separate s l i c e s approach each other. The f a c t that two s l i c e s are involved i m p l i e s that t h i s procedure w i l l n e c e s s a r i l y i n v o l v e the coordinate choice which induces the s l i c i n g , which i n turn i m p l i e s that the r e s u l t i n g 24 v e l o c i t y which i s t o c h a r a c t e r i z e a s i n g l e h y p e r s u r f a c e w i l l not n e c e s s a r i l y be a t e n s o r on t h e s l i c e . From a d y n a m i c a l p o i n t of v i e w , the 3- m e t r i c components C 5 ^ 9 i j d e t e r m i n e the c o n f i g u r a t i o n of a h y p e r s u r f a c e and may be i n t e r p r e t e d as g e n e r a l i z e d c o o r d i n a t e s which s p e c i f y the " p o s i t i o n " of the h y p e r s u r f a c e i n the space of a l l p o s s i b l e p o s i t i v e d e f i n i t e 3 - d i m e n s i o n a l m a n i f o l d s . L o o s e l y s p e a k i n g , the e x t r i n s i c c u r v a t u r e components K l J may then be thought of as the momenta d y n a m i c a l l y c o n j u g a t e t o t h e • T h i s i n t e r p r e t a t i o n i s c o n s i s t e n t w i t h s i m p l e r d y n a m i c a l t h e o r i e s such as Newtonian mechanics where the c o n j u g a t e momenta, a l t h o u g h c l o s e l y r e l a t e d t o t h e time d e r i v a t i v e s of the g e n e r a l i z e d c o o r d i n a t e s , do not demand f o r t h e i r s p e c i f i c a t i o n the n o t i o n of g i v i n g the s t a t e of the system a t two i n f i n i t e s i m a l l y s e p a r a t e d t i m e s . A more complete d e s c r i p t i o n of t h i s v i e w p o i n t i s found i n the 1962 ADM paper [ 1 ] . The s e t (l*^g ^  ,K 1 1 ) c o m p l e t e l y c h a r a c t e r i z e s the embedding of a h y p e r s u r f a c e i n a s p a c e t i m e , or "the s t a t e of a spacetime a t some i n s t a n t " and t h e r e b y c o n s t i t u t e s i n i t i a l d a t a f o r the s p a c e t i m e . 2.3 The i n i t i a l v a l u e e q u a t i o n s Having d e c i d e d t h a t the 3 - m e t r i c and e x t r i n s i c c u r v a t u r e a r e a p p r o p r i a t e q u a n t i t i e s f o r the complete d e s c r i p t i o n of the d y n a m i c a l s t a t e of a s p a c e t i m e , a t t e n t i o n i s now f o c u s s e d on the q u e s t i o n of the independence of t h i s i n i t i a l d a t a . As t h i s s e c t i o n w i l l show, f o u r components of the E i n s t e i n t e n s o r , 25 namely G / u L , may be w r i t t e n i n a 3+1 c o o r d i n a t e system e n t i r e l y i n terms of the q u a n t i t i e s ( t s > g i j ,K 4 J ). T h e r e f o r e the f o u r f i e l d e q u a t i o n s G V - 8 TT (2.38) r e p r e s e n t f o u r e q u a t i o n s of c o n s t r a i n t on the i n i t i a l d a t a s e t which i s extended t o i n c l u d e the q u a n t i t i e s T*^ . The c a l c u l a t i o n of the q u a n t i t e s G % i n the 3+1 f o r m a l i s m f i r s t r e q u i r e s the d e t e r m i n a t i o n of c e r t a i n components of R ^ v S . S p e c i a l i z i n g the 3+1 c o o r d i n a t e system by c h o o s i n g N=1 so t h a t the normal, n, t o the h y p e r s u r f a c e i s t h e t i m e l i k e b a s i s v e c t o r and u s i n g ( 2 . 2 1 ) , the components R^,j< a r e de t e r m i n e d by ^ ' R ^ K <V - Vs V k e- - V . V, e, (2.39) Now, from (2.34) and (2.35) Thus 26 v j v , e, - 7, 7j e; ( P « K ; j - p j K : * ) ( v<;,< K] - K * + " ' R 5 ^ * ) e. (2.40) Comparison w i t h (2.39) g i v e s the r e s u l t (2.41a) j - P< K i j - Dj K. (2.41b) The d e s i r e d components of the E i n s t e i n t e n s o r may now be c a l c u l a t e d u s i n g the r e l a t i o n (MTW, e q u a t i o n 14.6) 6^, ° °/5T<3" rs. I ju. v I i x cr I (2.42) where S ^ S M i s a p e r m u t a t i o n t e n s o r h a v i n g components +1 .4 rf/**> i s a-^  e v e n p t r m u i at i DA O*S p, x- <T -1 " o d d " 'l " and 11«/9 | denotes the r e s t r i c t i o n » >p . Then r ° - k . v i + l< i i + l < g i (2.43) 27 Now, the R i c c i s c a l a r i n t h r e e d i m e n s i o n s i s (2.44) where the p r o p e r t y T V R L 4 K J t = "^ R^W = + " J R JXK N A S B E E N used. D e f i n i n g T V K s ; j K ' J (2.45) 3 and n o t i n g t h a t ( T r k ) z - K ; i k ; j = ( k ' t ) 2 - K \ K\ = 2 ( k ' , k \ + KI k ? 3 + k j k ' , - k t k L - k ' , k ' , - k i K\ ) ( 2 - 4 6 ) a l l o w s (2.43) t o be w r i t t e n as -IG°0 = C 5 ) R +• ( T r K ) V - k ,- j 1<; J The f i e l d e q u a t i o n G ° = 87TT00 r e l a t e s the above e x p r e s s i o n t o t o the l o c a l energy d e n s i t y , ^ , as measured by an o b s e r v e r i n s t a n t a n e o u s l y a t r e s t i n the h y p e r s u r f a c e . Thus v ( T r K ) 1 - K.'j k L j = 1 6 * ^ (2.47) The o t h e r r e q u i r e d components of the E i n s e i n t e n s o r , G may be c a l c u l a t e d i n a s i m i l a r f a s h i o n u s i n g ( 2 . 4 2 ) . 28 For example G O t A ) o ° C.4 ) Q OS = - D , i<! + D 3 k5, - D , K 5 , = p , \<; - p a K*, + P i K5, - P, «'« - P . i<l -1?. i<i - P l V<l, - P, ( T r l < ) o r , i n g e n e r a l G ° l = \ < J 1 - D ! ( T r k ) (2.48) These e x p r e s s i o n s a r e t o be equated t o 8 n T ° ' which i s 871 t i m e s the momentum d e n s i t y , j 0 , a g a i n as measured by an o b s e r v e r m o m e n t a r i l y r e s t i n g i n the h y p e r s u r f a c e . Thus P ] l < 1 1 - p ; ( T r l < ) = 8 7 1 ] ' (2.49) E q u a t i o n s (2.47) and (2.49) a r e the i n i t i a l v a l u e e q u a t i o n s f o r g e n e r a l r e l a t i v i t y . They depend o n l y on q u a n t i t i e s d e f i n a b l e on a s i n g l e h y p e r s u r f a c e and a r e c o o r d i n a t e independent i n the sense t h a t a l l of the q u a n t i t i e s i n v o l v e d w i l l t r a n s f o r m as t e n s o r s under any r e l a b e l l i n g of the spacetime which l e a v e s the t = c o n s t a n t s u r f a c e s unchanged. The s i x r e m a i n i n g f i e l d e q u a t i o n s G l A = 8 7 1T'^ govern the e v o l u t i o n of the i n i t i a l d a t a ( ( % ) g r j f K u ) but a r e not e a s i l y d e t e r m i n e d by means of the p r o c e s s used above f o r the i n i t i a l v a l u e e q u a t i o n s . A r n o w i t t , Deser and M i s n e r [1] d e r i v e d the e v o l u t i o n e q u a t i o n s ( f o r t h e vacuum ca s e ) by w r i t i n g the u s u a l a c t i o n f o r g e n e r a l r e l a t i v i t y 29 - / P ^ M > R cT> ( 2 ' 5 0 ) i n the 3+1 f o r m a l i s m and demanding t h a t t h i s i n t e g r a l be s t a t i o n a r y w i t h r e s p e c t t o independent v a r i a t i o n s of the dy n a m i c a l v a r i a b l e s . These were taken t o be the s i x f u n c t i o n s ( w g ; j and t h e ADM c o n j u g a t e momenta ^' J c l o s e l y r e l a t e d t o the components of the e x t r i n s i c c u r v a t u r e TT' ^ H f7^ ( g ;i Tr K - O ) ( 2 ' 5 1 ) where t ?^g i s the d e t e r m i n a n t of the 3 - m e t r i c . The r e s u l t i s a set of 12 c o u p l e d e q u a t i o n s i n v o l v i n g f i r s t time d e r i v a t i v e s of ( i ) g ^ and T r l i . The e q u a t i o n s a r e q u i t e c o m p l i c a t e d and w i l l not be reproduced here s i n c e t h e i r e x a c t form i s of l i t t l e , i f any, r e l e v a n c e t o the i n i t i a l v a l u e problem. Moreover, the ADM form of the e v o l u t i o n e q u a t i o n s have not been used i n r e c e n t n u m e r i c a l c o n s t r u c t i o n s of spacetimes - the t r e n d b e i n g t o use the components K j as the c o n j u g a t e v a r i a b l e s t o the 3-me t r i c components [ 3 3 ] . One f u r t h e r f e a t u r e of the ADM a n a l y s i s sheds l i g h t on the n a t u r e of the i n i t i a l v a l u e e q u a t i o n s . When w r i t t e n i n the 3+1 f o r m a l i s m , the L a g r a n g i a n ^ ( x M ) i n (2.50) c o n t a i n s terms l i n e a r i n the l a p s e and s h i f t f u n c t i o n s . By c o n s i d e r i n g t h e l a p s e and the t h r e e components of the s h i f t t o be independent k i n e m a t i c a l v a r i a b l e s and r e q u i r i n g t h a t (2.50) remain s t a t i o n a r y when v a r i e d w i t h r e s p e c t t o them", the i n i t i a l v a l u e e q u a t i o n s a r e r e c o v e r e d . In t h i s way, the c o n s t r a i n t e q u a t i o n s a r e seen t o be a consequence of the f o u r - f o l d c o o r d i n a t e freedom of g e n e r a l r e l a t i v i t y . 30 2.4 P r e s e r v a t i o n of the c o n s t r a i n t s From the p o i n t of view of c o n s t r u c t i n g a s p a c e t i m e , e q u a t i o n s (2.47) and (2.49) r e p r e s e n t c o n s t r a i n t s which must be s a t i s f i e d i n s p e c i f y i n g an i n i t i a l d a t a s e t ( ( , )g,j ,K'i ) f o r a s p a c e l i k e h y p e r s u r f a c e , £ ( t ) . Assuming such a s e t of d a t a has been d e t e r m i n e d , one can, i n p r i n c i p l e a t l e a s t , e v o l v e the d a t a u s i n g the r e m a i n i n g E i n s t e i n e q u a t i o n s . At any l a t e r ( e a r l i e r ) time t 1 , t h a t i s on the h y p e r s u r f a c e H,(t'), the e v o l v e d d a t a ( < ? > >g ; j ' , KM ' ' , j ' ' ) w i l l a l s o s a t i s f y the c o n s t r a i n t e q u a t i o n s by v i r t u e of the B i a n c h i i d e n t i t i e s = O (2.52) and the c o n s e r v a t i o n of the s t r e s s - e n e r g y t e n s o r ~ r ^ , - 0 (2.53) To see t h a t t h i s i s so, the f i e l d e q u a t i o n s a re w r i t t e n i n the f o l l o w i n g form s - 8 - n T " ^ O (2.54) Then the c o n s t r a i n t e q u a t i o n s a r e s i m p l y (2.55) 31 Now H ^ j i / = G^-u - B i r T ^ i * -O ( 2 . 5 6 ) S O = H ^ . o • H^,i ^ T - H ^ • M T ; , H ^ (2.57) or vr,* *• - ( H M i . i * C 4 > r ^ H / O W ' <- C A , P ^ H ^ ) ( 2 . 5 8 ) The e v o l u t i o n e q u a t i o n s a r e - 0 ( 2 . 5 9 ) F u r t h e r m o r e , on any h y p e r s u r f a c e , the s p a t i a l d e r i v a t i v e s of H'"-1 w i l l a l s o v a n i s h H ^ L ' 1 ~~ 0 ( 2 . 6 0 ) E q u a t i o n s ( 2 . 5 4 ) , ( 2 . 5 8 ) and ( 2 . 6 0 ) now imply t h a t H 1 ° " ° ( 2 . 6 1 ) w i t h the consequence t h a t the c o n s t r a i n t e q u a t i o n s h o l d f o r the complete e v o l u t i o n , once they have been s a t i s f i e d on a 32 i n i t i a l s l i c e . T h i s p o i n t i s not t o o c r u c i a l f o r the i n i t i a l v a l u e problem i t s e l f , but i s e x t r e m e l y i m p o r t a n t i n the c o n t e x t of n u m e r i c a l e v o l u t i o n of the f i e l d e q u a t i o n s , where by c u r r e n t methods, c o n s t r a i n e d d a t a i s not a u t o m a t i c a l l y e v o l v e d i n t o c o n s t r a i n e d d a t a . In es s e n c e , the d i s c r e t e forms of the f i e l d e q u a t i o n s which a r e c u r r e n t l y used i n n u m e r i c a l r e l a t i v i t y a r e o v e r d e t e r m i n e d systems. P i r a n c o n s i d e r s t h i s o v e r d e t e r m i n i s m t o be the most i m p o r t a n t problem c u r r e n t l y f a c i n g n u m e r i c a l r e l a t i v i s t s [ 3 2 ] . T h i s completes the d e s c r i p t i o n of the o r i g i n s and b a s i c n a t u r e of the i n i t i a l v a l u e problem. The next s e c t i o n of t h i s c h a p t e r i s devoted t o the d e s c r i p t i o n of a p a r t i c u l a r approach to the i n i t i a l v a l u e problem, c h i e f l y due t o York and v a r i o u s c o l l a b o r a t o r s [ 1 1 ], [ 2 9 ] , [ 3 9 ] , [ 4 0 ] , [ 4 1 ] . 2.5 Y o r k ' s approach t o the i n i t i a l v a l u e problem Y o r k ' s a n a l y s i s of the i n i t i a l v a l u e problem i s based on a p a r t i c u l a r way of s e p a r a t i n g the i n i t i a l d a t a i n t o f r e e l y s p e c i f i a b l e and d e t e r m i n e d - f r o m - c o n s t r a i n t p i e c e s . P a r t of the m o t i v a t i o n f o r h i s approach was the d e s i r e t o i d e n t i f y the " t r u e " d y n a m i c a l g r a v i t a t i o n a l v a r i a b l e s - those which c o r r e s p o n d t o the r a d i a t i v e degrees of freedom - i n a c o v a r i a n t manner. From the v i e w p o i n t of c o n s t r u c t i n g i n i t i a l d a t a , however, the c h i e f s u c c e s s of Y o r k ' s work i s t h a t i t p u t s the i n i t i a l v a l u e e q u a t i o n s i n t o a form which a l l o w s them to be s o l v e d , a t l e a s t i n p r i n c i p l e , w i t h e x i s t i n g n u m e r i c a l methods and i n some c a s e s , by a n a l y t i c means. ( S i n c e t h i s 33 s e c t i o n d e a l s e n t i r e l y w i t h 3 - d i m e n s i o n a l q u a n t i t i e s , the s u p e r s c r i p t w i l l no l o n g e r be employed.) The s i m p l e s t p a r t of Y o r k ' s p r e s c r i p t i o n f o r the s p e c i f i c a t i o n of i n i t i a l d a t a i s t o g i v e f r e e l y on the h y p e r s u r f a c e the v a l u e of the mean e x t r i n s i c c u r v a t u r e , TrK T> \< = C j M l O (2.45) There a r e both p h y s i c a l and p r a c t i c a l m o t i v a t i o n s f o r c h o o s i n g t o do t h i s . From the p h y s i c a l p o i n t of view, TrK p r o v i d e s a measure of the l o c a l change i n volume of the h y p e r s u r f a c e as i t e v o l v e s i n t o a nearby s l i c e . For example, i f TrK v a n i s h e s , then the h y p e r s u r f a c e has maximal volume, as i s the case f o r the s t a n d a r d t = c o n s t a n t h y p e r s u r f a c e s of M i n k o w s k i i s p a c e t i m e . In a d d i t i o n , the v a l u e of TrK i s v e r y s e n s i t i v e t o the way i n which the h y p e r s u r f a c e i s embedded i n the spacetime - a l o c a l i z e d d e f o r m a t i o n ( e v o l u t i o n ) of the s l i c e w i l l cause TrK t o change somewhere i n the neighborhood of the d e f o r m a t i o n . In t h i s sense, TrK has been i n t e r p r e t e d as a n a t u r a l , l o c a l " t i m e " f u n c t i o n [ 2 3 ] , [ 2 9 ] . A l s o , c e r t a i n s i m p l e c h o i c e s of TrK l e a d t o s i g n i f i c a n t s i m p l i f i c a t i o n of the i n i t i a l v a l u e e q u a t i o n s as w i l l be shown l a t e r i n t h i s s e c t i o n . In the f o l l o w i n g a n a l y s i s , t h e n , TrK w i l l be assumed t o be g i v e n , so i t i s c o n v e n i e n t t o i s o l a t e the t r a c e - f r e e p a r t of the e x t r i n s i c c u r v a t u r e K lJ as f o l l o w s E L i 5 K l i ~ t 3 , J T r K (2.62a) Id E ; i = O K L s = E l i + i e J i Tr K (2.62b) 34 A key i d e a i n Y o r k ' s program which i s o r i g i n a l l y due t o L i c h n e r o w i c z i s t o s p e c i f y the i n i t i a l d a t a (g:j , E 1 4 , t r K ; ^ , j ) o n l y up t o a c o n f o r m a l t r a n s f o r m a t i o n . For example, the 3-m e t r i c i s w r i t t e n as <$L\ = ( 2 * 6 3 ) where " t i s a s t r i c t l y p o s i t i v e f u n c t i o n of p o s i t i o n on the h y p e r s u r f a c e , and the base m e t r i c , g t j i s assumed t o be g i v e n . I f V 1 and are v e c t o r s tangent t o the h y p e r s u r f a c e a t the same p o i n t , then the a n g l e 8 between them 9 = c o s " 1 — x (2.64) ( c ^ j V - V J C J K X w* w*; 1 i s c l e a r l y unchanged under such a t r a n s f o r m a t i o n - hence the term " c o n f o r m a l " . P h y s i c a l l y , the c o n f o r m a l f a c t o r d e t e r m i n e s the l o c a l l e n g t h s c a l e of the s u r f a c e . A s s o c i a t e d w i t h g;j i s i t s i n v e r s e g l i which s a t i s f i e s 3 { i § i * * o \ (2.65) Another u s e f u l r e l a t i o n i s ^ = J~<f (2.66) 35 I t i s seen from (2.64) and (2.66) t h a t the q u a n t i t y remains i n v a r i a n t under a c o n f o r m a l t r a n s f o r m a t i o n . A c o v a r i a n t d e r i v a t i v e D t c o m p a t i b l e w i t h g may be i n t r o d u c e d . " J ^ (2.67) The c o n n e c t i o n components P • < a r e e a s i l y shown t o be r e l a t e d t o P • * as f o l l o w s r\< - r U * + s i ^ K - JK t , ~ ) (2.68) In a d d i t i o n , a r a t h e r l e n g t h y but s t r a i g h t f o r w a r d c a l c u l a t i o n g i v e s the f o l l o w i n g r e l a t i o n s h i p between the s c a l a r c u r v a t u r e s of the p h y s i c a l and base m e t r i c s R = t ~ 4 R - 8 t ~ S i t (2.69) where A i s the L a p l a c i a n i n the base m e t r i c A s SL l ) 1 (2.70) The o t h e r i n i t i a l d a t a , w i t h the e x c e p t i o n of TrK, a r e a l s o t o be c o n f o r m a l l y s p e c i f i e d . The t r a c e - f r e e p a r t of the e x t r i n s i c c u r v a t u r e i s t r a n s f o r m e d as f o l l o w s C ( 2 . 7 1 ) 36 C l e a r l y , E i s a l s o t r a c e - f r e e . Moreover, t h i s p a r t i c u l a r s c a l i n g g i v e s the f o l l o w i n g r e s u l t Dj E M = i ) j E L i (2.72) s i n c e C T - j 4- Z-V'A X< * i H j ~ 3 ; Ag K j t , ) ) ( ^ - , e E "O + . - . V ' [ ( ^ E K S ^ E ; J - t x 5 1 4 T r § ' i ) + F i n a l l y , the q u a n t i t i e s p and j ' a r e assumed t o be s p e c i f i e d so t h a t and (2.73) (2.74) ar e the p h y s i c a l energy and momentum d e n s i t i e s r e s p e c t i v e l y . York has j u s t i f i e d the s c a l i n g f a c t o r s used i n (2.73) and (2.74) t h r o u g h d i m e n s i o n a l c o n s i d e r a t i o n s , but perhaps more i m p o r t a n t l y , i f the c o n f o r m a l d a t a s a t i s f i e s t he dominant  energy c o n d i t i o n as g i v e n by Hawking and E l l i s , [ 2 0 ] , then so 37 w i l l the p h y s i c a l d a t a , r e g a r d l e s s of the e x a c t n a t u r e o f " ^ . The dominant energy c o n d i t i o n a s s e r t s t h a t the l o c a l 4-momentum d e n s i t y T M = T ^ r v w (2.75) i s a n o n - s p a c e l i k e v e c t o r . For the c o n f o r m a l s t r e s s - e n e r g y t e n s o r T ^ t h i s i m p l i e s r ^ T > - q . i - * O (2.76) or Thus i f £ and j are chosen t o s a t i s f y the above, then the s c a l i n g g i v e n by e x p r e s s i o n s (2.73) and (2.74) gua r a n t e e s t h a t p >/ < g^ r TJ> (2-78) The second major i d e a i n Y o r k ' s approach t o the i n i t i a l v a l u e problem i s t o decompose the t r a c e - f r e e p a r t of the c o n f o r m a l e x t r i n s i c c u r v a t u r e i n t o t r a n s v e r s e - t r a c e l e s s (TT) and l o n g i t u d i n a l p a r t s - a p r o c e s s which can be performed on any symmetric t r a c e f r e e t e n s o r [ 3 9 ] , T h i s p r o c e d u r e i s analagous t o what may be done i n the study of the i n i t i a l v a l u e problem f o r e l e c t r o d y n a m i c s . There, the e l e c t r i c f i e l d E 38 may be s p l i t i n t o t r a n s v e r s e , E T , and l o n g i t u d i n a l , E L , p a r t s as f o l l o w s E - EL * ET V • E T - o EL - v u ( 2 ' 7 9 ) where U i s a s c a l a r f u n c t i o n . The e q u a t i o n of c o n s t r a i n t f o r the e l e c t r i c f i e l d yOe (2.80) where i s the charge d e n s i t y becomes a second o r d e r e l l i p t i c e q u a t i o n f o r the f u n c t i o n U X7Z U « 4 T T / 0 E ( 2 . 8 1 ) which i s the s i n g l e i n i t i a l v a l u e e q u a t i o n f o r e l e c t r o d y n a m i c s . The t r a n s v e r s e f i e l d E-p may be f r e e l y s p e c i f i e d and r e p r e s e n t s the r a d i a t i v e degrees of freedom of the e l e c t r o m a g n e t i c f i e l d (see [41] f o r more d e t a i l s ) . In a s i m i l a r f a s h i o n , the t r a n s v e r s e -t r a c e l e s s / l o n g i t u d i n a l d e c o m p o s i t i o n of E L i s e p a r a t e s the f r e e l y s p e c i f i a b l e p a r t of E c i from the p a r t which w i l l be de t e r m i n e d from the c o n s t r a i n t e q u a t i o n s (2.47) and ( 2 . 4 9 ) . Thus, put (2.82) 39 where the l o n g i t u d i n a l p a r t , E ^  i s d e r i v e d from d i f f e r e n t i a t i o n of some v e c t o r WL as f o l l o w s ^ (2.83) T h i s g u a r a n t e e s t h a t E j 1 i s both symmetric and t r a c e f r e e Now from (2.82) - .. ^ (2.84) - E l J - ( x w ) l J The t r a c e f r e e c o n d i t i o n on E T 1 r i s s a t i s f i e d by c o n s t r u c t i o n E r i = g t J E L J - gij ( i W ) l ) = o and the t r a n s v e r s a l i t y r equirement Dj E r r = O (2.85) l e a d s t o the f o l l o w i n g e q u a t i o n f o r the v e c t o r WL Dj (X w ) 1 J = ( A * W) 1 = E V J (2.86) 40 The o p e r a t o r & k d e f i n e d above, i s c a l l e d the " v e c t o r L a p l a c i a n " . ' I t s e x p l i c i t form, u s i n g (2.83) i s ( ^ w ) 1 = D ^ T > 1 W J + f)- i>> W L - | D ; D K W K = i>i . D * wL +• ( D ^ D 1 - D : D J ) wJ + ti), I), wJ = A W -v- ^ D ' D i W J +- R Lj W J where a form of the R i c c i i d e n t i t y has a l s o been used. The p r o p e r t i e s of are examined i n depth i n [11] and [39] a l o n g w i t h c o n d i t i o n s n e c e s s a r y f o r e x i s t e n c e and uniqueness of a s o l u t i o n WL of e q u a t i o n ( 2 . 8 6 ) . The t r a n s v e r s e - t r a c e l e s s p a r t of E L J may a l s o be decomposed i n t o a f r e e l y s p e c i f i a b l e symmetric, t r a c e f r e e p a r t T L i and a l o n g i t u d i n a l p a r t -(£v) lJ . Thus w r i t e , E ^ = "T l"i - ( I \l ) M (2.88) A g a i n , the t r a n s v e r s a l i t y of E^ 4 demands t h a t ( A x V ) L = D j f " ^ (2.89) U s i n g e q u a t i o n s ( 2 . 8 2 ) , (2.83) and ( 2 . 8 8 ) , the t r a c e - f r e e p a r t of the c o n f o r m a l l y s c a l e d e x t r i n s i c c u r v a t u r e i s E.^ =" T L J - ( i v ) L"J. 1- (Je w (2.90) = T ^ * c i x ) l J where y ; s w L - v (2.9D 41 The i n i t i a l v a l u e e q u a t i o n s may now be w r i t t e n i n terms of the c o n f o r m a l l y s c a l e d d a t a and the f u n c t i o n s t and X 1. R e c a l l e q u a t i o n (2.47) "R. + ( T f K ) 1 - K ; i K M = \b7ip (2.47) U s i n g e q u a t i o n s ( 2 . 6 2 ) , ( 2 . 6 9 ) , ( 2 . 7 1 ) , (2.73) and ( 2 . 9 0 ) , t h i s becomes ^ . - 4 * ( f l"> + f i x ) ' J )( f ;j + CJ?X h\, ) - i ( T r k ) : or j + ^ A ; i j / - a, - 8 - ~ •• - , t » (2.92) The o t h e r t h r e e i n i t i a l v a l u e e q u a t i o n s , Dj K ) C ~ D L CTV K ) = 8TT (2.49) may be r e w r i t t e n u s i n g ( 2 . 6 2 ) , ( 2 . 7 2 ) , (2.74) and (2.90) ^ 4 5 L TV k 0 -n ^  - l ° or ( A ^ X ) L c ~ i , T C i + H ^ D ^ T r K + B T T J 1 (2.93) 42 In summary, the f r e e l y s p e c i f i a b l e q u a n t i t i e s i n t h i s approach t o the i n i t i a l v a l u e problem a r e 1) the base 3-metric g:^ 2) the symmetric t r a c e - f r e e t e n s o r T' J 3) the mean e x t r i n s i c c u r v a t u r e TrK, and 4) the c o n f o r m a l l y s c a l e d energy and momentum d e n s i t i e s p and ] *•. Once t h e s e q u a n t i t i e s have been s p e c i f i e d , e q u a t i o n s (2.92) and (2.93) a r e t o be s o l v e d f o r and X u. The p h y s i c a l i n i t i a l d a t a i s then d e t e r m i n e d u s i n g the f o l l o w i n g e x p r e s s i o n s <} • i - A 3 ; J P = f> (2.94) In g e n e r a l , (2.92) and (2.93) r e p r e s e n t a system of f o u r c o u p l e d q u a s i - l i n e a r ( l i n e a r i n the second d e r i v a t i v e s ) e l l i p t i c 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 . In the s p e c i a l case of c o n s t a n t mean e x t r i n s i c c u r v a r t u r e , however, (2.93) d e c o u p l e s from (2.92) and a l s o becomes l i n e a r . P h y s i c a l l y , the f o u r f u n c t i o n s N' and X 1 may be i n t e r p r e t e d as g e n e r a l i z a t i o n s of the s i n g l e p o t e n t i a l , ^ , of Newtonian g r a v i t a t i o n . I n t h i s i n t e r p r e t a t i o n , the i n i t i a l v a l u e e q u a t i o n s c o n s t i t u t e the r e l a t i v i s t i c analogue of the P o i s s o n e q u a t i o n which r e l a t e s the Newtonian p o t e n t i a l t o the mass d e n s i t y y O M V * <f 4 TT p (2.95) 43 The q u e s t i o n s of e x i s t e n c e and uniqueness of s o l u t i o n s of the i n i t i a l v a l u e e q u a t i o n s have been examined by s e v e r a l r e s e a r c h e r s . Here, t o p o l o g i c a l c o n s i d e r a t i o n s p l a y an im p o r t a n t r o l e s i n c e the c o n s t r a i n t e q u a t i o n s do not det e r m i n e the t o p o l g y of the i n i t i a l s l i c e . Y o r k ' s o r i g i n a l work on the TT- d e c o m p o s i t i o n of symmetric t e n s o r s was r e s t r i c t e d t o c l o s e d p o s i t i v e d e f i n i t e m a n i f o l d s . C o n d i t i o n s f o r e x i s t e n c e and uniqueness i n t h i s case a r e d i s c u s s e d i n depth i n r e f e r e n c e The o t h e r major t o p o l o g i c a l c l a s s of i n i t i a l d a t a which has been examined i s t h a t of a s y m p t o t i c a l l y f l a t s l i c e s . Roughly s p e a k i n g , a s y m p t o t i c f l a t n e s s means t h a t the i n i t i a l d a t a (g.j ,K L J ) approaches t h a t of a s t a n d a r d t = c o n s t a n t s l i c e of M i n k o w s k i i spacetime a t l a r g e d i s t a n c e s from an a r b i t r a r i l y chosen o r i g i n on the s l i c e . In a d d i t i o n , the approach i s assumed r a p i d enough t o ensure t h a t the t o t a l energy and momentum of the spacetime a r e w e l l d e f i n e d and c o n s e r v e d q u a n t i t i e s . D e n o t i n g a f l a t 3 -metric by t:\ , and l e t t i n g r be the d i s t a n c e from some o r i g i n w i t h r e s p e c t t o t:i , then the f o l l o w i n g i s a t y p i c a l d e f i n i t i o n of a s y m p t o t i c f l a t n e s s [40] [1 1 ] . S ;3 u ^ h ^ y - 0 ( 7 ) r -* 0 0 c -* 0 0 r-+ oO (2.96) In the case of a s y m p t o t i c a l l y f l a t i n i t i a l d a t a , i t can 44 be shown [29] t h a t the a s y m p t o t i c b e h a v i o u r of de t e r m i n e s the t o t a l energy of the s p a c e t i m e . For t h i s r e a s o n , e q u a t i o n (2.92) i s o f t e n c a l l e d the H a m i l t o n i a n c o n s t r a i n t . S i m i l a r l y the b e h a v i o r of the " v e c t o r p o t e n t i a l " , X 1 , a t l a r g e d i s t a n c e s measures the t o t a l momentum of the s p a c e t i m e , and e q u a t i o n (2.93) i s t h e r e f o r e termed the momentum c o n s t r a i n t . The c o n d i t i o n s f o r e x i s t e n c e and uniqueness of s o l u t i o n s of the c o n s t r a i n t s have not been d e t e r m i n e d as c o m p l e t e l y f o r a s y m p t o t i c a l l y f l a t s l i c e s as they have f o r c l o s e d s l i c e s . Some known r e s u l t s are summarized i n [ 1 1 ] . 2.6 Past work on i n i t i a l v a l u e problems T h i s s e c t i o n p r e s e n t s a b r i e f r e v i e w of some of the i n i t i a l v a l u e problems which have been s t u d i e d t o t h i s d a t e . A r a t h e r s p e c i a l case of the i n i t i a l v a l u e problem r e s u l t s when the i n i t i a l s l i c e i s assumed t o have v a n i s h i n g e x t r i n s i c c u r v a t u r e Such a s l i c e i s o f t e n c a l l e d t i m e - s y m m e t r i c . In t h i s c a s e , the momentum c o n s t r a i n t s a r e a u t o m a t i c a l l y s a t i s f i e d and the H a m i l t o n i a n c o n s t r a i n t (2.92) becomes I f a t t e n t i o n i s f u r t h e r l i m i t e d t o the vacuum c a s e , the above becomes -8 A t = - R t + 16 TT (2.97) (2.98) 45 B r i l l [9] s t u d i e d t h i s e q u a t i o n f o r a r a t h e r u n p h y s i c a l , but e x a c t l y s o l u b l e , case i n v o l v i n g an a x i a l l y symmetric, "square w e l l " s c a l a r c u r v a t u r e . The s o l u t i o n s d e t e r m i n e d i n d i c a t e d the presence of g r a v i t a t i o n a l r a d i a t i o n ("time-symmetric B r i l l waves") h a v i n g a t o t a l energy r o u g h l y p r o p o r t i o n a l t o the square of the a m p l i t u d e of t h e r a d i a t i o n . I n a d d i t i o n , i t was found t h a t the da t a d e s c r i b e d a b l a c k h o l e when the a m p l i t u d e was s u f f i c i e n t l y l a r g e . E p p l e y [14] s o l v e d a s i m i l a r problem n u m e r i c a l l y u s i n g a more p h y s i c a l c u r v a t u r e s c a l a r and v e r i f i e d the q u a l i t a t i v e f e a t u r e s of B r i l l ' s a n a l y s i s . He then extended h i s study t o the case of non-time-symmetric d a t a [ 1 5 ] , i n which two of the momentum c o n s t r a i n t s , as w e l l as the H a m i l t o n i a n c o n s t r a i n t , were s o l v e d n u m e r i c a l l y . M i s n e r [26] s t u d i e d a n other vacuum time-symmetric problem which was e x a c t l y s o l u b l e . He demanded t h a t the m e t r i c of the i n t e r i o r of the a s y m p t o t i c a l l y f l a t s l i c e be c o n f o r m a l l y r e l a t e d t o t h a t of a 3 - d i m e n s i o n a l "doughnut". The s o l u t i o n s M i s n e r c o n s t r u c t e d d e s c r i b e d s l i c e s h a v i n g m u l t i p l y c o n n e c t e d t o p o l o g i e s and were e v e n t u a l l y i n t e r p r e t e d as r e p r e s e n t i n g b l a c k h o l e s . Smarr e t a l . [35] l a t e r used t h i s i n i t i a l d a t a i n t h e i r work on the head-on c o l l i s i o n of two b l a c k h o l e s . More r e c e n t l y , Nakamura e t a l . [28] have s t u d i e d the i n i t i a l v a l u e problem f o r the case of an a x i a l l y symmetric c o l l a p s i n g s t a r , which r e q u i r e d the n u m e r i c a l s o l u t i o n of both the H a m i l t o n i a n c o n s t r a i n t and one of the momentum c o n s t r a i n t s . The r e s u l t i n g d a t a has a l s o been e v o l v e d . In a d d i t i o n , t h e r e have been numerous o t h e r s t u d i e s r e c e n t l y 46 whose c h i e f c oncern i s i n e v o l v i n g i n i t i a l d a t a n u m e r i c a l l y , r a t h e r than g e n e r a t i n g i t . A review a r t i c l e by P i r a n [32] summarizes the s t a t e of th e s e s t u d i e s as of 1981. A l l of the problems s t u d i e d t o t h i s d a t e have had symmetries which have a l l o w e d the i n i t i a l v a l u e unknowns t o be w r i t t e n as f u n c t i o n s o f , a t most, two s p a t i a l v a r i a b l e s . In a d d i t i o n , most of the problems which have been examined have d e a l t w i t h a s y m p t o t i c a l l y f l a t s l i c e s . 47 CHAPTER 3 A S p e c i f i c I n i t i a l V a l u e Problem Working w i t h i n the f o r m a l i s m d e s c r i b e d i n the p r e v i o u s c h a p t e r , Bowen and York [3] and P i r a n and York [41] have r e c e n t l y d e t e r m i n e d s o l u t i o n s t o e q u a t i o n s (2.92) and (2.93) which have been i n t e r p r e t e d as p r o v i d i n g i n i t i a l d a t a f o r two new f a m i l i e s of b l a c k h o l e s . Bowen and York hope t h a t t h e i r method of a t t a c k i n g the problem w i l l e v e n t u a l l y prove u s e f u l i n p r o d u c i n g i n i t i a l d a t a f o r a non-head-on c o l l i s i o n of two or more b l a c k h o l e s - a problem which would appear t o be of s i g n i f i c a n t i n t e r e s t i n the c o n t e x t of the p r o d u c t i o n of g r a v i t a t i o n a l r a d i a t i o n . Due t o the i n t r o d u c t i o n of s e v e r a l s i m p l i f y i n g assumptions r e g a r d i n g the n a t u r e of the i n i t i a l s l i c e s , Bowen and York were a b l e t o f i n d a n a l y t i c s o l u t i o n s t o the momentum c o n s t r a i n t e q u a t i o n s ( 2 . 9 3 ) . E x a c t s o l u t i o n s o f ' the H a m i l t o n i a n c o n s t r a i n t c o u l d not be found, so n u m e r i c a l s o l u t i o n s were det e r m i n e d by P i r a n . The main g o a l of the c u r r e n t work was t o r e - s o l v e the H a m i l t o n i a n c o n s t r a i n t n u m e r i c a l l y , both t o check P i r a n ' s r e s u l t s and t o i n v e s t i g a t e a n u m e r i c a l t e c h n i q u e which may prove u s e f u l i n the s o l u t i o n of more c o m p l i c a t e d i n i t i a l v a l u e problems. T h i s c h a p t e r f i r s t r e v i e w s the work done by Bowen and York i n [3] ( S e c t i o n s 3.1 -3.2). Because the emphasis of t h i s t h e s i s i s on n u m e r i c a l 48 methods, no attempt has been made by the a u t h o r t o reproduce t h e i r a n a l y t i c r e s u l t s . S e c t i o n 3.3 then d e s c r i b e s how the H a m i l t o n i a n c o n s t r a i n t e q u a t i o n s f o r the new g e o m e t r i e s a r e posed as d i s c r e t e problems which may be s o l v e d n u m e r i c a l l y . 3.1.1 Topology of the s l i c e s and s i m p l i f y i n g a s sumptions As mentioned i n the p r e v i o u s c h a p t e r , t h e r e a r e two major t o p o l o g i c a l c l a s s e s of s p a c e l i k e h y p e r s u r f a c e s which have r e c e i v e d the most a t t e n t i o n i n the c o n t e x t of the i n i t i a l v a l u e problem - c l o s e d s l i c e s and a s y m p t o t i c a l l y f l a t s l i c e s . In the p r e s e n t c a s e , the h y p e r s u r f a c e s c o n s t r u c t e d a r e a s y m p t o t i c a l l y f l a t . T h i s a l l o w s q u a n t i t i e s such as the t o t a l e n ergy, l i n e a r momentum and a n g u l a r momentum of the s p a c e t i m e , as measured a t s p a c e l i k e i n f i n i t y , t o be d e f i n e d . In a d d i t i o n , the requirement of a s y m p t o t i c f l a t n e s s p r o v i d e s a boundary c o n d i t i o n f o r the c o n f o r m a l f a c t o r i n the s o l u t i o n of the H a m i l t o n i a n c o n s t r a i n t , as w i l l be seen below. To s i m p l i f y the a n a l y s i s of the i n i t i a l v a l u e e q u a t i o n s , Bowen and York r e s t r i c t e d a t t e n t i o n t o maximal volume, c o n f o r m a l l y f l a t , vacuum s l i c e s . R e c a l l t h a t a maximal volume h y p e r s u r f a c e has v a n i s h i n g mean e x t r i n s i c c u r v a t u r e T V K = 0 ( 3 - D Conformal f l a t n e s s means t h a t the p h y s i c a l m e t r i c , g ^ , of the h y p e r s u r f a c e may be w r i t t e n as q M = t 4 T.'j (3.2) 49 where f L j i s the u s u a l 3 - d i m e n s i o n a l E u c l i d e a n m e t r i c . Note t h a t t h i s i m m e d i a t e l y i m p l i e s t h a t the c o n f o r m a l l y s c a l e d c u r v a t u r e s c a l a r , R, i d e n t i c a l l y v a n i s h e s . F i n a l l y , t h e vacuum req u i r e m e n t s i m p l y means t h a t t h e r e a r e no mat t e r or momentum s o u r c e s on the h y p e r s u r f a c e - 8 P 0 (3.3) -\ o 0 (3.4) 3.1.2 S o l u t i o n of the momentum c o n s t r a i n t W i t h the above r e s t r i c t i o n s , the momentum c o n s t r a i n t s i m p l i f i e s t o C 2^ X ) ' - - 3i; T ; i (3.5) T h i s e q u a t i o n may be f u r t h e r s i m p l i f i e d by demanding t h a t T , which may be f r e e l y s p e c i f i e d , a l s o v a n i s h . Together w i t h the v a n i s h i n g of TrK, t h i s i m p l i e s t h a t the c o n f o r m a l l y s c a l e d e x t r i n s i c c u r v a t u r e , K l J and the p h y s i c a l e x t r i n s i c c u r v a t u r e a r e b o t h p u r e l y l o n g i t u d i n a l K l i ( i x ) l J (3.6) 50 K ' S • - + - ° K ' » (3 .7 ) E q u a t i o n (3.5) becomes (A,X) l= P j ( I X ) ' 1 - ^ l < " , 0 (3.8) U s i n g (2.87) w i t h R=0, and d e n o t i n g the f l a t - s p a c e L a p l a c i a n by Vz> the e x p l i c i t form of the above e q u a t i o n i s (A^XV = V 2 X L + -5 P 1 ^ X J * 0 ( 3 . 9 ) Bowen and York d e t e r m i n e d s o l u t i o n s t o t h i s e q u a t i o n by w r i t i n g X L as X l - - V - t D l X (3 .10) They then s o l v e d s u c c e s s i v e l y t h e e q u a t i o n s v z v 1 -- 0 - ; (3 .11) v 2 ^ D( v The r e a d e r may e a s i l y v e r i f y t h a t such a pr o c e d u r e y i e l d s v e c t o r s X which s a t i s f y ( 3 . 9 ) . Of the many p o s s i b l e K L J ' S t h a t c o u l d be c o n s t r u c t e d i n t h i s f a s h i o n , t he f o l l o w i n g were chosen T i 7 4 [ P ; n J ^ P ^ t ^ ^ j - S n i o ^ P ' o K ] (3 .12a) 51 r * t K i jg. J n i n j + £ K J A A ; (3.12b) Here, r i s the E u c l i d e a n d i s t a n c e from some a r b i t r a r y p o i n t of the h y p e r s u r f a c e , n 1 i s the u n i t normal of a r = c o n s t a n t 2-sp h e r e , P 1 and J 1 a r e c o n s t a n t v e c t o r s , i ^  v« i s t h e p e r m u t a t i o n t e n s o r and a i s a c o n s t a n t . These p a r t i c u l a r s o l u t i o n s were chosen f o r two main reasons which w i l l be d i s c u s s e d below. 3.2 The H a m i l t o n i a n c o n s t r a i n t and i n v e r s i o n t e c h n i q u e s The H a m i l t o n i a n c o n s t r a i n t ( 2 . 9 2 ) , i n the p r e s e n t c a s e , where TrK= r=R=0, i s 0 or v 2 ~ t * £ ^ 8 ^ 7 0 (3.13) where K i j w i l l be d e t e r m i n e d by (3.12a) or (3.12b). The requi r e m e n t t h a t a c o n f o r m a l l y f l a t h y p e r s u r f a c e be a s y m p t o t i c a l l y f l a t t r a n s l a t e s i n t o t h e f o l l o w i n g boundary c o n d i t i o n on ^ r A N- = 1 + 0 / 1 \ , ^oo I r ) (3.14) 52 As i t s t a n d s , the problem of d e t e r m i n i n g a u n i q u e , p o s i t i v e which s a t i s f i e s (3.13) everywhere on the h y p e r s u r f a c e , s u b j e c t t o the boundary c o n d i t i o n (3.14) i s not w e l l - p o s e d s i n c e the s c a l e d e x t r i n s i c c u r v a t u r e s of (3.12a) and (3.12b) d i v e r g e a t r=0. To c i r c u m v e n t t h i s problem, Bowen and York f u r t h e r r e s t r i c t e d a t t e n t i o n t o h y p e r s u r f a c e s which a r e i s o m e t r i c w i t h r e s p e c t t o a mapping t h r o u g h a 2-sphere of r a d i u s a. I n t r o d u c i n g the u s u a l s p h e r i c a l p o l a r c o o r d i n a t e s ( r , 6 , c / ) on the s l i c e , the mapping has the e x p l i c i t form (3.15) r or i n C a r t e s i a n c o o r d i n a t e s ( x , y , z ) z X - z = (3.16) T h i s t r a n s f o r m a t i o n i s d e f i n e d f o r x' 6 R 3-{0} and maps the r e g i o n 0<r<a onto r>a and v i c e v e r s a . Now a map x'c Cx l ) (3.17) i s an i s o m e t r y of a m e t r i c g.'i i f (3.18) In the p r e s e n t c a s e , the p h y s i c a l m e t r i c g M i s c o n f o r m a l l y f l a t . I f x *" a r e C a r t e s i a n c o o r d i n a t e s , then c ^ C x ' ) - C x 1 ) Si] (3.19) 53 and f o r the t r a n s f o r m a t i o n of (3.16) c>x l . g \ * (3.20) Thus (3.16) i s an i s o m e t r y of g;j i f q i j U O - ( S"; £ ) ( 6-J £ ) q . . Cx.'1) or 4 which i m m e d i a t e l y i m p l i e s D i f f e r e n t i a t i o n of the l a s t e x p r e s s i o n w i t h r e s p e c t t o r g i v e s 2 S - t x c ) t _ a C x."') +- a 9_r' ^ t ^ l l > CL „ , , / / , D^Cx'O (3.22) E v a l u a t i n g t h i s r e s u l t when r ( x l ) = r ' ( x ' L )=a y i e l d s 3r 2- r 0 (3.23) r.-a M a t h e m a t i c a l l y , t h i s e q u a t i o n p r o v i d e s a second boundary c o n d i t i o n f o r the c o n f o r m a l f a c t o r which a l l o w s one t o a v o i d h a v i n g t o d e a l : w i t h the s i n g u l a r b e h a v i o r of the s c a l e d e x t r i n s i c c u r v a t u r e a t the o r i g i n . G e o m e t r i c a l l y , the p r o c e s s of i m p o s i n g an i s o m e t r y c o n d i t i o n on t h e m e t r i c of t h e s l i c e i n t he above f a s h i o n l e a d s t o t h e i d e n t i f i c a t i o n of the 54 two r e g i o n s 0<r<a ; r>a as two s e p a r a t e , but i d e n t i c a l , a s y m p t o t i c a l l y f l a t s h e e t s , j o i n e d i n a smooth f a s h i o n a t r=a. Such a d e s c r i p t i o n i s o f t e n used t o c h a r a c t e r i z e t h e geometry of the time-symmetric ( K l i = 0 ) h y p e r s u r f a c e s of the S c h w a r z s c h i l d spacetime (see f o r example [ 2 7 ] , p.836) In t h i s c a s e , the " t h r o a t " which c o n n e c t s the two a s y m p t o t i c a l l y f l a t s h e e t s c o i n c i d e s w i t h the apparent h o r i z o n [20] of the S c h w a r z s c h i l d b l a c k h o l e . Bowen and York c o n c l u d e d t h a t the s l i c e s t hey were c o n s t r u c t i n g c o u l d a l s o be i n t e r p r e t e d as h y p e r s u r f a c e s of s p a c e t i m e s c o n t a i n i n g b l a c k h o l e s . S p e c i f i c a l l y , i t was argued t h a t a s l i c e h a v i n g the s c a l e d e x t r i n s i c c u r v a t u r e g i v e n by (3.12a) would r e p r e s e n t a "snapshot" of a b o o s t e d b l a c k h o l e h a v i n g l i n e a r momentum P L as measured by an o b s e r v e r a t s p a c e l i k e i n f i n i t y , and a h y p e r s u r f a c e w i t h the K i ' j of (3.12b) would c o r r e s p o n d t o a s p i n n i n g b l a c k h o l e h a v i n g i n t r i n s i c a n g u l a r momentum J c . The a b i l i t y t o i n t e r p r e t the r e s u l t s i n t h i s f a s h i o n p r o v i d e d p a r t of the m o t i v a t i o n f o r c h o o s i n g the p a r t i c u l a r s o l u t i o n s of the momentum c o n s t r a i n t s . In a d d i t i o n , i t was shown t h a t the K c j 's of (3.12) have the p r o p e r t y t h a t i f a c o n f o r m a l f a c t o r ^(r,e,9*) i s found which s a t i s f i e s (3.13) i n the r e g i o n r ^ a s u b j e c t t o the boundary c o n d i t i o n s ( 3 . 1 4 ) , and ( 3 . 2 3 ) , then the f u n c t i o n a u t o m a t i c a l l y s a t i s f i e s (3.13) i n t h e r e g i o n 0<r^a. That i s , the K i j 's of (3.12a) and (3.12b) were c o n s t r u c t e d so t h a t the p h y s i c a l e x t r i n s i c c u r v a t u r e would a l s o be i n v a r i a n t under the 55 i n v e r s i o n t r a n s f o r m a t i o n . ( A c t u a l l y , s i n c e the H a m i l t o n i a n c o n s t r a i n t i s q u a d r a t i c i n K.j , a change of s i g n i n Kcj under i n v e r s i o n i s a l l o w e d . The " + " s o l u t i o n g i v e n by (3.12a) t r a n s f o r m s w i t h o u t change i n s i g n ; t h e "-" s o l u t i o n t r a n s f o r m s w i t h a s i g n change. Bowen and York i n t e r p r e t the l a t t e r case as i n d i c a t i n g t h a t a momentum of - P L i s a s s o c i a t e d w i t h the "bottom" sheet of the s l i c e ( 0 < r ^ a ) , i f momentum +P 1 i s a s s o c i a t e d w i t h the " t o p " sheet ( r > a ) ) . 3.3 D i s c r e t i z a t i o n of the H a m i l t o n i a n c o n s t r a i n t W i t h t h e i n v e r s i o n t e c h n i q u e d e s c r i b e d i n the p r e v i o u s s e c t i o n , the problem of s o l v i n g the H a m i l t o n i a n c o n s t r a i n t becomes the .problem of s o l v i n g the f o l l o w i n g boundary v a l u e problem ,2 ~ ~ ' ^ t f K M * L J - O 8 t ? (3.13) JL\n -^ * \ (3.14) r ~> co V 4 * + S ; •3f 2.r O r r a . (3.23) w i t h K g i v e n by (3.12a) o r ' (3.12b). The purpose of t h i s s e c t i o n i s t o d e s c r i b e t h e manner i n w h i c h t h i s problem i s c o n v e r t e d t o a form s u i t a b l e f o r s o l u t i o n by n u m e r i c a l means. An e x a m i n a t i o n of the e x p l i c i t forms of (3.12a) and (3.12b) r e v e a l s t h a t the a s s o c i a t e d boundary v a l u e problems w i l l a l l have symmetries which can be e x p l o i t e d i n t h e i r 56 subsequent n u m e r i c a l s o l u t i o n . S p h e r i c a l c o o r d i n a t e s (r,6,0), w i t h the u s u a l r e l a t i o n t o C a r t e s i a n c o o r d i n a t e s ( x , y , z ) w i l l be employed. The ( f l a t ) m e t r i c i n s p h e r i c a l c o o r d i n a t e s i s (3.24) In t h e s e c o o r d i n a t e s , a u n i t normal, n 1 t o a 2-sphere c e n t r e d at r=0 has components ( 1 , 0 , 0 ) . C o n s i d e r e x p r e s s i o n (3.12a) f i r s t and t a k e P l i n the +z d i r e c t i o n . Thus P c P c o ^ e , - r o ) (3.25) Then, the o n l y n o n - v a n i s h i n g components of K 7 j a r e K r r K & e 3PcO50 - 3) Pco^ e 1 I ex. 1 t 2 , K 4-K " e r D e n o t i n g KV A K" k J by HB" (B -> " b o o s t e d " ) , then K r r <- r' k r e + r 4 '^ee + 2 £ Z r* (1 (3.26) S i m i l a r l y , t a k i n g J i n the +z d i r e c t i o n , the o n l y non-57 v a n i s h i n g component of e x p r e s s i o n (3.12b) i s r so H 5 (S -> " s p i n n i n g " ) i s r - U = _1 \Ccl = 18 T3- & i ^ e ' (3.27) S i n c e n e i t h e r Hg or H«, has any g$ dependence, the s o l u t i o n s of (3.13) w i l l be a x i a l l y symmetric (symmetric about the z - a x i s ) , as s h o u l d be e x p e c t e d , s i n c e the v e c t o r s P and J were taken a l o n g the z - a x i s . In a d d i t i o n , s i n c e c o s 2 ( 0 ) = c o s 2 ( * - & ) and s i n 2 ( 6 ) = s i n 2 ( n - 6) the s o l u t i o n s w i l l a l s o p o s s e s s a r e f l e c t i o n symmetry about the 0= H a x i s . Thus, i t w i l l s u f f i c e t o s o l v e (3.13) on a domain such as t h a t shown i n F i g u r e 3. The symmetries p r o v i d e the f o l l o w i n g a d d i t i o n a l boundary c o n d i t i o n s f o r -2> e & = o (3.28) 0 (3.29) The unboundedness of the domain p r e s e n t s a problem i n the s o l u t i o n of (3.13) by n u m e r i c a l means, s i n c e a c o m p u t a t i o n a l domain must be f i n i t e . York and P i r a n h a n d l e d t h i s d i f f i c u l t y by u s i n g the known a s y m p t o t i c b e h a v i o r of t o produce an approximate boundary c o n d i t i o n which c o u l d be imposed a t a f i n i t e r a d i u s . T h i s scheme was o r i g i n a l l y used i n the p r e s e n t work u n t i l i t was s u g g e s t e d by Gus Gassmann t h a t the i n f i n i t e Domain of F i g u r e Boundary 3 V a l u e Problem 59 d o m a i n be t r a n s f o r m e d t o a f i n i t e d o m a i n i n t h e f o l l o w i n g f a s h i o n . I n t r o d u c e a new r a d i a l c o o r d i n a t e s , d e f i n e d b y S = 1 - a ( 3 . 3 0 ) r T h i s t a k e s t h e i n f i n i t e d o m a i n { P ( r , & ) | &'<v<<x> • 0<B<;L} i n t o t h e b o u n d e d d o m a i n { P ( s , 6 ) | 0<s^:1 ; O^e^j}. The u s e o f t h i s c o o r d i n a t e c h a n g e p r o v e d t o be q u i t e a d v a n t a g e o u s n u m e r i c a l l y . The L a p l a c i a n o f i n (s,e ) c o o r d i n a t e s i s v ^ = ~~dF ^ - + c ^ r e - ^ ( ( 3 . 3 1 ) T h i s e x p r e s s i o n i s s i n g u l a r when s=1 o r when & = 0 . The b e h a v i o r a t s=1 p r e s e n t s n o p r o b l e m n u m e r i c a l l y , s i n c e a s w i l l be s e e n b e l o w , t h e d i s c r e t e a p p r o x i m a t i o n o f t h e o p e r a t o r d o e s n o t h a v e t o b e e v a l u a t e d a t s = 1 . H o w e v e r , t h e d i s c r e t e f o r m o f Sir, & 20 I -7>6 m u s t be e v a l u a t e d a t & = 0 . T h u s , a t 9=0 t h e a b o v e t e r m i s r e p l a c e d by t h e f o l l o w i n g e x p r e s s i o n w h i c h r e s u l t s f r o m a n a p p l i c a t i o n o f L ' H o p i t a l ' s r u l e a n d t h e b o u n d a r y c o n d i t i o n ( 3 . 2 8 ) & -*• o -a© V 1 = Z , ( 3 . 3 2 ) The b o u n d a r y c o n d i t i o n s ( 3 . 2 8 ) a n d ( 3 . 2 9 ) a r e u n c h a n g e d b y t h e 60 i n t r o d u c t i o n of the s c o o r d i n a t e . The i n n e r boundary c o n d i t i o n (3.23) becomes 3& 1 2. (3.33) and the o u t e r boundary c o n d i t i o n (3.14) i s me r e l y y | 6 r , - 1 (3.34) To s o l v e the boundary v a l u e problem n u m e r i c a l l y , the well-known t e c h n i q u e s of f i n i t e d i f f e r e n c i n g a r e used. (The re a d e r who i s u n f a m i l i a r w i t h t h i s t o p i c i s r e f e r r e d t o Appendix A and the r e f e r e n c e s g i v e n t h e r e i n . ) The c o n t i n u o u s domain i s r e p l a c e d by a u n i f o r m g r i d as shown i n F i g u r e 4. The p o i n t s marked w i t h a 'x' i n the diagram, which comprise the s e t | ( s;, e 3 ) I s; -- i A S , L « O. 1, • • •, n s - 1 ; 0i - s A G , j = o,^, • - • , n e | n e = 71 • ( Z A £> ) ~ 1 a r e t h e p o i n t s a t which a d i f f e r e n c e analogue of the i n t e r i o r e q u a t i o n (3.13) w i l l be s a t i s f i e d . The p o i n t s marked w i t h a [ ( S - , 0 j ) i j = 0,1 • •• , A e ] U [( s i , G ; i -• o, \ • , m s-l ] u 61 e cp a ffl m B Si e 9 TV a — ^ * * * * * * 5k d * * * * * * # * T '-±-1-* — * — * — # — * — * — * — * — * O - # * * * * * * * i—*—*—*—*—*—*—* Q a a a B a s a F i g u r e 4 U n i f o r m G r i d f o r D i s c r e t i z a t i o n of Boundary V a l u e Problem 62 which l i e o u t s i d e of the c o n t i n u o u s domain a r e i n t r o d u c e d so t h a t c e n t r e d d i f f e r e n c e a p p r o x i m a t i o n s may be employed f o r the boundary c o n d i t i o n s as w e l l as f o r the i n t e r i o r e q u a t i o n a t g r i d p o i n t s l y i n g on the a c t u a l boundary of the domain. U s i n g e x p r e s s i o n s (3.31) and ( 3 . 3 2 ) , and r e p l a c i n g a l l d e r i v a t i v e s by second o r d e r c e n t r e d d i f f e r e n c e s , the d i s c r e t e v e r s i o n of (3.13) i s 4 . M - S i ^ a 1 A S * H i , J O = 1,2-, ' • , ^ <? (3.35) The boundary e q u a t i o n s ( 3 . 3 3 ) , ( 3 . 2 8 ) , and (3.29) a r e d i s c r e t i z e d i n the same f a s h i o n 2 A S 5L (3.36) t L, i - t , - , - 1 Z AG 0 O,-,^-! (3.37) i i = D, ••• j As -1 (3.38) 63 F i n a l l y , (3.34) s i m p l y becomes A 3 , j 1 (3.39) These l a s t f i v e e x p r e s s i o n s d e f i n e a system of n o n - l i n e a r a l g e b r a i c e q u a t i o n s i n the unknowns ^ i . j , which when s o l v e d , p r o v i d e a p p r o x i m a t i o n s t o the v a l u e s of the t r u e s o l u t i o n t a t the d i s c r e t e g r i d p o i n t s . There i s no theorem r e g a r d i n g the uniqueness of a s o l u t i o n t o such a n o n - l i n e a r system. E x i s t e n c e i s demonstrated by a c t u a l l y s o l v i n g the system ( a p p r o x i m a t e l y ) n u m e r i c a l l y . Some of the more well-known t e c h n i q u e s f o r s o l v i n g n o n - l i n e a r systems of d i f f e r e n c e e q u a t i o n s a r e d e s c r i b e d i n s e c t i o n A.4 of Appendix A. The next c h a p t e r d e t a i l s the r e l a t i v e l y new method which was used t o s o l v e the above problems. A q u a n t i t y of p a r t i c u l a r p h y s i c a l i n t e r e s t which may be c a l c u l a t e d once ^ has been ( a p p r o x i m a t e l y ) d e t e r m i n e d i s the t o t a l energy of the a s y m p t o t i c a l l y f l a t s p a c e t i m e [29] d e f i n e d where the i n t e g r a t i o n i s performed over a 2-sphere a t s p a c e l i k e i n f i n i t y . A p p l y i n g Gauss's theorem i n the r e g i o n r>a, the above becomes by (3.40) v2^ ao 64 U s i n g e q u a t i o n s (3.13) and (3.23) and the f a c t t h a t H has no ^ dependence, t h i s becomes b - ^ J ^ - - J ^ s i o G ae ( 3 . 4 D In p r a c t i c e , b o th of t h e s e i n t e g r a l s must be e v a l u a t e d by n u m e r i c a l means and as d i s c u s s e d i n Chapter 5, do n o t , i n g e n e r a l , y i e l d i d e n t i c a l n u m e r i c a l r e s u l t s . One f i n a l r e s u l t of Bowen and Y o r k ' s a n a l y s i s was v e r y u s e f u l i n t e s t i n g and comparing the n u m e r i c a l p r o c e d u r e s used i n t h i s t h e s i s . They c o n s i d e r e d t h e f o l l o w i n g "model" H, s i m i l a r i n form t o (3.26) but h a v i n g no a n g u l a r dependence - ( 1 - t )^ <3.42» They were then a b l e t o produce the f o l l o w i n g e x a c t s o l u t i o n of the boundary v a l u e problem g i v e n by ( 3 . 1 3 ) , (3.14) and (3.23) TrAODEL = ( 1 + ~ + — + ~T * ~A > (3.43) <~ v r » where the t o t a l energy E, d e t e r m i n e d by (3.40) or (3.41) i s E = ( P 2 + 4 a 2 ) ' (3.44) 65 CHAPTER 4 The M u l t i - g r i d Method T h i s c h a p t e r d e s c r i b e s a g e n e r a l n u m e r i c a l t e c h n i q u e known as the m u l t i - g r i d method, which may be used t o s o l v e boundary v a l u e problems such as the one f o r m u l a t e d i n the p r e v i o u s c h a p t e r . The method can a l s o be a p p l i e d t o o t h e r c o n t i n u o u s problems such as f u n c t i o n a l i n t e g r a l s or m i n i m i z a t i o n problems, but t h e s e a s p e c t s w i l l not be d i s c u s s e d h e r e . S e c t i o n 4.1 i n t r o d u c e s t h e b a s i c p r o c e s s e s i n v o l v e d i n any m u l t i - g r i d a l g o r i t h m . A d e s c r i p t i o n of the p a r t i c u l a r a l g o r i t h m which was implemented f o r the purpose of s o l v i n g t h e problem d e s c r i b e d i n Chapter 3 f o l l o w s i n s e c t i o n s 4.2 - 4.7. (A l i s t i n g of the a c t u a l program i s c o n t a i n e d i n Appendix B, a l o n g w i t h d e t a i l s of the i m p l e m e n t a t i o n . ) The m a j o r i t y of the m a t e r i a l of t h i s c h a p t e r i s o r i g i n a l l y due t o Brandt and v a r i o u s c o l l a b o r a t o r s [ 4 ] , [ 5 ] , [ 6 ] , [8] and t h e p r e s e n t a t i o n c l o s e l y p a r a l l e l s t h a t of the above r e f e r e n c e s . 4.1 B a s i c m u l t i - g r i d p r o c e s s e s C o n s i d e r the f o l l o w i n g g e n e r a l t w o - d i m e n s i o n a l boundary v a l u e problem L t u ( x ) \ ~- K x ) • x € J l c (4.1a) 66 B x u C O * = c^Cx) ; x e ^ J l (4.1b) where L and B a r e d i f f e r e n t i a l o p e r a t o r s c o r r e s p o n d i n g t o the i n t e r i o r and boundary e q u a t i o n s f o r u ( x ) . Assume t h a t t h i s p roblem has been d i s c r e t i z e d i n some p r e s c r i b e d f a s h i o n u s i n g f i n i t e d i f f e r e n c e t e c h n i q u e s (see Appendix A) r e s u l t i n g i n the d i s c r e t e system The g o a l i s t o s o l v e t h i s d i s c r e t e system f o r the f u n c t i o n u h which i s an a p p r o x i m a t i o n of u. For the sake of s i m p l i c i t y , i t w i l l be assumed here t h a t the f i n i t e domain Si i s a r e c t a n g u l a r g r i d , G , h a v i n g a u n i f o r m s p a c i n g , h, i n both d i r e c t i o n s . In a d d i t i o n , because the m u l t i - g r i d method c o n s i d e r s the d i f f e r e n c e d i n t e r i o r and boundary e q u a t i o n s as s e p a r a t e systems t o be s o l v e d u s i n g the same t e c h n i q u e , o n l y the s o l u t i o n of the i n t e r i o r e q u a t i o n s w i l l be d e s c r i b e d e x p l i c i t l y . Suppose the system (4.2) i s t o be s o l v e d i t e r a t i v e l y , w hich w i l l always be the case i f L ( o r B ) i s n o n - l i n e a r . Then a good i n i t i a l e s t i m a t e of u^ i s h i g h l y d e s i r a b l e . One r a t h e r o b v i o u s way of g e n e r a t i n g such an e s t i m a t e i s t o s o l v e the problem l _ H u H » * H on G H (4.3) where G H i s a c o a r s e r g r i d , h a v i n g , f o r example, a mesh 67 s p a c i n g H=2h. T h i s problem w i l l i n v o l v e fewer e q u a t i o n s than (4.2) and t h e r e f o r e s h o u l d r e q u i r e l e s s work t o s o l v e . Once u H has been d e t e r m i n e d , then ~ w . _ - r r w , H (4.4) may be used as the i n i t i a l e s t i m a t e of u U . (The o p e r a t o r := means " i s a s s i g n e d the v a l u e " . ) Here JLH i s an o p e r a t o r which i n t e r p o l a t e s a f i n e g r i d f u n c t i o n from a c o a r s e g r i d f u n c t i o n . The l o g i c a l e x t e n s i o n of t h i s i d e a i s t o s o l v e d i s c r e t i z e d v e r s i o n s of (4.1) on an e n t i r e sequence of s u c c e s s i v e l y f i n e r g r i d s . W h i l e t h i s t e c h n i q u e i s a p a r t of the m u l t i - g r i d s o l u t i o n p r o c e s s , i t i s commonly used i n c o n j u n c t i o n w i t h many o t h e r schemes f o r the s o l u t i o n of d i f f e r e n c e e q u a t i o n s . However, i n the m u l t i - g r i d method, the c o a r s e g r i d s p l a y a second i m p o r t a n t r o l e which w i l l now be d e s c r i b e d . Assume f o r the time b e i n g t h a t L h i s l i n e a r and t h a t an e s t i m a t e u h of the f i n e - g r i d unknown has been d e t e r m i n e d . Then the r e s i d u a l v e c t o r , r " , of the system i s g i v e n by w _ , ^ ~ ^ f w (4.5) and (4.2a) i s s o l v e d by f i n d i n g the c o r r e c t i o n , v ^ , which s a t i s f i e s L W V K - ~ r U (4.6a) - u > + v K (4.6b) Of c o u r s e , n o t h i n g s i g n i f i c a n t has been a c c o m p l i s h e d i n r e w r i t i n g t h e problem i n t h i s form - (4.6a) i s j u s t as 68 d i f f i c u l t t o s o l v e as ( 4 . 2 a ) . However, i f t h e r e was some way of r e p r e s e n t i n g (4.6a) on a c o a r s e r g r i d G , then an e s t i m a t e of v w c o u l d be o b t a i n e d i n a f a s h i o n a n a l a g o u s t o the d e t e r m i n a t i o n of u K from the s o l u t i o n of the c o a r s e g r i d system ( 4 . 3 ) . In g e n e r a l , (4.6a) can not be r e p r e s e n t e d a c c u r a t e l y on a c o a r s e r g r i d s i n c e r K may be h i g h l y o s c i l l a t o r y from p o i n t t o p o i n t on G ; t h a t i s , r may have h i g h f r e q u e n c y components t h a t c o u l d not be r e p r o d u c e d on a c o a r s e r g r i d . In f a c t , t h i s i s t y p i c a l l y the case when the f i n e g r i d s o l u t i o n e s t i m a t e i s g e n e r a t e d from i n t e r p o l a t i o n of a c o a r s e g r i d s o l u t i o n . T h e r e f o r e , b e f o r e (4.6a) can be r e p r e s e n t e d on G H, the r e s i d u a l must be smoothed i n some f a s h i o n , so as t o remove the h i g h f r e q u e n c y components. F o r t u n a t e l y , f o r most problems, t h i s smoothing may be a c c o m p l i s h e d u s i n g well-known r e l a x a t i o n t e c h n i q u e s . (See Appendix A, s e c t i o n A.3.1) A s i m p l e example b e s t i l l u s t r a t e s t h i s . C o n s i d e r the f o l l o w i n g o n e - d i m e n s i o n a l boundary v a l u e problem c>-g. J ( x ) ; ( o, i ) (4.7a) u ( o l = a ( ! ) = O (4.7b) D i s c r e t i z e t h i s problem by i n t r o d u c i n g the u n i f o r m g r i d G G - i x L ) x; -- L h ; i = 0, 1 , • • , w s (4.8) D e n o t i n g u ( x t ) by u w the second d e r i v a t i v e i s r e p l a c e d by the 69 second-order c e n t r e d d i f f e r e n c e a p p r o x i m a t i o n y i e l d i n g the s e t of d i f f e r e n c e e q u a t i o n s -a; v i t 2 u ; - a i - i - $ • . t - 1, • • , M - 1 (4.9) which may be w r i t t e n i n the form | * K r W (4.10) Here L h i s an N-1 x N-1 t r i d i a g o n a l m a t r i x ; u h and f H a r e (N-l)-component v e c t o r s . C o n s i d e r the s o l u t i o n of the above system by the f o l l o w i n g i t e r a t i v e scheme. Le t u be the a p p r o x i m a t i o n t o u a t t h e k - t h i t e r a t i o n . Then U- - e x - <x> P ( L. <x - * ; LA. - co P r C> . ^ .co <4'11> Here, D i s the main d i a g o n a l of L , so - \S H ( 4 - 1 2 ) co i s a parameter which w i l l be chosen from the i n t e r v a l [ 0 , 1 ] . T h i s method i s c a l l e d a damped J a c o b i i t e r a t i o n . I f 60 = 1, f o r example, then t h i s i t e r a t i o n i n v o l v e s making a sweep th r o u g h t h e p o i n t s of G , a l t e r i n g the v a l u e of each g r i d - f u n c t i o n v a l u e so t h a t the l o c a l d i f f e r e n c e e q u a t i o n i s s a t i s f i e d u s i n g n e i g h b o r i n g f u n c t i o n v a l u e s from the p r e v i o u s i t e r a t i o n . Now examine th e e f f e c t of t h i s i t e r a t i o n on the r e s i d u a l v e c t o r 70 C K v O I ^  I K * O C W G r ( ° (4.13) wi t h G = 1 - c o L W P ^ T N -1 (4.14) C l e a r l y C*) Q * r Co) (4.15) r where G i s t h e k-th power of the m a t r i x G. Now, G p o s s e s s e s a complete s e t of o r t h o n o r m a l e i g e n v e c t o r s , m=1,...,N-l w i t h •A. /• \ c o r r e s p o n d i n g e i g e n v a l u e s . Thus r may be e x p r e s s e d as NJ - 1 r ( o ) * Z c~ ^ (4.16) where t h e c w a r e c o e f f i c i e n t s . I t f o l l o w s from (4.15) t h a t M • 1 C^ - T. C~ (4.17) From (4.12) and ( 4 . 1 3 ) , i t can be seen t h a t the e i g e n v e c t o r s of G a r e the same as thos e o f h L , and t h a t \ . _ co (4.18) where ^ - « A a r e the e i g e n v a l u e s of h z . A s t r a i g h t f o r w a r d 71 c a l c u l a t i o n s h o w s t h a t t h e e i g e n v e c t o r s a n d e i g e n v a l u e s o f h a L h a r e K ) , s i n C Z n m W ) s'.^CM-iWmK)) ( 4 . 1 9 a ) = 4 S I H 1 - ) ( 4 . 1 9 b ) A 2_ . • • M - 1 T a k i n g oo= 1 / 2 , t h i s g i v e s $U = ( s ' m ( T I ^ W ) , s ^ C 2 n ^ , - - , s i ^ - ' W ^ ) ) ( 4 . 2 0 a ) ) X„ - c o s * ' — " ^ ( 4 ' 2 0 b ) Now c o n s i d e r t h e e i g e n v a l u e c o r r e s p o n d i n g t o t h e l o w e s t f r e q u e n c y e i g e n v e c t o r o f G X , = c « l ( t ) - 1 " + <5(^4) (4 .21) . - 1 - 0CKx) F r o m ( 4 . 1 7 ) , i t c a n be s e e n t h a t t h e c o r r e s p o n d i n g l o w f r e q u e n c y c o m p o n e n t o f t h e r e s i d u a l w i l l be damped v e r y s l o w l y , f o r s m a l l h , by t h i s i t e r a t i o n , r e s u l t i n g i n t h e s l o w a s y m p t o t i c c o n v e r g e n c e r a t e c h a r a c t e r i s t i c o f r e l a x a t i o n m e t h o d s . H o w e v e r , t h i s i s n o t t h e c a s e f o r t h e h i g h f r e q u e n c y c o m p o n e n t s o f t h e r e s i d u a l v e c t o r , w h i c h w i l l be d e f i n e d a s t h o s e c o m p o n e n t s w h i c h c a n n o t be r e p r e s e n t e d on a c o a r s e r g r i d G w w i t h H = 2 h . T h e s e c o m p o n e n t s c o r r e s p o n d t o e i g e n v e c t o r s h a y i n g w a v e l e n g t h s l e s s t h a n 4 h , t h a t i s w i t h mh i . The c o r r e s p o n d i n g e i g e n v a l u e s s a t i s f y < C o * 2 ( 5 ) - O.S ( 4 - 2 2 ) 72 Thus, t h e h i g h f r e q u e n c y components of the r e s i d u a l a r e damped v e r y e f f i c i e n t l y - a few r e l a x a t i o n sweeps w i l l v i r t u a l l y e l i m i n a t e them. In a d d i t i o n , each i t e r a t i o n reduces h i g h f r e q u e n c y components by an amount which i s independent of the mesh s p a c i n g , t h a t i s , the smoothing r a t e i s independent of h. In a s i m i l a r f a s h i o n i t can be shown t h a t the e r r o r v e c t o r Q_ ~ UL ~ UL (4.23) i s a l s o smoothed by t h i s r e l a x a t i o n p r o c e s s . The smoothing of t h e r e s i d u a l and e r r o r v e c t o r s i s a c h a r a c t e r i s t i c p r o p e r t y of r e l a x a t i o n methods. However, the smoothing r a t e of a g i v e n method depends on the n a t u r e of the f i n i t e d i f f e r e n c e problem t o which i t i s a p p l i e d , and the c h o i c e of an a p p r o p r i a t e scheme f o r a p a r t i c u l a r problem i s , i n g e n e r a l , a n o n - t r i v i a l t a s k . T h i s t o p i c i s d i s c u s s e d e x t e n s i v e l y ' i n r e f e r e n c e s [4] and [ 8 ] , where both t h e o r e t i c a l and n u m e r i c a l r e s u l t s f o r v a r i o u s t y p e s of r e l a x a t i o n are p r e s e n t e d f o r a f a i r l y b r o ad c l a s s of problems. Assume t h a t a r e l a x a t i o n scheme w i t h s a t i s f a c t o r y smoothing p r o p e r t i e s has been d e t e r m i n e d f o r the c u r r e n t problem ( 4 . 2 a ) . Then a f t e r a few r e l a x a t i o n sweeps on G w, the h i g h f r e q u e n c y components of r 1" w i l l e s s e n t i a l l y be l i q u i d a t e d . At t h i s p o i n t the system (4.6a) may be r e p r e s e n t e d on the c o a r s e r g r i d G H, s i n c e the d e s i r e d c o r r e c t i o n v h w i l l a l s o be a smooth f u n c t i o n . Thus, on G , the f o l l o w i n g problem i s s o l v e d L. v - _L w r (4.24) 73 Here, i s a r e s t r i c t i o n o p e r a t o r which produces a c o a r s e g r i d f u n c t i o n from a f i n e g r i d f u n c t i o n . U s u a l l y , the p o i n t s of G H w i l l be a subset of t h o s e of , and I * i n v o l v e s a s t r a i g h t f o r w a r d t r a n s f e r r a l of the a p p r o p r i a t e f u n c t i o n v a l u e s . Once (4.24) has been s o l v e d , the a p p r o x i m a t i o n t o the f i n e g r i d unknown i s updated as f o l l o w s . K H (4.25) Here I H i s a n o t h e r i n t e r p o l a t i o n o p e r a t o r w h i c h , i n p r a c t i c e , u s u a l l y p e r forms l i n e a r i n t e r p o l a t i o n . A g a i n , t h i s i n t e r p o l a t i o n p r o c e s s may i n t r o d u c e h i g h f r e q u e n c y components i n the r e s i d u a l but t h e s e may be e f f e c t i v e l y e l i m i n a t e d by a few more r e l a x a t i o n sweeps on the f i n e g r i d . The p r o c e s s of u s i n g a c o a r s e g r i d t o compute an a p p r o x i m a t i o n t o v i s c a l l e d a c o a r s e g r i d c o r r e c t i o n . T h i s same t e c h n i q u e may be used t o s o l v e the c o a r s e g r i d system ( 4 . 2 4 ) . That i s , r e l a x a t i o n sweeps a r e performed over G1"1, u p d a t i n g the a p p r o x i m a t i o n v of v u n t i l the c o r r e s p o n d i n g r e s i d u a l H -r * h | w ~ H (4.26) i s smoothed. At t h i s p o i n t , an even c o a r s e r g r i d may be employed t o compute an a p p r o x i m a t i o n t o the d e f e c t v H - v H. The p r o c e s s c o n t i n u e s u s i n g c o a r s e r and c o a r s e r g r i d s u n t i l e v e n t u a l l y , on the c o a r s e s t g r i d , a problem r e s u l t s which may be s o l v e d v e r y i n e x p e n s i v e l y w i t h o u t the a i d of a n o t h e r g r i d . Once t h i s problem has been s o l v e d , a d e s c e n t towards the f i n e s t g r i d i s begun, u s i n g a s e r i e s of i n t e r p o l a t i o n s of the v a r i o u s computed c o a r s e g r i d c o r r e c t i o n s , each f o l l o w e d by a, 74 few more r e l a x a t i o n sweeps t o remove r e s i d u a l components i n t r o d u c e d by the i n t e r p o l a t i o n s . T h i s e n t i r e p r o c e s s i s c a l l e d a c o a r s e g r i d c o r r e c t i o n c y c l e . At t h e end of such a c y c l e , a l l components of r h w i l l e s s e n t i a l l y have been reduced by the same f a c t o r , and i f the i n i t i a l a p p r o x i m a t i o n o f . u ^ was good, the f i n e g r i d problem may be s o l v e d t o w i t h i n the u d e s i r e d t o l e r a n c e . I f an even b e t t e r a p p r o x i m a t i o n of u i s d e s i r e d , another c o a r s e g r i d c o r r e c t i o n c y c l e may be performed. Even though a t t e n t i o n has been r e s t r i c t e d t o the case of l i n e a r d i f f e r e n c e e q u a t i o n s , the p r e c e d i n g d e s c r i p t i o n i l l u s t r a t e s t he key f e a t u r e s common t o any m u l t i - g r i d a l g o r i t h m . There a r e t h r e e major i d e a s i n v o l v e d : 1) a sequence of g r i d s w i t h g e o m e t r i c a l l y d e c r e a s i n g mesh s i z e s i s employed. On each s u c c e s s i v e g r i d , the f i n i t e d i f f e r e n c e e q u i v a l e n t of (4.1a) i s s o l v e d t o produce an i n i t i a l e s t i m a t e f o r t h e unknown f u n c t i o n on the next f i n e r g r i d . 2) i n the p r o c e s s of s o l v i n g any system on any p a r t i c u l a r g r i d , r e l a x a t i o n sweeps are a p p l i e d s o l e l y f o r the purpose of smoothing the r e s i d u a l of the system. (An e x c e p t i o n i s made f o r the c o a r s e s t g r i d where r e l a x a t i o n sweeps may be employed t o a c t u a l l y s o l v e the system.) 3) once the r e s i d u a l of a g i v e n system i s s u f f i c i e n t l y smooth, the problem of computing the n e c e s s a r y c o r r e c t i o n t o the g r i d f u n c t i o n i s t r a n s f e r r e d t o a c o a r s e r g r i d . Q u i t e a number of i m p o r t a n t d e t a i l s , such as how t o de t e r m i n e when a r e s i d u a l i s " s u f f i c i e n t l y smooth", or when a g i v e n system has been " s o l v e d " , have been n e g l e c t e d i n the 75 d e s c r i p t i o n of the m u l t i - g r i d method thus f a r . R a t h e r than c o m p l e t i n g the d e s c r i p t i o n of a m u l t i - g r i d a l g o r i t h m f o r a l i n e a r problem, a t t e n t i o n i s now d i r e c t e d t o the more g e n e r a l case of n o n - l i n e a r problems. 4•2 The f u l l a p p r o x i m a t i o n s t o r a g e scheme As b e f o r e , l e t the c o n t i n u o u s domain of the boundary v a l u e problem be a p p r o x i m a t e d by a sequence of g r i d s denoted by G^, k=0,1,...,m where G° i s the c o a r s e s t g r i d and G^ i s the f i n e s t g r i d . A g a i n assume t h a t each g r i d i s u n i f o r m w i t h mesh s p a c i n g h< which s a t i s f i e s h* = 2 h K * i so t h a t e v e r y o t h e r " l i n e " of G*^ i s a " l i n e " of G K. I t s h o u l d be noted t h a t t h i s p a r t i c u l a r c h o i c e f o r s u c c e s s i v e mesh s p a c i n g s seems t o be o p t i m a l f o r most problems which have been s o l v e d by the mul* t i -g r i d method so f a r [ 8 ] , The f i r s t p a r t of t h e m u l t i - g r i d a l g o r i t h m d e s c r i b e d i n the p r e v i o u s s e c t i o n remains i n t a c t f o r the n o n - l i n e a r c a s e . That i s , f o r each k, k=0,1,...m, the problem i s ( a p p r o x i m a t e l y ) s o l v e d , and the s o l u t i o n i s i n t e r p o l a t e d t o p r o v i d e an i n i t i a l e s t i m a t e f o r the f u n c t i o n on the next g r i d (4.27) (4.28) where, as i n ( 4 . 4 ) , 32* i s a c o a r s e - t o - f i n e i n t e r p o l a t i o n 76 operator. In p r a c t i c e , JET ^ u s u a l l y performs polynomial i n t e r p o l a t i o n . Brandt suggests general r u l e s for determining an appropriate order of i n t e r p o l a t i o n , the aim being to e x p l o i t any known "smoothness p r o p e r t i e s " of the a c t u a l s o l u t i o n u to as great an extent as p o s s i b l e . In p a r t i c u l a r , i f L i s a second order d i f f e r e n t i a l operator, and the d i f f e r e n c e system i s second order i n t r u n c a t i o n e r r o r , then i t i s suggested that cubic i n t e r p o l a t i o n be used. To solve any of the systems (4.27), with the exception of k=0, the techniques of smoothing by r e l a x a t i o n and coarse g r i d c o r r e c t i o n s are again employed. Because the systems are non-l i n e a r , a good non-linear r e l a x a t i o n scheme must be found. (A few examples of such methods are described i n Section A.4 of Appendix A.) Again, the choice of an appropriate r e l a x a t i o n scheme i s problem-dependent. I f the m u l t i - g r i d a l g orithm i s to perform e f f i c i e n t l y , the r e l a x a t i o n method must r e s u l t i n a r e l a t i v e l y high smoothing rate which i s independent of h<, and must be f a i r l y inexpensive to perform. In p a r t i c u l a r , the work req u i r e d to complete a r e l a x a t i o n sweep on G K should be 0(n<) where n< i s the number of p o i n t s of G*. The reader i s again r e f e r r e d to reference [4] for a more d e t a i l e d d i s c u s s i o n of t h i s important t o p i c . Assume that a good r e l a x a t i o n method has been found, and that the problem on G has been solved. Then, on G , r e l a x a t i o n sweeps are performed u n t i l the r e s i d u a l r * of the system has been smoothed. In p r a c t i c e , t h i s smoothing i s most conveniently monitored by c a l c u l a t i n g , at each sweep, the norm of the dynamic r e s i d u a l vector r , which i s the vector of 77 defects which are normally computed in the course of the relaxation i t e r a t i o n . The quantity 1 ^ (4.29) r \\ ? Ml ^ i< where f i s the dynamic residual vector of the previous i t e r a t i o n and II * II i s some discrete norm, w i l l remain f a i r l y small ( t y p i c a l l y ~ 0.5) as long as r K s t i l l contains high frequency components which are being e f f i c i e n t l y damped by the relaxations. Once r k i s smooth, convergence slows, andjX quickly approaches a value which, for small h<, i s very close to unity. Thus r k i s assumed to be smooth when /X exceeds some value i which i s normally supplied as an adjustable parameter to the multi-grid algorithm. (See Appendix A of Reference [4] for more d e t a i l s . ) The implementation of coarse g r i d corrections described in section 4.1 must be modified since expressions (4.6a) and (4.6b) are not v a l i d in the non-linear case. However, given an approximation u of u , the problem i s s t i l l to determine the correction v K such that (4.30) Subtract the quantity L^u*from both sides of the above to get L k ( a % v K ) - L kCX < = ^ - L ' a * - - r * (4.31) Now, when r^ has been smoothed, t h i s problem may be transferred to Gk"' . That i s , on the coarse gr i d , the 78 following problem i s posed L " ~ V ^ L^Z*" - i T r * (4.32) where 11"1 i s a fine-to-coarse r e s t r i c t i o n operator (as in (4.24)). A f u l l explanation of this expression w i l l be delayed u n t i l the discussion of l o c a l truncation error. For the time being, i t i s simply stated that the solution u * M of the above is not the same as the solution of the o r i g i n a l G*~l problem , fc-l vc-l r Ve-l 1— U. ~ •> Rather, i t i s the coarse gri d equivalent of the function which (nearly) s a t i s f i e s the fine g r i d equations. When an approximate solution, u of (4.32) has been determined, the quantity i s the coarse grid approximation to the desired correction v K . Thus, the fine gr i d function i s updated as follows otK C L k + xS-, (a.*-1 - Z . V a * ) (4 .34) where I H i s another interpolation operator (as in ( 4 . 2 5 ) ) . Note that, in general i f i s not equal to the identity operator, so that t h i s i s not equivalent to LL := _ l _ K - ( OL 79 The former e x p r e s s i o n i s t o be p r e f e r r e d s i n c e h i g h f r e q u e n c y i n f o r m a t i o n about u which e x i s t s p r i o r t o the c o a r s e g r i d c o r r e c t i o n i s r e t a i n e d . A g a i n , a f t e r the c o a r s e g r i d c o r r e c t i o n has been co m p l e t e d , a few more r e l a x a t i o n sweeps over G a r e made t o remove the r a p i d f l u c t u a t i o n s i n r which a r e i n t r o d u c e d by i n t e r p o l a t i o n of the c o r r e c t i o n . At t h i s p o i n t , the G problem may be s a t i s f a c t o r i l y s o l v e d . I f n o t , another c o a r s e g r i d c o r r e c t i o n i s i n i t i a t e d . As s h o u l d be e x p e c t e d , the same t e c h n i q u e i s used t o s o l v e ( 4 . 3 2 ) . A few It -I r e l a x a t i o n sweeps on G a r e f o l l o w e d by a c o a r s e g r i d c o r r e c t i o n on G , and so on. Thus i n , a complete c o a r s e g r i d c o r r e c t i o n c y c l e , f o r the s o l u t i o n of the G^ problem, the f o l l o w i n g systems of e q u a t i o n s a r e a p p r o x i m a t e l y s o l v e d F l i • (4.35) The above e q u a t i o n s d e f i n e what Brandt c a l l s t h e f u l l  a p p r o x i m a t i o n s t o r a g e (FAS) scheme, s i n c e a t any stage of the s o l u t i o n p r o c e s s , the g r i d f u n c t i o n u j , f o r any j , i s an a p p r o x i m a t i o n on G J of the unknown on the f i n e s t g r i d . To more f u l l y u n d e r s t a n d the o p e r a t i o n of the FAS a l g o r i t h m , i t i s u s e f u l t o i n t r o d u c e the concept of t r u n c a t i o n e r r o r . G i v e n the d i f f e r e n t i a l o p e r a t o r L of (4.1a) and a c o r r e s p o n d i n g d i f f e r e n c e o p e r a t o r L K , the l o c a l t r u n c a t i o n  e r r o r , r * , d e f i n e d on G , i s g i v e n by I K s L T ^ u . - I ' L u . (4.36) 80 where u i s the exact s o l u t i o n of (4.1) and I i s an o p e r a t o r which r e s t r i c t s a f u n c t i o n of the c o n t i n u o u s domain t o the g r i d G . The v a l u e of ~cK a t any p o i n t of G p r o v i d e s a measure of how w e l l the d i f f e r e n c e o p e r a t o r a p p r o x i m a t e s the d i f f e r e n t i a l o p e r a t o r a t t h a t p o i n t . Note t h a t (4.36) may be r e w r i t t e n as i * T * 1 * • -r (4.37) I _L CL = 7 + T. and, i f i t were p o s s i b l e t o approximate then the s o l u t i o n k r u of L*OL* * S K * -LK ( 4 ' 3 8 ) would be a b e t t e r a p p r o x i m a t i o n of u than the s o l u t i o n of L V " - ( 4- 3 9 ) In a s i m i l a r f a s h i o n , a n o t h e r type of t r u n c a t i o n e r r o r may be i n t r o d u c e d . D e f i n e k , k -p. k K + i - j - k . K+I K+1 /A 4Q\ where u * + l s a t i s f i e s , k-v* i<+\ r k + l (4.41 ) L_ UL - J rk+i i s c a l l e d t h e r e l a t i v e l o c a l t r u n c a t i o n e r r o r of L w i t h r e s p e c t t o L l < + 1 , and i s s i m p l y the q u a n t i t y which must be added t o the r i g h t hand of (4.39) so t h a t the s o l u t i o n , u k , of the r e s u l t i n g system c o i n c i d e s w i t h I £+( u k + l . Now, a l t h o u g h 81 t r £ M c a n n o t be d e t e r m i n e d e x a c t l y w i t h o u t s o l v i n g ( 4 . 4 1 ) e x a c t l y , i f a n a p p r o x i m a t i o n u w + s o f u**[ i s k n o w n , t h e n T I<:+I £ I— l ^ i u. ~ _L k + i L_ (4.42) a p p r o x i m a t e s T£+I . The c o a r s e g r i d c o r r e c t i o n e q u a t i o n ( 4 . 3 2 ) may now be r e w r i t t e n w i t h t h e a i d o f ( 4 . 3 1 ) a n d t h e l a s t e x p r e s s i o n I . o_ = l _ J _ k u . - _L « L_ u. + _L J K - I r *-t ( 4 . 4 3 ) By t h e a b o v e r e a s o n i n g , t h e s o l u t i o n , u K ' , o f t h i s s y s t e m w i l l c o i n c i d e ( n e a r l y ) w i t h I K " ' u K , a s was p r e v i o u s l y c l a i m e d . The r e l a t i v e t r u n c a t i o n e r r o r e s t i m a t e s , w h i c h a m u l t i -g r i d p r o c e d u r e p r o d u c e s , a r e a l s o u s e f u l f o r p r o v i d i n g n a t u r a l  c o n v e r g e n c e c r i t e r i a , t h a t i s , q u a n t i t i e s u s e d f o r d e t e r m i n i n g when a f i n e g r i d p r o b l e m s h o u l d be c o n s i d e r e d s o l v e d . The r u l e a d o p t e d h e r e i s t h e f o l l o w i n g : a p r o b l e m i s s o l v e d when . t h e n o r m o f t h e ( d y n a m i c ) r e s i d u a l s i s r o u g h l y t h e same s i z e a s t h e n o r m o f t h e l o c a l t r u n c a t i o n e r r o r o f t h e d i f f e r e n c e s c h e m e . I t c a n e a s i l y be shown [ 6 ] , t h a t i f t h e d i f f e r e n c e e q u a t i o n s a r e o f o r d e r p a n d hi<=2h«r+i , t h e n i s ' r e l a t e d t o t h e r e l a t i v e l o c a l t r u n c a t i o n e r r o r a s f o l l o w s k K 1 ~" k x = 1 _ Z - P = ^ L . p r ^ ' ( 4 . 4 4 ) s o t h a t f o r t h e p u r p o s e s o f d e t e r m i n i n g when t o s t o p s o l v i n g t h e p r o b l e m , c a l c u l a t i o n o f u z i U w i l l s u f f i c e . H o w e v e r , 82 £ £ + l i s not d e f i n e d i n the m u l t i - g r i d a l g o r i t h m u n t i l the s o l u t i o n on G ( has begun, by which time the problem on G K has presumably been s o l v e d . F o r t u n a t e l y , the f o l l o w i n g r e l a t i o n may a l s o be e a s i l y d e r i v e d K ^ _L ^ 9- (4.45) "C K + I = ^ ~ £ K- — 4 '< The q u a n t i t y T f may be c a l c u l a t e d as soon as a c o a r s e g r i d c o r r e c t i o n i s i n i t i a t e d on G K . Thus on G K, the s o l u t i o n p r o c e s s s t o p s when tl r "ll 4 « II r V II U * 4 6 ) where cx i s t y p i c a l l y 0.25, but i n g e n e r a l i s a t u n a b l e parameter of the a l g o r i t h m . 4.3 S o l u t i o n on the c o a r s e s t g r i d Any problem e n c o u n t e r e d on the c o a r s e s t g r i d G°, be i t the f i r s t of the systems a p p r o x i m a t i n g the c o n t i n u o u s problem, or a c o a r s e g r i d c o r r e c t i o n of one of the subsequent problems, must, of c o u r s e , be s o l v e d w i t h o u t the a i d of c o a r s e g r i d c o r r e c t i o n s . T h i s u s u a l l y p r e s e n t s no problem, s i n c e G° w i l l t y p i c a l l y have so few p o i n t s , t h a t even i f many r e l a x a t i o n sweeps must be performed t o a t t a i n c o nvergence, the net work expended w i l l be e s s e n t i a l l y n e g l i g i b l e i n comparison t o t h a t i n v o l v e d i n making even a s i n g l e sweep over t h e f i n e s t g r i d , G M. A more c r u c i a l p o i n t t h a t must be noted i s t h a t , as mentioned i n Appendix A, a system of e q u a t i o n s may be s o l v e d by r e l a x a t i o n o n l y i f the system (or more c o r r e c t l y f o r the 83 n o n - l i n e a r case - the l i n e a r i z e d system) i s p o s i t i v e d e f i n i t e (or n e g a t i v e d e f i n i t e ) . I f t h i s i s not the c a s e , t h a t i s , i f f o r any of the m+1 problems, the system has i e i g e n v a l u e s l e s s ( g r e a t e r ) than 0, then the c o r r e s p o n d i n g Jl l o w e s t f r e q u e n c y components of the r e s i d u a l may a c t u a l l y be m a g n i f i e d by r e l a x a t i o n sweeps. Now, on any g r i d but G°, t h e s e low f r e q u e n c y components a r e t o be a n n i h i l a t e d by c o a r s e g r i d c o r r e c t i o n s , but s i n c e t h e m u l t i - g r i d method d i c t a t e s t h a t r e l a x a t i o n a l s o be employed i n the c o r r e c t i o n p r o c e s s s , t h e s e components may c o n t i n u e t o be m a g n i f i e d . The o n l y p l a c e where they may be e f f e c t i v e l y damped i s on G° and o n l y then by employing a d i r e c t method f o r the s o l u t i o n of the c o a r s e s t g r i d systems. (See Appendix A, s e c t i o n A.3.1) A g a i n , t h i s w i l l not u s u a l l y degrade the e f f i c i e n c y of the m u l t i g r i d a l g o r i t h m s i g n i f i c a n t l y , p r o v i d e d the number of p o i n t s n 0 i n G° i s s m a l l enough so t h a t the O ( n J ) o p e r a t i o n s t y p i c a l l y r e q u i r e d t o s o l v e the s p a r s e l i n e a r i z e d systems d i r e c t l y , r e p r e s e n t s o n l y a f r a c t i o n of the o p e r a t i o n s needed t o r e l a x t h e f i n e s t g r i d . On the o t h e r hand, G° must c o n t a i n enough p o i n t s t o be a b l e t o e f f e c t i v e l y r e p r e s e n t a l l of the components of any of the m+1 problems which may be m a g n i f i e d by r e l a x a t i o n sweeps. 4.4 Treatment of boundary c o n d i t i o n s As mentioned i n s e c t i o n 4.1, the m u l t i - g r i d method t r e a t s the i n t e r i o r and boundary d i f f e r e n c e e q u a t i o n s i n d e p e n d e n t l y . T h i s means t h a t the systems 84 are solved i n the same manner as the i n t e r i o r equations, using the techniques of smoothing and coarse g r i d c o r r e c t i o n . Of course, i n the case of D i r i c h l e t boundary c o n d i t i o n s , there i s nothing to " s o l v e " ; the boundary values of u are simply determined from r e s t r i c t i o n of the given boundary values of u. The f o l l o w i n g d i s c u s s i o n a p p l i e s to the case of Neumann or Robbins (mixed) c o n d i t i o n s . As discussed i n s e c t i o n A.2 of Appendix A, i t i s a common p r a c t i c e i n the s o l u t i o n of d i f f e r e n c e systems to combine the boundary equations and i n t e r i o r equations to form a s i n g l e system. A l t e r n a t i v e l y , i f an i t e r a t i v e technique i s employed, and the boundary equations are l i n e a r , then the appropriate g r i d f u n c t i o n values might be updated a f t e r each pass over the i n t e r i o r equations, so as to s a t i s f y the boundary equations e x a c t l y . However, such a process, which reduces the boundary r e s i d u a l s to zero, tends to introduce l a r g e r e s i d u a l s i n the neighboring i n t e r i o r d i f f e r e n c e equations. This may r e s u l t i n a s e r i o u s d e t e r i o r a t i o n of the smoothing rate of the i n t e r i o r r e l a x a t i o n scheme which w i l l reduce the o v e r a l l convergence r a t e . Noting t h a t , j u s t as i n the i n t e r i o r case, the boundary r e s i d u a l s need only be smoothed, Brandt suggests that a more appropriate procedure i s to sweep over the g r i d p o i n t s a s s o c i a t e d with the boundary equations (see Figure 20), a d j u s t i n g each f u n c t i o n value so that the r e s u l t i n g boundary r e s i d u a l i s the average of the r e s i d u a l s of the two neighboring equations. Brandt a l s o claims that making two such passes per i n t e r i o r r e l a x a t i o n sweep u s u a l l y ensures that the boundary r e s i d u a l s are smoothed as e f f i c i e n t l y as the i n t e r i o r 85 C r e s i d u a l s . When t r e a t e d i n t h i s f a s h i o n , the c o a r s e g r i d c o r r e c t i o n p r o c e s s may then be a p p l i e d t o the boundary systems u s i n g f o r m u l a e e x a c t l y analagous t o ( 4 . 3 5 ) . 4.5 Implementation of t h e FAS a l g o r i t h m A pseudo-code form of a FAS m u l t i - g r i d a l g o r i t h m s u i t a b l e f o r the s o l u t i o n of a n o n - d e f i n i t e problem i s shown i n F i g u r e 5. T h i s a l g o r i t h m , w i t h a few m o d i f i c a t i o n s which a r e d e s c r i b e d i n s e c t i o n 4.7 and Appendix B, was used t o s o l v e the boundary v a l u e problem f o r m u l a t e d i n the p r e v i o u s c h a p t e r . The main r o u t i n e - PROCEDURE MULTI_GRID - i s r e s p o n s i b l e f o r s u c c e s s i v e l y s o l v i n g t h e d i s c r e t e problems (L u = f ; B u = qK) on g r i d s Gfc, k=0,1,...,m. The number of g r i d s t o be used i s s u p p l i e d as a parameter, a l o n g w i t h the c o a r s e g r i d s p a c i n g h 0 . A l s o g i v e n a r e t h r e e c o n t r o l parameters and ^ , an i n i t i a l e s t i m a t e , u°, of the unknown on the c o a r s e s t g r i d , and a convergence c r i t e r i o n t° f o r the i n i t i a l c o a r s e g r i d p roblem. The parameters cx and vj have been d i s c u s s e d p r e v i o u s l y , £ i s used t o p r o v i d e convergence c r i t e r i a f o r c o a r s e g r i d c o r r e c t i o n s . In p r a c t i c e a l l t h r e e of thes e p a rameters may be used t o tune t h e a l g o r i t h m . T h i s t o p i c i s d i s c u s s e d e x t e n s i v e l y i n Appendix A of [4]. The second r o u t i n e - PROCEDURE SOLVE_ON_GRID - i s a f u n c t i o n of two p a r a m e t e r s : k i n d i c a t e s which of the m+1 problems i s b e i n g s o l v e d , X i s the g r i d on which the problem i s c u r r e n t l y b e i n g s o l v e d . The f l o w of t h e r o u t i n e i s s t r a i g h t f o r w a r d . Any system on the c o a r s e s t g r i d i s s o l v e d by 8 6 PROCEDURE MULTI_GRID(m,ho ,oc, , £ ° ) PERFORM SOLVE_ON_GRID(0,0) FOR k = 1 .. m h,<:= i h . K - i u": = JI-u*-' F 1 4: = f K Gk:= g* PERFORM SOLVE_ON_GRID(k,k) END FOR END PROCEDURE PROCEDURE SOLVE_ON_GRID(k,X) IF % = 0 THEN s o l v e L e u * = F^; B*u*= G ^ d i r e c t l y ELSE II f II :«= co II r l| : = RELAX U ) WHILE llrll > 6 * DO IF H r l l / urn > ^  THEN {convergence i s slow - s t a r t c o a r s e g r i d c o r r e c t i o n } rf-' := u<- I i " L*u* • := B ^ ' l i - ' u * - IV B'u* IF X = k THEN := « U r t l II END IF ux-':= i f u * F-t-' .= F*-<+ t*-' G*-':= 0*-'+ r l ' 1 eA-':= S llrll { i n v o k e r o u t i n e r e c u r s i v e l y t o pe r f o r m c o a r s e g r i d c o r r e c t i o n } PERFORM SOLVE_ON_GRID(k,A - 1) { i n t e r p o l a t e c o r r e c t i o n } END IF n f u : = " <rll llrii := RELAX(-2) END WHILE END IF END PROCEDURE F i g u r e 5 FAS a l g o r i t h m 87 d i r e c t means. In any o t h e r c a s e , r e l a x a t i o n sweeps a r e performed over G u n t i l the problem i s s o l v e d or slow convergence i s d e t e c t e d . The r o u t i n e RELAX performs the r e l a x a t i o n of both the i n t e r i o r and boundary systems and r e t u r n s the norm of the dynamic r e s i d u a l s . I f convergence i s slow, a c o a r s e g r i d c o r r e c t i o n i s s t a r t e d . The r i g h t hand s i d e s , F and G£"' , of the c o a r s e g r i d systems a r e updated u s i n g the c o a r s e g r i d c o r r e c t i o n e q u a t i o n s . I t s h o u l d be noted t h a t the q u a n t i t i e s r^' 1 and r^" 1 r e p r e s e n t r e l a t i v e t r u n c a t i o n e r r o r e s t i m a t e s of the o r i g i n a l i n t e r i o r and boundary d i f f e r e n c e e q u a t i o n s o n l y i n the c a s e ^ = k . I n t h i s c a s e , H t u"' II i s used t o c a l c u l a t e the convergence c r i t e r i o n £ as d e s c r i b e d i n s e c t i o n 4.2. The convergence parameter f o r the c o a r s e g r i d system, , i s updated u s i n g & and then SOLVE_ON_GRID i s in v o k e d r e c u r s i v e l y t o a c t u a l l y p e r f o r m the c o a r s e g r i d c o r r e c t i o n . When the c o r r e c t i o n has been completed, u i s updated and r e l a x a t i o n sweeps a r e resumed. When the norm of the dynamic r e s i d u a l s i s l e s s than the t o l e r a n c e € , the r o u t i n e t e r m i n a t e s . 4.6 Work e s t i m a t e s When p r o p e r l y t u n e d , the m u l t i - g r i d method p r o v i d e s a ve r y e f f i c i e n t means of s o l v i n g q u i t e g e n e r a l boundary v a l u e problems. In t h i s s e c t i o n , a rough a n a l y s i s of the work r e q u i r e d t o s o l v e a t y p i c a l problem u s i n g a m u l t i - g r i d a l g o r i t h m i s made. The m a j o r i t y of the work expended i n a m u l t i - g r i d 88 s o l u t i o n i s used t o p e r f o r m r e l a x a t i o n sweeps over the v a r i o u s g r i d s . Brandt c l a i m s t h a t n u m e r i c a l e x p e r i m e n t s have shown t h a t o t h e r p r o c e s s e s , such as i n t e r p o l a t i o n s or i n j e c t i o n s , a ccount f o r no more than 30% of the t o t a l c o m p u t a t i o n a l work. A s i n g l e r e l a x a t i o n sweep over any g r i d G s h o u l d t a k e 0(n<) o p e r a t i o n s where nK i s the number of p o i n t s of G K. F u r t h e r m o r e , the c o e f f i c i e n t of t h i s o r d e r e s t i m a t e w i l l be the same f o r a l l g r i d s i f , as i s u s u a l , the d i f f e r e n c e e q u a t i o n s a r e of the same form on a l l g r i d s . Denote by w K, the number of o p e r a t i o n s needed t o p e r f o r m a s i n g l e G* r e l a x a t i o n sweep. I f the problem i s t w o - d i m e n s i o n a l and h*-i=2hi<, then n^-i'S 4 n,< and w^ "1^ i w*. Now, assuming t h a t the r e l a x a t i o n method used has a smoothing r a t e which i s independent of the mesh s i z e , then on any g r i d , an e s s e n t i a l l y c o n s t a n t number of sweeps, p, must be made b e f o r e a c o a r s e g r i d c o r r e c t i o n i s s t a r t e d . F o l l o w i n g a c o a r s e g r i d c o r r e c t i o n , a n o t h e r q sweeps w i l l , i n g e n e r a l , be r e q u i r e d t o smooth out components i n t r o d u c e d by i n t e r p o l a t i o n of the c o r r e c t i o n . At t h i s p o i n t , the problem may be s o l v e d ; i f n o t , a n o t h e r c o a r s e g r i d c o r r e c t i o n i s made. Each such c o r r e c t i o n s h o u l d reduce the magnitude of the r e s i d u a l v e c t o r by a r o u g h l y c o n s t a n t f a c t o r . Suppose t h a t , a t most, 6 c o a r s e g r i d c o r r e c t i o n s a r e r e q u i r e d t o s o l v e any problem. Then the t o t a l r e l a x a t i o n work, WK, used t o s o l v e on G i s W * ^ ( p + cj- q ) w « +. 6 W (4.47) where W*~l i s the r e l a x a t i o n work needed t o s o l v e the G k _ l 89 problem. C l e a r l y , W*4 ( P + « q ) w K (1 + A « * f i " 3 ' * + + tfkW° (4.48) where W° i s the work r e q u i r e d t o compute a d i r e c t s o l u t i o n on G° which w i l l be assumed t o be c o n s t a n t . Now i f 6" < 4, then l 4 S O 1 - £ S i n c e w K~cn f c and 5 1 4 < \ ~* 5: n K / n 0 , t h i s becomes W ^ < A , 1 - c/4 n 0 (4.49) w i t h the c o n c l u s i o n , t h a t the r e l a x a t i o n work, and t h e r e f o r e the t o t a l work n e c e s s a r y t o s o l v e the problem on any g r i d i s p r o p o r t i o n a l t o the number of p o i n t s i n the g r i d . T h i s b e h a v i o r has been v e r i f i e d i n n u m e r i c a l e x p e r i m e n t s i n v o l v i n g a wide v a r i e t y of boundary v a l u e p r o b l e m s , and i s p r o b a b l y the most c o m p e l l i n g reason f o r e m p l o y i n g the m u l t i - g r i d method. None of the methods commonly used f o r the s o l u t i o n of f i n i t e d i f f e r e n c e systems p r o v i d e 0(n«) performance f o r g e n e r a l p roblems. For example, as d i s c u s s e d i n s e c t i o n s A.3.2 and A.4 of Appendix A, the commonly used s u c c e s s i v e - o v e r - r e l a x a t i o n (SOR) t e c h n i q u e r e q u i r e s a t l e a s t 0(N,<) o p e r a t i o n s i n g e n e r a l . 9 0 T h i s r e s u l t assumes t h a t the number of g r i d p o i n t s i n e i t h e r d i r e c t i o n i s r o u g h l y the same, so t h a t * *. The work e s t i m a t e f o r the m u l t i - g r i d method i s then O(N^). 4 . 7 A d a p t i v e d i s c r e t i z a t i o n In the d e s c r i p t i o n of the m u l t i - g r i d a l g o r i t h m so f a r , the sequence of g r i d s employed have a l l a p p r o x i m a t e d the e n t i r e c o n t i n u o u s domain of the boundary v a l u e problem. In a d d i t i o n , because the c o a r s e s t g r i d s p a c i n g and number of g r i d s t o be used are s u p p l i e d as parameters t o the a l g o r i t h m , i t i s i m p l i c i t l y assumed t h a t the user has some i d e a of how f i n e a mesh i s r e q u i r e d t o p r o v i d e a s a t i s f a c t o r y s o l u t i o n of the problem. In p r a c t i c e , t h i s assumption may be f a r from r e a l i s t i c , s i n c e i t presumes a p r i o r i i n f o r m a t i o n about the unknown f u n c t i o n . Moreover, the g r i d s p a c i n g needed t o y i e l d a s o l u t i o n Of u n i f o r m a c c u r a c y w i l l i n g e n e r a l v a r y from p l a c e t o p l a c e i n t h e s o l u t i o n domain. I f each g r i d used c o v e r s the e n t i r e domain, much work may e s s e n t i a l l y be wasted by u s i n g f i n e g r i d s i n r e g i o n s where t h e y a r e not r e a l l y n e c e s s a r y . One a t t r a c t i v e c o n c e p t , which can be i n c o r p o r a t e d i n t o a m u l t i - g r i d scheme w i t h r e l a t i v e e a se, i s t o a l l o w t h e u l t i m a t e d i s c r e t i z a t i o n of the c o n t i n u o u s domain t o be d e t e r m i n e d i n t h e c o u r s e of t h e s o l u t i o n p r o c e s s . Such a t e c h n i q u e i s c a l l e d a d a p t i v e d i s c r e t i z a t i o n and, when p r o p e r l y implemented, can enhance the e f f e c t i v e n e s s and e f f i c i e n c y of the a l g o r i t h m c o n s i d e r a b l y . A g e n e r a l d i s c u s s i o n of a d a p t i v e t e c h n i q u e s i n the c o n t e x t of the m u l t i - g r i d method would r e q u i r e a n o t h e r c h a p t e r . The c u r r e n t d i s c u s s i o n w i l l be l i m i t e d t o a b r i e f 91 d e s c r i p t i o n of the s i m p l e t e c h n i q u e used i n the s o l u t i o n of the problem f o r m u l a t e d i n the p r e v i o u s c h a p t e r . The i n t e r e s t e d r e a der i s r e f e r r e d t o [ 4 ] , [ 5 ] , or [ 6 ] f o r c o n s i d e r a b l y more d e t a i l e d d i s c u s s i o n s . The b a s i c g o a l of a d a p t i v e d i s c r e t i z a t i o n i s t o p r o v i d e a s o l u t i o n of u n i f o r m l y h i g h a c c u r a c y w i t h a minimum of wasted work. To a c h i e v e t h i s , i t i s f i r s t n e c e s s a r y t o have some means of e s t i m a t i n g t h e a c c u r a c y of a computed s o l u t i o n a t a g i v e n p l a c e i n the problem domain. In a d d i t i o n , g e n e r a t i n g such an e s t i m a t e s h o u l d not r e q u i r e so much work t h a t the c o m p u t a t i o n a l advantage the a d a p t i v e t e c h n i q u e i s supposed t o p r o v i d e i s l o s t . In g e n e r a l , e s t i m a t i n g the a c c u r a c y of an a pproximate s o l u t i o n d i r e c t l y i s d i f f i c u l t , i f not i m p o s s i b l e , s i n c e the e x a c t s o l u t i o n i s u s u a l l y unknown. However, i n the m u l t i - g r i d method, the l o c a l a c c u r a c y of the d i f f e r e n c e scheme can be e s t i m a t e d u s i n g the l o c a l t r u n c a t i o n e r r o r e s t i m a t e s t h a t the a l g o r i t h m r o u t i n e l y g e n e r a t e s . For p r o p e r l y c o n s t r u c t e d d i f f e r e n c e schemes, as the l o c a l t r u n c a t i o n e r r o r becomes v e r y s m a l l , so does the l o c a l e r r o r i n t h e computed s o l u t i o n . Thus, an a b i l i t y t o c o n t r o l the l e v e l of t r u n c a t i o n e r r o r a l s o p r o v i d e s a way of c o n t r o l l i n g the a c c u r a c y of the s o l u t i o n . One might a l s o expect t h a t a u n i f o r m l e v e l of t r u n c a t i o n e r r o r i n a l l p a r t s of the domain might a l s o imply a r e l a t i v e l y u n i f o r m l y a c c u r a t e s o l u t i o n . These i d e a s are i n c o r p o r a t e d i n t o the m u l t i - g r i d program which i s l i s t e d i n Appendix B. The program a c c e p t s a parameter, Tc , which r e p r e s e n t s an upper bound on the d e s i r e d l e v e l of t r u n c a t i o n e r r d r . When the s u b m i t t e d problem has been 92 s o l v e d on some g r i d G x, a l o c a l t r u n c a t i o n e r r o r e s t i m a t e z*~ on i s g e n e r a t e d . T h i s e s t i m a t e i s then examined t o d e termine i f t h e r e i s some r e g i o n c o v e r e d by G where z c o n s i s t e n t l y exceeds zc . i f such a r e g i o n i s d i c o v e r e d , then a new, f i n e r g r i d , G i s i n t r o d u c e d , o n l y i n t h a t r e g i o n . As a r e s u l t , t h e sequence of g r i d s used do n o t , i n g e n e r a l , a l l e x t e n d over the e n t i r e domain. F i n e r and f i n e r g r i d s a r e i n t r o d u c e d , o n l y where i t i s f e l t they a r e r e q u i r e d , u n t i l the e s t i m a t e d l o c a l t r u n c a t i o n e r r o r i s l e s s than "Cc everywhere i n the domain. The m o d i f i c a t i o n s t o the FAS a l g o r i t h m a r e s t r a i g h t f o r w a r d and a r e d i s c u s s e d i n Appendix B. The i m p l e m e n t a t i o n of t h i s t e c h n i q u e was a l s o a i d e d by knowledge of the r e s u l t s P i r a n had p r e v i o u s l y c a l c u l a t e d . Most n o t a b l y , i t was known t h a t t h e s o l u t i o n s sought were a l l n e a r l y s p h e r i c a l l y symmetric. T h e r e f o r e , the . f i n e s t l e v e l of d i s c r e t i z a t i o n r e q u i r e d i n the a n g u l a r d i r e c t i o n was s u p p l i e d as a parameter t o the program, and a d a p t i v e d i s c r e t i z a t i o n was performed o n l y i n the r a d i a l d i r e c t i o n . In a d d i t i o n , i t was d e t e r m i n e d e x p e r i m e n t a l l y t h a t the t r u n c a t i o n e r r o r e s t i m a t e s on any g r i d tended t o d e c r e a s e m o n o t o n i c a l l y w i t h i n c r e a s i n g r a d i a l d i s t a n c e which a l l o w e d f u r t h e r s i m p l i f i c a t i o n s of the a d a p t a t i o n p r o c e s s . In c o n c l u s i o n , the m u l t i - g r i d method can produce s o l u t i o n s t o d i s c r e t i z e d boundary v a l u e problems i n an e f f i c i e n t manner, w h i l e a l s o p r o v i d i n g an e s t i m a t e of the a c c u r a c y of the d i f f e r e n c e scheme used. The major drawback t o the method would seem t o be i n i t s i m p l e m e n t a t i o n . Coding a m u l t i - g r i d a l g o r i t h m i s not a t r i v i a l t a s k and t h i s may 9 3 e x p l a i n why the method has not y e t e n j o y e d w i d e s p r e a d use. 94 CHAPTER 5 N u m e r i c a l E x p e r i m e n t s and R e s u l t s T h i s c h a p t e r p r e s e n t s the r e s u l t s of some n u m e r i c a l t e s t s p erformed t o e v a l u a t e the m u l t i - g r i d a l g o r i t h m as w e l l as the r e s u l t s of the a p p l i c a t i o n of the method t o the boundary v a l u e problem of Chapter 3. 5.1 Comparison of m u l t i - g r i d method w i t h o t h e r methods The f i r s t s e r i e s of n u m e r i c a l t e s t s were d e s i g n e d t o compare the performance of the m u l t i - g r i d a l g o r i t h m w i t h two SOR-Newton methods and a N e w t o n - d i r e c t scheme which used a n e s t e d d i s s e c t i o n o r d e r i n g . The r e a d e r i s r e f e r r e d t o Appendix A f o r d e s c r i p t i o n s of the l a t t e r t h r e e methods. I t was o r i g i n a l l y i n t e n d e d t o c a r r y out the t e s t s on the boundary v a l u e problem d e s c r i b e d i n c h a p t e r 3 u s i n g the "model" H g i v e n by e q u a t i o n (3.42) so t h a t the s o l u t i o n s o b t a i n e d from the v a r i o u s methods c o u l d be checked a g a i n s t the e x a c t s o l u t i o n ( 3 . 4 3 ) . However, the i n n e r boundary c o n d i t i o n (3.33) makes the system of d i f f e r e n c e e q u a t i o n s n o n - d e f i n i t e w i t h the r e s u l t t h a t . the r e l a x a t i o n methods f a i l t o converge t o a s o l u t i o n . T h e r e f o r e , a m o d i f i e d t e s t problem, which was d e f i n i t e , was used so t h a t the e f f i c i e n c y of t h e r e l a x a t i o n methods, which a r e s i m i l a r t o tho s e which have been p r e v i o u s l y used i n the 95 s o l u t i o n of e l l i p t i c problems i n n u m e r i c a l r e l a t i v i t y [ 1 5 ] , [ 4 1 ] , c o u l d be e v a l u a t e d w i t h r e s p e c t t o the m u l t i - g r i d method. For each method, the m o d i f i e d t e s t problem, w i t h P=2 and a=1, was s o l v e d on 4 u n i f o r m g r i d s h a v i n g mesh s p a c i n g s A. s -A 0 - z • Z"" . ( 5 - 1 ) ^- 3, 4 , 5, 6 (In a d d i t i o n , the problem was s o l v e d f o r n=7 u s i n g the m u l t i -g r i d method.) Each r o u t i n e was s u p p l i e d w i t h the same i n i t i a l e s t i m a t e s f o r the unknown f u n c t i o n on each g r i d , and the same convergence c r i t e r i a . These were d e t e r m i n e d i n the f o l l o w i n g way. For each n, the m u l t i - g r i d program was used t o s o l v e the a c t u a l , n o n - d e f i n i t e t e s t problem t o l e v e l n-1. The i n t e r p o l a t e d e s t i m a t e f o r l e v e l n was then s t o r e d t o be used as the i n i t i a l e s t i m a t e f o r the o t h e r t h r e e methods. At t h i s p o i n t , the f l o w of the m u l t i - g r i d r o u t i n e was a l t e r e d i n a f a s h i o n which e f f e c t i v e l y r e p l a c e d the Neumann and Robbins boundary c o n d i t i o n s of the problem w i t h D i r i c h l e t c o n d i t i o n s , making the problem d e f i n i t e . The m u l t i - g r i d program proceeded t o s o l v e the l e v e l n problem u n t i l the norm of the dynamic r e s i d u a l s was l e s s than the norm of the e s t i m a t e d r e l a t i v e t r u n c a t i o n e r r o r . When t h i s had been a c c o m p l i s h e d , the norm of the a c t u a l r e s i d u a l was computed t o s e r v e as the convergence c r i t e r i o n f o r the o t h e r t h r e e methods. The o t h e r t h r e e r o u t i n e s took the e s t i m a t e g e n e r a t e d by the m u l t i - g r i d program and s o l v e d the m o d i f i e d problem ( D i r i c h l e t boundary c o n d i t i o n s ) u n t i l the norm of the t r u e 96 r e s i d u a l was l e s s than t h e convergence c r i t e r i o n o u t p u t by the m u l t i - g r i d method. (The norm of the t r u e r e s i d u a l was m o n i t o r e d , because the dynamic r e s i d u a l s g e n e r a t e d f o r a g i v e n f u n c t i o n e s t i m a t e v a r y from method t o method). A l l of the r o u t i n e s used were w r i t t e n i n " s t a n d a r d " FORTRAN and c o m p i l e d u s i n g the IBM H o p t i m i z i n g c o m p i l e r . The t e s t s were performed on an Amdahl 470 V/8 CPU o p e r a t i n g under the MTS t i m e - s h a r i n g o p e r a t i n g system. Two q u a n t i t i e s were measured t o c h a r a c t e r i z e each run - e x e c u t i o n time and memory s t o r a g e . The e x e c u t i o n t i m e s were d e t e r m i n e d u s i n g a system t i m i n g r o u t i n e and were r e p r o d u c a b l e t o w i t h i n a p e r c e n t or so even under v a r y i n g degrees of system l o a d . The memory s t o r a g e f i g u r e s were hand c a l c u l a t e d from the d i m e n s i o n s of the major a r r a y s used f o r each method and do not i n c l u d e o b j e c t code r e q u i r e m e n t s . A l l c a l c u l a t i o n s were performed u s i n g IBM double p r e c i s i o n (8 by t e ) v a r i a b l e s and a r i t h m e t i c . T a b l e I shows the t e s t r e s u l t s f o r the m u l t i - g r i d method. The t o t a l r e l a x a t i o n work performed on a l l l e v e l s i s l i s t e d i n terms of the e q u i v a l e n t number of sweeps, n^, on the f i n e s t l e v e l . Not i n c l u d e d i n t h i s t o t a l i s the work expended i n s o l v i n g the c o a r s e s t g r i d (5x5) systems d i r e c t l y which r e p r e s e n t s a s i g n i f i c a n t f r a c t i o n of the t o t a l e x e c u t i o n time f o r the c o a r s e r systems ( 9 x 9 , 17x17), but an e s s e n t i a l l y n e g l i g i b l e amount f o r t h e f i n e s t system. The c o n s t a n c y of n ^ , as w e l l as the t i m i n g f i g u r e s t h e m s e l v e s , show t h a t the work r e q u i r e d f o r s o l u t i o n i s b a s i c a l l y p r o p o r t i o n a l t o the number of g r i d p o i n t s . A l s o note t h a t the s t o r a g e r e q u i r e m e n t s are a l s o e s s e n t i a l l y l i n e a r i n the number of g r i d p o i n t s . 97 G r i d nc. Time (msec) S t o r a g e ( k b y t e s ) ll r l l 9x9 7.0 80 25 8.7(-4) 17x17 7.0 145 40 3.1(-4) 33x33 7.5 390 90 4.4(-5) 65x65 8.5 1 540 260 9.4(-6) 127x127 7.5 5200 960 1.4(-6) T a b l e I M u l t i - g r i d r e s u l t s f o r m o d i f i e d t e s t problem A c t u a l l y , f o r any but the s m a l l e s t problem, about 40% of t h i s s t o r a g e i s u n n e c c e s s a r y , as i t was used t o m a i n t a i n s e v e r a l a d d i t i o n a l f u n c t i o n s per g r i d which c o u l d have been c a l c u l a t e d when needed, r a t h e r than b e i n g s t o r e d , a t the expense of a 5-10% i n c r e a s e i n e x e c u t i o n t i m e . T a b l e I I shows the r e s u l t s f o r the point-SOR method. N e a r - o p t i m a l v a l u e s of the r e l a x a t i o n parameter, <x) , were d e t e r m i n e d e x p e r i m e n t a l l y f o r each g r i d and a r e l i s t e d i n the t a b l e . A l s o g i v e n a r e the number of r e l a x a t i o n sweeps, ng, needed f o r convergence which would appear t o i n c r e a s e a t l e a s t l i k e the number of p o i n t s on a s i d e of the g r i d . On the c o a r s e r g r i d s , t h i s method o u t p e r f o r m s the m u l t i - g r i d a l g o r i t h m due t o the l a t t e r ' s use of a d i r e c t method on the c o a r s e s t g r i d . On the f i n e r g r i d s however, the m u l t i - g r i d method i s c l e a r l y s u p e r i o r . Note t h a t the s t o r a g e r e q u i r e m e n t s 98 G r i d 00 Time (msec) S t o r a g e ( k b y t e s ) II r |( 9x9 1 .55 10 20 1 5.6(-4) 17x17 1 .60 10 64 4 2.8(-4) 33x33 1 .60 48 1 170 16 4.3(-5) 65x65 1 .60 170 17000 64 9.4(-6) T a b l e I I Point-SOR r e s u l t s f o r m o d i f i e d t e s t problem here v a r y l i n e a r l y w i t h the number of g r i d p o i n t s . The r e s u l t s f o r the l i n e - S O R method a r e g i v e n i n T a b l e I I I . The l i n e r e l a x a t i o n method used h e r e , i n the s p e c i a l case G r i d oJ Time (msec) S t o r a g e ( k b y t e s ) II r II 9x9 1 .35 8 16 1 2.9(-4) 17x17 1 .45 9 71 4 3.1(-4) 33x33 1 .45 38 1 100 16 4.4(-5) 65x65 1 .00 [1 00] [1 1300] 64 [1 .8(-5)] * 5 e e Text T a b l e I I I Line-SOR r e s u l t s f o r m o d i f i e d t e s t problem of O J =1, i s i d e n t i c a l t o the r e l a x a t i o n scheme used i n the m u l t i - g r i d a l g o r i t h m . The r e s u l t s a r e q u i t e s i m i l a r t o th o s e 99 of the point-SOR method, except t h a t on the 65 x 65 g r i d , d i v e r g e n c e o c c u r e d f o r any co > 1. The convergence w i t h oJ =1 on t h i s g r i d was v e r y slow and the s o l u t i o n p r o c e s s was a b o r t e d a f t e r 100 i t e r a t i o n s . From the a s y m p t o t i c convergence r a t e , i t was e s t i m a t e d t h a t a t l e a s t 1500 more i t e r a t i o n s would be r e q u i r e d t o s o l v e the system. F i n a l l y , the r e s u l t s of t h e Newton-nested d i s s e c t i o n method a r e shown i n T a b l e IV. On a l l g r i d s , because the i n i t i a l e s t i m a t e s were good, o n l y one Newton i t e r a t i o n was G r i d Time (msec) S t o r a g e ( k b y t e s ) 9x9 100 18 2.1(-5) '17x17 370 48 3.4(-7) 33x33 1 170 240 7.6(-8) 65x65 17000 1400 1.0(-8) T a b l e IV Newton-ND r e s u l t s f o r m o d i f i e d t e s t problem r e q u i r e d f o r convergence. T h i s meant t h a t o n l y a s i n g l e LU d e c o m p o s i t i o n , which a c c o u n t s f o r the m a j o r i t y of the work i n t h i s method, had t o be performed f o r each problem. The e x e c u t i o n time performance of t h i s method i s about the same as the r e l a x a t i o n methods on the 2 f i n e s t g r i d s , a l t h o u g h the memory r e q u i r e m e n t s a r e much g r e a t e r . I t must be emphasized t h a t t h i s method, u n l i k e the r e l a x a t i o n methods, c o u l d be used 100 t o s o l v e a n o n - d e f i n i t e problem such as the o r i g i n a l t e s t p roblem. However, the m u l t i - g r i d method shows a c l e a r advantage over the d i r e c t method, b o t h i n e x e c u t i o n time and s t o r a g e r e q u i r e m e n t s . 5.2 T e s t i n g of the a d a p t i v e m u l t i - g r i d a l g o r i t h m The second s e t of n u m e r i c a l t e s t s performed was i n t e n d e d t o t e s t t h e a c c u r a c y of the m u l t i - g r i d a l g o r i t h m o p e r a t i n g i n an a d a p t i v e mode as d e s c r i b e d i n the p r e v i o u s c h a p t e r . For the s e t e s t s , the prop e r boundary c o n d i t i o n s were used i n c o n j u n c t i o n w i t h the model H of (3.42) so t h a t the c a l c u l a t e d s o l u t i o n c o u l d be compared w i t h the e x a c t s o l u t i o n ( 3 . 4 3 ) . The i n p u t t o the a l g o r i t h m c o n s i s t e d of the momentum P, and the parameter T c which c o n t r o l l e d the d i s c r e t i z a t i o n i n the s d i r e c t i o n . Because the t e s t problem has a s p h e r i c a l l y symmetric s o l u t i o n , a maximum of two l e v e l s of d i s c r e t i z a t i o n i n the 0 d i r e c t i o n was employed. When the a l g o r i t h m c o mpleted the s o l u t i o n a t a g i v e n l e v e l , a new, f i n e r g r i d was i n t r o d u c e d o n l y i n the r e g i o n where i t was e s t i m a t e d t h a t the r e l a t i v e l o c a l t r u n c a t i o n e r r o r exceeded T0. I f t h i s r e g i o n was n u l l , the a l g o r i t h m t e r m i n a t e d . The t o t a l e n e r g i e s of the computed s o l u t i o n s were a l s o c a l c u l a t e d n u m e r i c a l l y f o r comparison w i t h t h e e x a c t e n e r g i e s g i v e n by e q u a t i o n ( 3 . 4 4 ) . R o u t i n e s f o r e v a l u a t i n g b oth e x p r e s s i o n s (3.40) and (3.41) n u m e r i c a l l y were coded. Romberg i n t e g r a t i o n [13] was employed i n b o t h c a s e s . To t e s t the i n t e g r a t i o n r o u t i n e s , the e x a c t v a l u e s of "f - and H K O O £ L were 101 s u p p l i e d on a 65 x 65 g r i d f o r v a r i o u s v a l u e s of P. Ta b l e V shows the r e s u l t s . C l e a r l y , the volume i n t e g r a l f o r m u l a f o r p E e x a c t E s u r f . E r r o r (%) E v o l . E r r o r (%) 2 2.8284 2.8242 0.1 5 2.8284 0.00 6 6.3246 6.2559 1 .1 6.3246 0.00 10 10. 198 9.9436 2.5 10.199 0.01 14 14. 142 13.558 4. 1 14.152 0.06 18 18.111 17.051 5.8 18.143 0.20 Tab l e V T e s t i n g of i n t e g r a t i o n r o u t i n e s the energy g i v e s s u p e r i o r n u m e r i c a l r e s u l t s . (The p o o r e r performance of the s u r f a c e i n t e g r a l form i s p r i m a r i l y due t o the i n a c c u r a c y i n v o l v e d i n computing V n u m e r i c a l l y ) . As a r e s u l t , t he volume i n t e g r a l form was used t o c a l c u l a t e the e n e r g i e s a s s o c i a t e d w i t h a l l of the s o l u t i o n s which a r e d e s c r i b e d i n t h i s , and subsequent s e c t i o n s . The r e s u l t s of a s e r i e s of t e s t runs w i t h f i x e d momentum (P=4, a=1) and v a r y i n g v a l u e s of ~C0 a r e l i s t e d i n T a b l e V I . The f i r s t column l i s t s t he s u p p l i e d v a l u e of T c. The next 6 columns show the e x t e n t of the v a r i o u s g r i d s used i n the c o u r s e of each s o l u t i o n . The number a t the head of each of t h e s e columns i s the maximum number of g r i d p o i n t s a v a i l a b l e i n the s - d i r e c t i o n a t each l e v e l , and the numbers beneath 102 E x t e n t of g r i d s Time (msec) E r r o r (%) 9 17 33 65 129 265 H- II OO II • II, 9 ~ — -- 360 .707 .698 10- 4 9 1 7 21 1 130 .178 .170 10- 5 9 1 7 33 53 73 3200 .041 .036 10- 6 9 17 33 65 113 201 6900 .009 .007 T a b l e VI E f f e c t of v a r y i n g t r u n c a t i o n e r r o r parameter i n d i c a t e the p o s i t i o n of the o u t e r boundary of the c o r r e s p o n d i n g g r i d f o r each r u n . The e x e c u t i o n t i m e s l i s t e d a r e f o r s o l u t i o n of the e n t i r e problem, which, i n each c a s e , began w i t h a s o l u t i o n on a 5x5 g r i d . In the f i n a l two columns, the r e l a t i v e p e r c e n t a g e e r r o r s i n the computed s o l u t i o n s , c a l c u l a t e d as E X A C T LOMPaTEO yj, E X A C T T M X 1 0 O (5.2) u s i n g b o t h L<[ and norms a r e l i s t e d . These l a s t two columns show how the e r r o r i n the computed s o l u t i o n u n i f o r m l y d e c r e a s e s as the convergence c r i t e r i a becomes more s t r i n g e n t , even though the e n t i r e domain i s not b e i n g d i s c r e t i z e d i n a u n i f o r m f a s h i o n . T a b l e V I I shows the r e s u l t s of a s e r i e s of runs i n which "Co was h e l d c o n s t a n t a t 10~ 5 and the momentum was v a r i e d . The 103 format of t h i s t a b l e i s the same as T a b l e V I . The e x e c u t i o n time needed i n c r e a s e s w i t h the momentum, but remains e s s e n t i a l l y l i n e a r i n the t o t a l number of g r i d p o i n t s used a t the v a r i o u s f i n e s t l e v e l s . The e r r o r i n the computed s o l u t i o n , as a f u n c t i o n of momentum, i s q u i t e c o n s t a n t , v a r y i n g from p E x t e n t of g r i d s Time (msec) E r r o r (%) 9 17 33 65 129 H * Hex, II • \\^  2 9 17 25 45 — 2000 .058 .052 4 9 17 33 53 73 3225 .041 .036 6 9 17 33 57 93 3700 .081 .062 8 9 17 33 57 101 3750 . 1 32 .090 10 9 17 33 57 105 3800 .196 .121 12 9 17 33 65 109 4200 .081 .046 14 9 17 33 65 1 13 4300 .106 .053 16 9 17 33 65 113 4500 .133 .062 18 9 17 33 65 1 17 4400 .165 .070 20 9 17 33 65 1 17 4850 .194 .078 T a b l e V I I M u l t i - g r i d t e s t r e s u l t s f o r v a r y i n g momentum .04% t o .12% i n the L 1 norm. A g a i n , t h i s demonstrates the e f f e c t i v e n e s s of the a d a p t i v e p r o c e d u r e i n p r o d u c i n g s o l u t i o n s of u n i f o r m a c c u r a c y w i t h a minimum of wasted work. F i g u r e 6 shows the e n e r g i e s computed and a p l o t of the e x a c t v a l u e s E - C ?2 - 4 a 1 ) V z = ( P 2 + 4 ) ^  (5.3) 1 04 At P=20, the d e v i a t i o n of the c a l c u l a t e d energy from the e x a c t v a l u e i s about 1%. The energy e s t i m a t e t e n d s t o be l e s s a c c u r a t e than the s o l u t i o n e s t i m a t e f o r l a r g e r v a l u e s of P, s i n c e as P i n c r e a s e s , the volume i n t e g r a l term of (3.41) c o n t r i b u t e s more t o t h e t o t a l energy and the e x p o n e n t i a t i o n of ^ tends t o magnify the e r r o r i n "-f . F i n a l l y , t o p r o v i d e a v i s u a l r e p r e s e n t a t i o n of some of the f e a t u r e s of the m u l t i - g r i d a l g o r i t h m d i s c u s s e d i n the p r e v i o u s c h a p t e r , the e f f e c t s of r e l a x a t i o n sweeps and c o a r s e g r i d c o r r e c t i o n s on the r e s i d u a l s of a t y p i c a l (P=4) problem ar e shown i n F i g u r e s 7 and 8. The s o l i d l i n e i n F i g u r e 7 c o n n e c t s the v a l u e s of the r e s i d u a l s r;,j , i = 0, ... , 32; j = 9 as computed on a 33x17 g r i d i m m e d i a t e l y f o l l o w i n g i n t e r p o l a t i o n of the s o l u t i o n computed on a 17x17 g r i d . The h i g h l y o s c i l l a t o r y n a t u r e of the r e s i d u a l s i s c l e a r l y v i s i b l e . However, a f t e r 5 r e l a x a t i o n sweeps, the r e s i d u a l s have e v i d e n t l y been smoothed. At t h i s p o i n t , a c o a r s e g r i d c o r r e c t i o n i s performed. The s o l i d l i n e of F i g u r e 8 shows the r e s i d u a l s as computed i m m e d i a t e l y a f t e r t h e i n t e r p o l a t i o n of the c o r r e c t i o n . A g a i n , the r e s i d u a l s show s t r o n g f l u c t u a t i o n s on the s c a l e of the g r i d s p a c i n g . F o l l o w i n g t h e a p p l i c a t i o n of 3 more r e l a x a t i o n sweeps, t h e r e s i d u a l s a r e once a g a i n smoothed and a n o t h e r c o a r s e g r i d c o r r e c t i o n can now be performed. For a l l of the above t e s t p roblems, as w e l l as the problems t o be d e s c r i b e d i n t h e next two s e c t i o n s , 3 c o a r s e g r i d c o r r e c t i o n c y c l e s were g e n e r a l l y r e q u i r e d a t each l e v e l t o s o l v e w i t h i n the l e v e l of t r u n c a t i o n e r r o r . I t seemed t h a t 105 F i g u r e 6 Energy v s . Momentum f o r Model Problem 106 in F i g u r e 7 Pre-CGC Smoothing of R e s i d u a l s F i g u r e 8 Post-CGC Smoothing of R e s i d u a l s 108 the n o n - d e f i n i t e n e s s of the problem tended t o slow the o v e r a l l convergence r a t e , because when the problems were m o d i f i e d by imposing D i r i c h l e t c o n d i t i o n s as d e s c r i b e d i n the p r e v i o u s s e c t i o n , o n l y a s i n g l e c o r r e c t i o n c y c l e was u s u a l l y r e q u i r e d . I t i s p o s s i b l e t h a t a d i f f e r e n t r e l a x a t i o n scheme, or a r e f o r m u l a t i o n of the boundary c o n d i t i o n s might improve the performance, but thes e p o s s i b i l i t i e s have not y e t been i n v e s t i g a t e d . 5.3 N u m e r i c a l r e s u l t s f o r boosted b l a c k h o l e s T h i s s e c t i o n p r e s e n t s the r e s u l t s o b t a i n e d from a p p l i c a t i o n of the a d a p t i v e m u l t i - g r i d a l g o r i t h m t o the d i s c r e t i z e d boundary v a l u e problem (3 . 35) - ( 3 . 38) , w i t h Hi.] det e r m i n e d by one of HB of e q u a t i o n ( 3 . 2 6 ) . For both HB and , the problem was s o l v e d f o r a s e r i e s of l i n e a r momentum v a l u e s chosen t o c o r r e s p o n d w i t h t h o s e used by P i r a n i n h i s work on t h e problem. In a l l c a s e s the r a d i u s a was chosen t o be 1 and r c f o r a l l of the runs was 10~ 5. T a b l e V I I I l i s t s the c a l c u l a t e d t o t a l e n e r g i e s a s s o c i a t e d w i t h each momentum v a l u e u s i n g He and Hg . Note t h a t t h e r e i s v e r y l i t t l e d i f f e r e n c e i n the c a l c u l a t e d e n e r g i e s f o r a g i v e n momentum between the two c a s e s . F i g u r e 9 shows a p l o t of the da t a f o r Ha . Q u a l i t a t i v e l y , the p l o t i s v e r y s i m i l a r t o the one i n F i g u r e 6. J u d g i n g from the r e s u l t s of the p r e v i o u s s e c t i o n , the c a l c u l a t e d e n e r g i e s a r e p r o b a b l y a c c u r a t e t o w i t h i n a p e r c e n t or two. Both s e t s of r e s u l t s agree w i t h those o b t a i n e d by P i r a n t o w i t h i n two p e r c e n t . ( P i r a n does not s t a t e which of 109 F i g u r e 9 Energy v s . Momentum f o r Boosted B l a c k H o l e s 110 He or Hg was used i n the c a l c u l a t i o n of h i s r e s u l t s ) . F i g u r e p E (H^) E (H B) 1 .0 2.35 2.33 2.5 3.59 3.55 5.0 6.15 6.09 7.5 8.87 8.81 10.0 11.6 11.5 12.5 14.4 14.3 15.0 17..2 17.1 17.5 20.0 • 20.0 Ta b l e V I I I T o t a l energy v s . momentum f o r b o o s t e d h o l e s 10 shows the r a d i a l b e h a v i o r of "H' ( u s i n g s c o o r d i n a t e s ) f o r - f . ' t h r e e v a l u e s of P, u s i n g H B . P o r t i o n s of the same d a t a a r e p l o t t e d i n F i g u r e 11 a g a i n s t the s t a n d a r d r a d i a l c o o r d i n a t e (r=8.0 -> s=0.875). A l l of the s o l u t i o n s o b t a i n e d were v e r y n e a r l y s p h e r i c a l l y symmetric. F i g u r e 12 shows the s m a l l d e p a r t u r e from s p h e r i c a l symmetry a t s=0 of the same 's p l o t t e d i n the p r e v i o u s two f i g u r e s . P i r a n a l s o used the s o l u t i o n s he d e t e r m i n e d n u m e r i c a l l y t o c a l u l a t e the p o s i t i o n s of the apparent h o r i z o n s of the v a r i o u s b o o s t e d b l a c k h o l e s . T h i s i n f o r m a t i o n was then used t o e s t i m a t e the amount of g r a v i t a t i o n a l r a d i a t i o n which might be F i g u r e 10 S Dependence of Conformal F a c t o r of Boosted H o l e s F i g u r e 11 R Dependence of Conformal F a c t o r of B o o s t e d H o l e s 6 (TT/32) F i g u r e 12 A n g u l a r Dependence of Conformal F a c t o r of B o o s t e d H o l e s 114 p r e s e n t i n the systems. These c a l c u l a t i o n s have not y e t been performed w i t h the c u r r e n t d a t a . 5.4 N u m e r i c a l r e s u l t s f o r s p i n n i n g b l a c k h o l e s Another s e r i e s of m u l t i - g r i d runs were performed u s i n g the squared c o n f o r m a l e x t r i n s i c c u r v a t u r e g i v e n by e q u a t i o n ( 3 . 2 7 ) , a g a i n w i t h a=1 and T C = 1 0 - 5 . A p l o t of the c a l c u l a t e d t o t a l energy v e r s u s a n g u l a r momentum appears i n F i g u r e 13. The range of J v a l u e s used was a g a i n chosen t o f a c i l i t a t e c omparison w i t h P i r a n ' s r e s u l t s . R a d i a l c r o s s s e c t i o n s of the c o n f o r m a l f a c t o r f o r 3 v a l u e s of J a r e d i s p l a y e d u s i n g s and r c o o r d i n a t e s i n F i g u r e s 14 and 15 r e s p e c t i v e l y . The computed s o l u t i o n s f o r l a r g e v a l u e s of J show s i g n i f i c a n t d e p a r t u r e from s p h e r i c a l symmetry near the i n n e r boundary of the s o l u t i o n domain as can be seen i n F i g u r e 16. A p l o t of ~^ , f o r J=1000, i n t h e r e g i o n 1<:r<8, 0^9<x appears i n F i g u r e 17. The asymmetry near r=1 i s c l e a r l y v i s i b l e . Bowen and York have shown t h a t t h e sp a c e t i m e s g e n e r a t e d from e v o l u t i o n of the i n i t i a l d a t a s e t [ t ^ f . - j ,N-~'°KlJ ( J ) ] w i l l not be K e r r s p a c e t i m e s . However, they h y p o t h e s i z e t h a t the s p a c e t i m e s w i l l be a s y m p t o t i c a l l y K e r r ; t h a t i s , i t i s ex p e c t e d t h a t the r o t a t i n g b l a c k h o l e s w i l l emit g r a v i t a t i o n a l r a d i a t i o n and e v e n t u a l l y " s e t t l e down" t o a K e r r c o n f i g u r a t i o n . P i r a n and York used t h i s r e a s o n i n g t o e s t i m a t e the amount of g r a v i t a t i o n a l r a d i a t i o n which c o u l d be p r e s e n t i n the new s p a c e t i m e s . The C h r i s t o d o l o u f o r m u l a [12] r e l a t e s the t o t a l energy M of the ( n e u t r a l ) K e r r spacetime t o the 1<f 1tf F i g u r e 13 Energy v s . Momentum f o r S p i n n i n g B l a c k H o l e s F i g u r e 14 S Dependence of Conformal F a c t o r of S p i n n i n g H o l e s F i g u r e 15 R Dependence of Conformal F a c t o r of S p i n n i n g H o l e s 118 F i g u r e 16 A n g u l a r Dependence of Conformal F a c t o r of S p i n n i n g H o l e s 8-0 F i g u r e 17 Conformal F a c t o r of S p i n n i n g B l a c k Hole (J=1000) 120 i r r e d u c i b l e mass, M , and the a n g u l a r momentum J , of the h o l e M i 2 = +• S ^ (5.4) The i r r e d u c i b l e mass i s r e l a t e d t o the a r e a , A, of the apparent h o r i z o n of the K e r r h o l e as f o l l o w s / A \ ^  ^ -- (5.5) For the case of one of the new s p i n n i n g b l a c k h o l e s , Bowen and York argue t h a t the apparent h o r i z o n c o i n c i d e s w i t h the r=a s u r f a c e . The mass, M A H, a s s o c i a t e d w i t h the apparent h o r i z o n i s d e f i n e d as K A H = ( — ] ^  (5.6) V Iferv / The a r e a of the apparent h o r i z o n , A A H » can be e a s i l y c a l c u l a t e d n u m e r i c a l l y ( o n c e " ^ has been de t e r m i n e d ) from the f o l l o w i n g e x p r e s s i o n A AH ^ J J °t & , ^ ) s . n 6 c J c 3 d p / (5.7) o o which i n the c u r r e n t c a s e , i s s i m p l y AAM= I n ^ A ( a, 0 ) s\rs6 (5.8) Jo Making the assumptions t h a t the b l a c k h o l e w i l l not l o s e a n g u l a r momentum as i t s e t t l e s t o a K e r r c o n f i g u r a t i o n , and 121 t h a t M w i l l not d e c r e a s e , York and P i r a n c o n c l u d e t h a t the the q u a n t i t y M ( K A „ J T ) > (MA2;, +• J[ ) (5.9) 4 r~AA H r e p r e s e n t s an a s y m p t o t i c lower bound on t h e sum of the i r r e d u c i b l e and r o t a t i o n a l c o n t r i b u t i o n s t o the t o t a l energy of the system. I f t h i s i s the c a s e , then the q u a n t i t y AE E - K ( KAH , J ) (5.10) i s an upper l i m i t on the amount of g r a v i t a t i o n a l r a d i a t i o n p r e s e n t i n the system. F o l l o w i n g York and P i r a n , the v a l u e s of t h r e e parameters c h a r a c t e r i z i n g the new r o t a t i n g b l a c k h o l e s a r e p l o t t e d i n F i g u r e 18. The q u a n t i t i e s GJI = (5.11) and tV^ =- — (5.12) a r e a n a l a g o u s t o the K e r r a n g u l a r momentum parameter s e s which tends t o u n i t y as t h e a n g u l a r momentum of t h e K e r r h o l e 122 F i g u r e 18 A n g u l a r Momentum and Energy Parameters f o r S p i n n i n g H o l e s 123 becomes v e r y l a r g e . I f the assumptions of the p r e v i o u s p a r a g r a p h a r e v a l i d , then £^ and £u. r e p r e s e n t lower and upper bounds on the v a l u e of € f o r the K e r r b l a c k h o l e t o which the new h o l e i s a s y m p t o t i c . I t appears t h a t approaches an a s y m p t o t i c l i m i t of a p p r o x i m a t e l y .92; a l s o seems t o approach a l i m i t i n g v a l u e of about .98. For l a r g e v a l u e s of J , the computed e n e r g i e s a r e p r o b a b l y o n l y a c c u r a t e t o 1 or 2 p e r c e n t , so i t i s p o s s i b l e t h a t t h i s f a m i l y of h o l e s has an extreme K e r r l i m i t . These r e s u l t s d i f f e r from those d i s c u s s e d by P i r a n and York. They f i n d ^ and € u . t e n d i n g t o l i m i t i n g v a l u e s of about .33 and .55 r e s p e c t i v e l y . An e x a m i n a t i o n of P i r a n ' s code f o r the s o l u t i o n of the c o n s t r a i n t e q u a t i o n ( f o r the r o t a t i n g h o l e s o n l y ) s u g g e s t s t h a t the f a c t o r of 1/8 a p p e a r i n g i n the n o n - l i n e a r term of e q u a t i o n (3.13) may have been o m i t t e d i n the c a l c u l a t i o n . Because H s i s q u a d r a t i c i n J , the v a l u e s of J s u p p l i e d t o P i r a n ' s a l g o r i t h m p r o b a b l y c o r r e s p o n d e d t o a c t u a l v a l u e s which were a f a c t o r t lmes l a r g e r . Among o t h e r t h i n g s , t h i s i m p l i e s t h a t h i s a s y m p t o t i c l i m i t f o r 6^ s h o u l d a l s o be m u l t i p l i e d by / iT, which y i e l d s a v a l u e of about .93, which i s i n good agreement w i t h t h e c u r r e n t r e s u l t . ( I n f a c t a l l of t h e p r e v i o u s r e s u l t s f o r 6^ , and J/M^ H agree w i t h the c u r r e n t c a l c u l a t i o n s t o w i t h i n two p e r c e n t when the f a c t o r of / tT i s t a k e n i n t o a c c o u n t ) . F i n a l l y , i t would appear t h a t AE, as g i v e n by e q u a t i o n (5.10) i s c o n s i d e r a b l y s m a l l e r f o r l a r g e v a l u e s of J than had p r e v i o u s l y been c a l c u l a t e d . From T a b l e I X , i t can be seen t h a t the maximum v a l u e of A.E i s about 3% f o r the t h r e e c o n f i g u r a t i o n s of l a r g e s t a n g u l a r momentum. P i r a n and York had 124 J E M AH M(J,M ) AH A E (%) 1.0 2.05 2.03 2.05 0 3.0 2.33 2.23 2.33 0 10.0 3.48 3.03 3.45 1 30.0 5.76 4.67 5.67 2 100.0 10.4 8.07 10.2 2 300.0 18.0 13.7 17.5 3 1000.0 32.9 24.7 31.9 3 10000.0 1 04. 77.6 101. 3 Table IX V a r i o u s e n e r g i e s of s p i n n i n g b l a c k h o l e s o b t a i n e d a f i g u r e of about 25% f o r J=1000. T h i s would seem t o i n d i c a t e t h a t t h i s f a m i l y of b l a c k h o l e s may r a d i a t e l e s s than had been p r e v i o u s l y e x p e c t e d . However, no f i r m c o n c l u s i o n s can be made about the r a d i a t i v e n a t u r e of t h e s e systems u n t i l the i n i t i a l d a t a has a c t u a l l y been e v o l v e d . 125 CHAPTER 6 C o n c l u s i o n s and P o s s i b l e F u t u r e A p p l i c a t i o n s The r e s u l t s p r e s e n t e d i n the p r e v i o u s c h a p t e r show t h a t the m u l t i - g r i d method was q u i t e s u c c e s s f u l i n p r o d u c i n g n u m e r i c a l s o l u t i o n s of the H a m i l t o n i a n c o n s t r a i n t f o r the new f a m i l i e s of b l a c k h o l e s . A l t h o u g h the performance of the method was not o p t i m a l i n the sense t h a t t h r e e c o a r s e g r i d c o r r e c t i o n c y c l e s were g e n e r a l l y needed t o s o l v e a problem a t any g i v e n l e v e l of d i s c r e t i z a t i o n , the amount of work expended was s t i l l l i n e a r i n the number of f i n e g r i d p o i n t s used. Thus, the c o m p u t a t i o n a l advantage of u s i n g the method, as opposed t o a more t r a d i t i o n a l scheme such as SOR, i n c r e a s e d as f i n e r g r i d s were used. F u r t h e r m o r e , because of t h e n o n - d e f i n i t e n e s s of the problems, s t r a i g h t f o r w a r d r e l a x a t i o n methods c o u l d not have been used t o s o l v e the d e s i r e d e q u a t i o n s . The o n l y o t h e r method examined which was c a p a b l e of p r o d u c i n g s o l u t i o n s t o the " r e a l " problems - the Newton-nested d i s s e c t i o n scheme -was a l s o o u t p e r f o r m e d by the m u l t i - g r i d method. F i n a l l y , the f e a s i b i l i t y of u s i n g a d a p t i v e t e c h n i q u e s i n c o n j u n c t i o n w i t h t h e m u l t i - g r i d method was d e m o n s t r a t e d . I t i s d i f f i c u l t t o imagine how a d a p t i v e d i s c r e t i z a t i o n c o u l d be e a s i l y i n c o r p o r a t e d i n t o the o t h e r t h r e e methods. Of a l l the problems e n c o u n t e r e d i n t h e c o u r s e of 126 d e s i g n i n g and debugging the m u l t i - g r i d program used t o o b t a i n the r e s u l t s of Chapter 5 , the n o n - d e f i n i t e n e s s of the d i f f e r e n c e e q u a t i o n s and the t r e a t m e n t of the n o n - D i r i c h l e t boundary c o n d i t i o n s p r o v e d t o be the most t r o u b l e s o m e . The f a c t t h a t the d i f f e r e n c e systems were n o n - d e f i n i t e went u n d e t e c t e d t h r o u g h the i n i t i a l t e s t i n g of the program, where o n l y two or t h r e e r e l a t i v e l y c o a r s e l e v e l s of d i s c r e t i z a t i o n were used. In t h e s e t e s t s , the d i f f e r e n c e e q u a t i o n s were " s o l v e d " on the c o a r s e s t l e v e l by r e l a x a t i o n . Because the s o l u t i o n p r o c e s s on a g i v e n g r i d was t e r m i n a t e d when the norm of the r e s i d u a l s was l e s s than some convergence c r i t e r i o n , the f a c t t h a t a few components were b e i n g m a g n i f i e d by the r e l a x a t i o n sweeps w h i l e the r e s t were b e i n g damped, was not o b v i o u s . I t was o n l y when r e l a t i v e l y f i n e g r i d s were f i n a l l y used t h a t i t was seen t h a t the c o a r s e g r i d c o r r e c t i o n s were not e f f e c t i v e and t h a t o v e r a l l d i v e r g e n c e was a c t u a l l y o c c u r i n g . I t was a t t h i s p o i n t t h a t the r o u t i n e s f o r s o l v i n g the c o a r s e s t systems d i r e c t l y were coded and added t o the program. A l t h o u g h B r a n d t ' s s u g g e s t i o n s f o r the smoothing of the n o n - D i r i c h l e t boundary c o n d i t i o n s were implemented i n t h e program, i t i s not c l e a r whether the t r e a t m e n t i s c o m p l e t e l y s a t i s f a c t o r y . Because of the i n t r o d u c t i o n of " o f f - b o u n d a r y " p o i n t s t o a l l o w the use of c e n t r e d d i f f e r e n c e a p p r o x i m a t i o n s everywhere, the c o a r s e g r i d c o r r e c t i o n p r o c e s s i n v o l v e d an e x t r a p o l a t i o n of the f i n e g r i d f u n c t i o n v a l u e s near the boundary t o produce a c o a r s e g r i d r e p r e s e n t a t i o n of the f i n e g r i d f u n c t i o n . I t i s p o s s i b l e t h a t the i n a c c u r a c y i n v o l v e d i n 127 t h i s e x t r a p o l a t i o n c o n t r i b u t e d t o the s l o w e r o v e r a l l r a t e of convergence of the method when the p r o p e r boundary c o n d i t i o n s were used as compared t o t h a t o b t a i n e d when D i r i c h l e t c o n d t i o n s were imposed (as i n the f i r s t s e r i e s of n u m e r i c a l t e s t s d e s c r i b e d i n the p r e v i o u s c h a p t e r ) . The t r e a t m e n t of the boundary c o n d i t i o n s c e r t a i n l y w a r r a n t s f u r t h e r i n v e s t i g a t i o n . I t would p r o b a b l y be i n s t r u c t i v e t o i n v e s t i g a t e the e f f e c t of u s i n g n o n - c e n t r e d d i f f e r e n c e e q u a t i o n s near b o u n d a r i e s , so as t o a v o i d the i n t r o d u c t i o n of g r i d p o i n t s e x t r a n e o u s t o the s o l u t i o n domain. As noted i n the i n t r o d u c t o r y c h a p t e r of t h i s t h e s i s , one of the main purposes of a p p l y i n g the m u l t i - g r i d method t o the H a m i l t o n i a n c o n s t r a i n t f o r the new b l a c k h o l e f a m i l i e s was t o i n v e s t i g a t e i t s e f f e c t i v e n e s s as a p o s s i b l e t o o l f o r the s o l u t i o n of more c o m p l i c a t e d i n i t i a l v a l u e problems. G e n e r a l i n i t i a l v a l u e problems i n v o l v i n g a l l t h r e e s p a t i a l d i m e n s i o n s , as w e l l as most t w o - d i m e n s i o n a l problems w i l l u ndoubtedly r e q u i r e the n u m e r i c a l s o l u t i o n of a l l f o u r of the i n i t i a l v a l u e e q u a t i o n s . The m u l t i - g r i d method has p r e v i o u s l y been a p p l i e d t o n o n - l i n e a r systems of e l l i p t i c e q u a t i o n s [ 4 ] , [ 8 ] . The b a s i c p r o c e s s e s i n v o l v e d a r e the same as t h o s e f o r a s c a l a r boundary v a l u e problem. A g a i n , i t has been the e x p e r i e n c e of r e s e a r c h e r s t h a t the c o n s t r u c t i o n of a p r o p e r l y smoothing r e l a x a t i o n scheme f o r t h e g i v e n system of e q u a t i o n s i s the f i r s t p r i o r i t y f o r the e f f i c i e n t o p e r a t i o n of the m u l t i - g r i d a l g o r i t h m . Moreover, i t seems t h a t t h e r e are v e r y few g e n e r a l " r u l e s " f o r d e s i g n i n g good r e l a x a t i o n schemes f o r e q u a t i o n s r e s u l t i n g from the d i s c r e t i z a t i o n of an a r b i t r a r y 128 e l l i p t i c system. In a d d i t i o n t o the d e s i g n of a p p r o p r i a t e r e l a x a t i o n methods, o t h e r f a c t o r s which would have t o be d e a l t w i t h i n the' m u l t i - g r i d s o l u t i o n of a g e n e r a l i n i t i a l v a l u e problem i n c l u d e the t r e a t m e n t of boundary c o n d i t i o n s and the unboundedness of the domain of the problem i n the case of a s y m p t o t i c a l l y f l a t i n i t i a l d a t a . One of the b a s i c p r e m i s e s of the m u l t i - g r i d p h i l o s o p h y i s t h a t u s e f u l i n f o r m a t i o n can be e x t r a c t e d from a v e r y c o a r s e a p p r o x i m a t i o n of the c o n t i n u o u s problem of i n t e r e s t . For v e r y l a r g e domains, i t may not always be p o s s i b l e t o m e a n i n g f u l l y a p p r o x i m a t e the problem on a g r i d w i t h so few unknowns t h a t the s o l u t i o n of the r e s u l t i n g system of a l g e b r a i c e q u a t i o n s may be performed w i t h an e s s e n t i a l l y n e g l i g i b l e amount of work. In such c a s e s , the work expended i n the m u l t i - g r i d s o l u t i o n p r o c e s s may not be p r o p o r t i o n a l t o the number of f i n e g r i d p o i n t s . On the p o s i t i v e s i d e , one might expect t h a t the m u l t i -g r i d method, combined w i t h a d a p t i v e d i s c r e t i z a t i o n , would a l l o w a more a c c u r a t e s o l u t i o n of a g i v e n i n i t i a l v a l u e problem than SOR, f o r example, f o r a g i v e n amount of c o m p u t a t i o n a l work. As s t a t e d a t the end of Chapter 4, the p r i c e t h a t must be p a i d f o r the c o m p u t a t i o n a l e f f i c i e n c y of the m u l t i - g r i d method i s the s u b s t a n t i a l human e f f o r t r e q u i r e d t o implement i t . In comparison t o the development of the program l i s t e d i n Appendix B, t h e c o d i n g o f t h e p o i n t - and l i n e - S O R methods was almost t r i v i a l . I t must be noted, however, t h a t the the major p o r t i o n of the m u l t i - g r i d program was d e s i g n e d so t h a t i t c o u l d be e a s i l y m o d i f i e d f o r the 129 s o l u t i o n of a d i f f e r e n t boundary v a l u e problem. Brandt makes the o b s e r v a t i o n , which would appear t o be f a i r l y a c c u r a t e , t h a t most of the r o u t i n e s used i n a w e l l w r i t t e n m u l t i - g r i d program f o r the s o l u t i o n of a p a r t i c u l a r boundary v a l u e problem can be used w i t h l i t t l e or no m o d i f i c a t i o n f o r o t h e r s i m i l a r p r oblems. T h i s i s l i t t l e c o n s o l a t i o n , however, t o a p r o s p e c t i v e " m u l t i - g r i d d e r " who may not have a c c e s s t o an e x i s t i n g p i e c e of s o f t w a r e which c o u l d be e a s i l y m o d i f i e d f o r her use. " G e n e r a l purpose" m u l t i - g r i d s o f t w a r e has been w r i t t e n , but i t i s p r o b a b l e t h a t a f a i r amount of e f f o r t and e x p e r t i s e would be r e q u i r e d t o use i t e f f e c t i v e l y . F i n a l l y , a p a r t from the i n i t i a l v a l u e e q u a t i o n s , o t h e r e l l i p t i c e q u a t i o n s , which c o u l d be s o l v e d u s i n g the m u l t i - g r i d method, a r i s e i n some c u r r e n t and proposed approaches t o the n u m e r i c a l e v o l u t i o n of g r a v i t a t i o n a l i n i t i a l d a t a [ 3 2 ] , [ 4 0 ] . In a d d i t i o n , t h e r e has been some s u c c e s s i n t h e a p p l i c a t i o n of the m u l t i - g r i d scheme t o time dependent problems. For example, Brandt e t a l [ 7 ] , have a p p l i e d the method t o h y d r o d y n a m i c a l problems w h e r e i n some of the v a r i a b l e s a r e t r e a t e d i m p l i c i t l y i n time t o a l l o w the use of a r e l a t i v e l y l a r g e time s t e p . The p o s s i b i l i t y of u s i n g a s i m i l a r t e c h n i q u e f o r g r a v i t a t i o n a l e v o l u t i o n problems would seem t o be worthy of i n v e s t i g a t i o n . B i b l i o g r a p h y Ames, W i l l i a m F. N u m e r i c a l Methods f o r P a r t i a l  D i f f e r e n t i a l Equat i o n s . New York: Academic P r e s s , 1964. A r n o w i t t , R., S. Deser and C.W. M i s n e r . "The Dynamics of G e n e r a l R e l a t i v i t y . " In G r a v i t a t i o n : An I n t r o d u c t i o n t o C u r r e n t  R e s e a r c h . Ed. L o u i s W i t t e n . New York: John W i l e y and Sons, 1962, pp. 227-265. Bowen, J e f f r e y M. and James W. York, J r . "Time-asymmetric I n i t i a l Data f o r B l a c k H o l e s and B l a c k - H o l e C o l l i s i o n s . " P h y s i c a l Review Dj_ 21 (1980), 2047-2056. B r a n d t , A. " M u l t i - l e v e l A d a p t i v e S o l u t i o n s t o Boundary-value Problems." Mathematics of  Computation, 31 ( 1 9 7 7 ) , 333-390. ; . " M u l t i - l e v e l A d a p t i v e S o l u t i o n s 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 . " In N u m e r i c a l S o f t w a r e I I I . Ed. John RiTce. New York: Academic P r e s s , 1977, pp. 277-318. " M u l t i - l e v e l A d a p t i v e T e c h n i q u e s (MLAT) f o r S i n g u l a r P e r t u r b a t i o n Problems." In N u m e r i c a l A n a l y s i s of S i n g u l a r P e r t u r b a t i o n  Problems. Eds. P.W. Hemker and J . J . M i l l e r . London: Academic P r e s s , 1979, pp. 53-147. B r a n d t , A., J.E. Dendy, J r . and Hans Ruppel "The M u l t i - g r i d Method f o r S e m i - i m p l i c i t Hydrodynamics Codes." J o u r n a l of C o m p u t a t i o n a l  P h y s i c s , 34 (1980), 348-370. B r a n d t , A. and Nathan D i n a r . " M u l t i - g r i d S o l u t i o n s t o E l l i p t i c Flow Problems." In N u m e r i c a l  Methods f o r 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 . Ed. S.V. P a r k e r . New York: Academic P r e s s , 1979, pp. 53-147. B r i l l , D.R. "General R e l a t i v i t y : S e l e c t e d T o p i c s of C u r r e n t I n t e r e s t . " S u p p l . Nuovo Cimento, 2 (1964), 1-56. 131 10. B r u h a t , Yvonne. "The Cauchy Problem." In G r a v i t a t i o n : An I n t r o d u c t i o n t o C u r r e n t R e s e a r c h . Ed. L o u i s W i t t e n . New York: John W i l e y and Sons, 1962, pp. 130-168. 11. Choquet-Bruhat, Yvonne and James W. York, J r . "The Cauchy Problem." In G e n e r a l R e l a t i v i t y  and G r a v i t a t i o n . Ed. A. H e l d . New York: Plenum P r e s s , 1980, pp. 99-172. 12. C h r i s t o d o l o u , D. " R e v e r s i b l e and I r r e v e r s i b l e T r a n s f o r m a t i o n s i n B l a c k H o l e P h y s i c s . " P h y s i c a l  Review L e t t e r s , 25 (1970), 1596-1597. 13. D a v i s , P h i l i p J . and P h i l i p R a b i n o w i t z . Methods of N u m e r i c a l I n t e g r a t i o n . New York: Academic P r e s s , 1975. 14. E p p l e y , Kenneth. " E v o l u t i o n of Time-symmetric G r a v i t a t i o n a l Waves: I n i t i a l Data and Apparent H o r i z o n s . " P h y s i c a l Review D, 16 (1977), 1609-1614. 15. . "Pure G r a v i t a t i o n a l Waves." In Sources of G r a v i t a t i o n a l R a d i a t i o n . Ed. L a r r y Smarr. Cambridge: Cambridge U n i v e r s i t y P r e s s , 1979, pp. 275-291. 16. Fox, L. " F i n i t e D i f f e r e n c e Methods f o r E l l i p t i c Boundary V a l u e Problems." In The S t a t e of the A r t i n  N u m e r i c a l A n a l y s i s . Ed. D. J a c o b s . New York: Academic P r e s s , 1977, pp. 799-881. 17. George, J.A. "Nested D i s s e c t i o n of a R e g u l a r F i n i t e Element Mesh." SIAM J o u r n a l of N u m e r i c a l  A n a l y s i s , 10 (1973), 345-363 18. George, A l a n and Joseph W-H. L i u . Computer S o l u t i o n of L arge Sparse P o s i t i v e D e f i n i t e Systems. Englewood C l i f f s , N.J.: P r e n t i c e - H a l l , 1981. 19. G r i f f i t h s , D a v i d F. and A.R. M i t c h e l l . The F i n i t e D i f f e r e n c e Method i n 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 . New York: John W i l e y and Sons, 1980. 20. Hawking, S.W. and G.F.R. E l l i s . The Large S c a l e S t r u c t u r e of Spacetime. Cambridge: Cambridge U n i v e r s i t y P r e s s , 1973. 21. Hoffman, A . J . , S.M. M i c h a e l and D.J. Rose. " C o m p l e x i t y Bounds f o r R e g u l a r F i n i t e D i f f e r e n c e and F i n i t e - e l e m e n t G r i d s . " SIAM J o u r n a l of  N u m e r i c a l A n a l y s i s , 10 (1973), 364-369. 132 22. James, M.L., G.M. Smith and J.C. W o l f o r d . A p p l i e d N u m e r i c a l Methods f o r D i g i t a l Computation. New York: Harper Row, 1972. 23. Kuchar, K a r l . " C a n o n i c a l Q u a n t i z a t i o n of C y l i n d r i c a l G r a v i t a t i o n a l Waves." P h y s i c a l Review D, 4 (1971), 955-986. 24. . " C a n o n i c a l Q u a n t i z a t i o n of G r a v i t y " In R e l a t i v i t y , A s t r o p h y s i c s and Cosmology. Ed. Werner I s r a e l . B o s t o n : D. R e i d e l , 1973. 25. Marchuk, G.I. Methods of N u m e r i c a l M a t h e m a t i c s . New York: S p r i n g e r - V e r l a g , 1975. 26. M i s n e r , C h a r l e s W. "Wormhole I n i t i a l C o n d i t i o n s . " P h y s i c a l Review, 118 (1964), 1110-1111. 27. M i s n e r , C h a r l e s W., K i p S. Thorne and John A. Wheeler. G r a v i t a t i o n . San F r a n c i s c o : W.H. Freeman and Co., 1973. 28. Nakamura, T., K. Maeda, S. Miyami and M. S a s a k i . " R e l a t i v i s t i c C o l l a p s e of an A x i a l l y Symmetric S t a r . I . " P r o g r e s s of T h e o r e t i c a l P h y s i c s , 63 (1980), 1229-1244. 29. 0 Murchadha, N. and James W. York, J r . " I n i t i a l V a l u e Problem of G e n e r a l R e l a t i v i t y . I . G e n e r a l F o r m u l a t i o n and P h y s i c a l I n t e r p r e t a t i o n . " P h y s i c a l Review D, 10 (1974), 428-436. 30. O r t e g a , J.M. and W.C. R h e i n b o l d t . I t e r a t i v e S o l u t i o n of Non-1inear E q u a t i o n s i n S e v e r a l  V a r i a b l e s . New Y o r k : Academic P r e s s , 1970 31. P i r a n , T s v i . " N u m e r i c a l Codes f o r C y l i n d r i c a l G e n e r a l R e l a t i v i s t i c Systems." J o u r n a l of C o m p u t a t i o n a l  P h y s i c s , 35 (1980), 254-283. 32. . "Problems and S o l u t i o n s i n N u m e r i c a l R e l a t i v i t y . " ( u n p u b l i s h e d ) . 33. . " N u m e r i c a l R e l a t i v i t y . " ( u n p u b l i s h e d ) . 34. Smarr, L a r r y . " B a s i c Concepts i n F i n i t e D i f f e r e n c i n g 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 . " I n Sources of  G r a v i t a t i o n a l R a d i a t i o n . Ed. L a r r y Smarr. Cambridge: Cambridge U n i v e r s i t y P r e s s , 1979, pp. 139-159 133 35. Smarr, L a r r y , A. Cadez, B. d e W i t t and K. E p p l e y . " C o l l i s i o n of Two B l a c k H o l e s : T h e o r e t i c a l Framework." P h y s i c a l Review D, 14 (1976), 2443-2452. 36. V a r g a , R i c h a r d S. M a t r i x I t e r a t i v e A n a l y s i s . Englewood C l i f f s , N.J.: P r e n t i c e - H a l l , 1962. 37. Weinberg, S t e v e n . G r a v i t a t i o n and Cosmology: P r i n c i p l e s and A p p l i c a t i o n s of the G e n e r a l  Theory of R e l a t i v i t y . New York: John W i l e y and Sons, 1972. 38. Young, D.M. I t e r a t i v e S o l u t i o n of Large L i n e a r Systems. New York: Academic P r e s s , 1971. 39. York, James W., J r . " C o n f o r m a l l y I n v a r i a n t O r t h o g o n a l D e c o m p o s i t i o n of Symmetric Tensors on Riemannian M a n i f o l d s and the I n i t i a l V a l u e Problem of G e n e r a l R e l a t i v i t y . " J o u r n a l of M a t h e m a t i c a l P h y s i c s , 14 (1973), 456-464. 40. . " K i n e m a t i c s and Dynamics of R e l a t i v i t y . " In Sources of G r a v i t a t i o n a l R a d i a t i o n . Ed. L a r r y Smarr. Cambridge: Cambridge U n i v e r s i t y P r e s s , 1979, pp. 83-126 41. York, James W., J r . and T s v i P i r a n . "The I n i t i a l V a l u e Problem and Beyond." ( u n p u b l i s h e d ) . 134 APPENDIX A B a s i c N u m e r i c a l Techniques f o r E l l i p t i c PDE's T h i s appendix r e v i e w s some of the b a s i c c o n c e p t s and methods used i n the n u m e r i c a l s o l u t i o n of e l l i p t i c boundary-v a l u e problems u s i n g f i n i t e d i f f e r e n c e s . No attempt i s made t o d e s c r i b e a l l of the a v a i l a b l e t e c h n i q u e s , nor i s any p a r t i c u l a r method p r e s e n t e d i n d e t a i l . The m a t e r i a l i n s e c t i o n s A.1 - A.2 i s i n t e n d e d t o g i v e the r e a d e r a g e n e r a l i d e a of the methods used t o c o n v e r t boundary-value problems, such as the one d e s c r i b e d i n Chapter 3, t o f i n i t e d i f f e r e n c e form. S e c t i o n s A.3 - A.4 d e s c r i b e some of the " c l a s s i c a l " methods of s o l v i n g the systems of a l g e b r a i c e q u a t i o n s which r e s u l t from such problems, w i t h emphasis p l a c e d on the p a r t i c u l a r methods which were used i n t h i s t h e s i s f o r the purpose of comparison w i t h the m u l t i - g r i d method d e t a i l e d i n Chapter 4. 135 A. 1 D i s c r e t i z a t i o n A g e n e r a l boundary-value problem i n d d i m e n s i o n s may be w r i t t e n i n the form L l u ( x ) j - £ Cx) j x= Cx, ,...!,)) £ JL c |Rd- ( A . la) 8 S u t x ) ^ = c^CO • X e 3J1 (A. 1b) Here L and B a r e l i n e a r or n o n - l i n e a r d i f f e r e n t i a l o p e r a t o r s , f ( x ) and g ( x ) are s p e c i f i e d f u n c t i o n s and u ( x ) i s the unknown f u n c t i o n . ; For s i m p l i c i t y , assume t h a t the domain ^ i s a r e c t a n g u l a r r e g i o n of R2" as d e p i c t e d i n F i g u r e 19a. Thus = [ U A J ) I a < X v < b j c ^ .< d \ (A. 2) The f i r s t s t e p i n the d i s c r e t i z a t i o n p r o c e s s i s t o r e p l a c e the c o n t i n u o u s domain by a d i s c r e t e domain. The s i m p l e s t way t o do t h i s i s t o i n t r o d u c e a g r i d on the c o n t i n u o u s domain as shown i n F i g u r e 19b. In t h i s case the g r i d has u n i f o r m s p a c i n g s h i n the x d i r e c t i o n and k i n the y d i r e c t i o n and i s t h e r e f o r e c a l l e d a u n i f o r m g r i d . The f i n i t e s e t of g r i d p o i n t s i s the d i s c r e t e domain S)-K Xl h -  \ Cxi a « x ; * b J c < \ \ 4 d \ (A.3) CL (a) X o (b) F i g u r e 19 D i s c r e t i z a t i o n of Domain of Boundary V a l u e Problem 137 where % t B OL + t K j u -- o, A , • •• , Kl , M = h""( b -ex) « « s { A * 4 ) ^ ^ c v j K • j * <v, • ••, M • n * k ( d - e j The boundary of the d i s c r e t e domain, 2>-£L , i s made up of those g r i d p o i n t s which l i e on the boundary of the c o n t i n u o u s domain. A r e s t r i c t i o n o p e r a t o r I i s now i n t r o d u c e d which o p e r a t e s on an a r b i t r a r y f u n c t i o n d e f i n e d on -ft , y i e l d i n g the f i n i t e s e t of f u n c t i o n v a l u e s c o r r e s p o n d i n g t o the elements of f l h . The second s t e p of the d i s c r e t i z a t i o n p r o c e s s i n v o l v e s r e p l a c i n g the c o n t i n u o u s d i f f e r e n t i a l o p e r a t o r s L and B w i t h d i s c r e t e o p e r a t o r s and B^ such t h a t the s o l u t i o n u h of the system - an f l * (A.5a) 3 h L A H = X ^ G j o n 7)^ (A. 5b) i s an a p p r o x i m a t i o n t o I K u . One method f o r c o n s t r u c t i n g such d i s c r e t e o p e r a t o r s , the t e c h n i q u e of f i n i t e d i f f e r e n c i n g , i s the t o p i c of the next s e c t i o n . A.2 F i n i t e D i f f e r e n c e s Assume t h a t the s o l u t i o n u ( x , y ) t o the above problem i s s u f f i c i e n t l y w e l l behaved t h a t i t may be expanded i n a T a y l o r s e r i e s about any p o i n t i n the domain -O. . 138 S p e c i f i c a l l y , c o n s i d e r the f o l l o w i n g e x p a n s i o n f o r the v a l u e u(a+( i + 1 )h,c +jk) 5 u ( x ^ i , y j ) S o l v i n g the above f o r ^ f ^ ( x < ' f Y j ) y i e l d s I n t r o d u c i n g the s h o r t - h a n d n o t a t i o n , the f i r s t o r d e r f o r w a r d d i f f e r e n c e a p p r o x i m a t i o n f o r i s L A U A / - u;,j ( A . 8 ) h The a p p r o x i m a t i o n i s c a l l e d f i r s t o r d e r s i n c e the t r u n c a t i o n  e r r o r of the e x p r e s s i o n , r e s u l t i n g from t r u n c a t i o n of the T a y l o r s e r i e s , i s of o r d e r h. In a s i m i l a r f a s h i o n , an e x p a n s i o n f o r u L - i 0 may be performed ( A . 9 ) T h i s y i e l d s the f i r s t o r d e r backward d i f f e r e n c e f o r m u l a f o r ULl, I — U. , j ( A . 1 0 ) 139 Another approximation may be constructed by s u b t r a c t i n g (A.9) from (A.6) (A.11) y i e l d i n g the second order c e n t r a l d i f f e r e n c e approximation f o r •ax LA w + , 3 - L.L j (A.12) D i f f e r e n c e approximations f o r higher d e r i v a t i v e s may be obtained i n a s i m i l a r f a s h i o n . For example, the second order c e n t r a l d i f f e r e n c e approximation of * i s U U - L \ - 2. ,\ * Uj-i, j (A.13) Second order c e n t r a l d i f f e r e n c e formulae are probably the most commonly used approximations i n f i n i t e d i f f e r e n c i n g of e l l i p t i c boundary value problems. Higher order approximations, while more accurate f o r a given g r i d spacing, i n general re q u i r e more work n u m e r i c a l l y , both for t h e i r e v a l u a t i o n and for the s o l u t i o n of the a l g e b r a i c equations r e s u l t i n g from t h e i r use. No elementary d i s c u s s i o n of d i f f e r e n c e techniques fo r e l l i p t i c equations would be complete without mention of the 5-point approximation f o r the L a p l a c i a n , V^u, i n the s p e c i a l case when the g r i d spacing i n the y d i r e c t i o n i s a l s o h UC.-i,^ » U i-->, j <Ui,j tuU.;,j.) - 4u:,] 140 This, too, i s a second order central difference formula. As a f i n a l example of simple d i f f e r e n c i n g techniques, consider the following expression V' [ o-Cx.Oj) V U L(>L , L J)] (A.14) where a(x,y) i s a s p e c i f i e d function. Expressions of t h i s sort often a r i s e in physical applications which involve some sort of macroscopic conservation law. For example, using Gauss's theorem V ' [ aCx,u>) V ixCx.cp] d x diu^ C L C * , ^ ) 0 U L C X.^> • dfn (A.15) n If (A.14) i s d i s c r e t i z e d in the following manner, using second order central differences ^ [_ a ; + v2 , \ - UL.-, j ) - a £ . V t , j Cu,-, j - a.;-, j ) ] a . - . j i ' / ! ( u ; , - r > - u . £ , j ) - a i. i-vz ( u ; , j - Ur , j ->) j ^ A 1 g j i where a l r i / j =. a(xi±i ,y ); a;,j*i = a(.xj ,y4 + £ ), then the solution of the difference equations w i l l automatically s a t i s f y a discrete version of (A.15) u M t 1 - 1 - H. \ ^  C f^M.Vt, i ( u ^ , j - u^.j) ] " h [ a.j,; duo,-, - a - i , j )] J K . 0 141 However, i f (A.14) i s r e w r i t t e n as ex Q ) 2 I J L •» ^o- ^Ly-and d i f f e r e n c e d as f o l l o w s L l L*\ • j 2, u;,i * U then the s o l u t i o n of the d i f f e r e n c e e q u a t i o n s w i l l not obey a d i s c r e t e v e r s i o n of ( A.15). D i f f e r e n c e e x p r e s s i o n s such as (A.16) a r e c a l l e d c o n s e r v a t i v e , s i n c e the d i s c r e t e unknowns d i f f e r e n c e schemes a r e d i s c u s s e d i n d e pth i n [ 2 5 ] . G e n e r a l and [ 1 9 ] . There a r e t h r e e major t y p e s of boundary c o n d i t i o n s e n c o u n t e r e d i n b o undary-value problems: 1) D i r i c h l e t c o n d i t i o n s , where the v a l u e of the unknown f u n c t i o n i s g i v e n on the boundary, 2) Neumann c o n d i t i o n s , where the normal d e r i v a t i v e of the f u n c t i o n on t h e boundary i s s p e c i f i e d , and 3) Robbins or mixed c o n d i t i o n s , where some c o m b i n a t i o n of the f u n c t i o n and i t s f i r s t d e r i v a t i v e on the boundary a r e g i v e n . D i r i c h l e t c o n d i t i o n s p r e s e n t l i t l e d i f f i c u l t y when f o r m u l a t i n g f i n i t e d i f f e r e n c e schemes. However, t o implement the o t h e r two t y p e s of c o n d i t i o n s , i t i s o f t e n u s e f u l t o i n t r o d u c e e x t r a l i n e s of g r i d p o i n t s which l i e o u t s i d e o f the a c t u a l c o n t i n u o u s domain, as d e p i c t e d i n F i g u r e 20. T h i s a l l o w s a c e n t r e d d i f f e r e n c e a p p r o x i m a t i o n t o the normal d e r i v a t i v e t o obey a d i s c r e t e v e r s i o n of a c o n s e r v a t i o n law. C o n s e r v a t i v e f i n i t e d i f f e r e n c i n g t e c h n i q u e s a r e t r e a t e d i n d e t a i l i n [1] 142 i i i *-- - *-- B - -Q I i i * 7 *----/ ! / Q : I M T B O O u C e o T O I M P L E M E N T E x t r a G r i d P o i n t s F i g u r e 20 - D i s c r e t i z a t i o n of Boundary C o n d i t i o n s 143 be employed. The net r e s u l t of r e p l a c i n g the o p e r a t o r s L and B of (A.1) w i t h f i n i t e d i f f e r e n c e a p p r o x i m a t i o n s L and B i s t o c o n v e r t the c o n t i n u o u s d i f f e r e n t i a l problem i n t o a f i n i t e system of a l g e b r a i c e q u a t i o n s . In g e n e r a l , t h e r e w i l l be one such e q u a t i o n f o r each g r i d p o i n t i n Xl" (XI*" - "^Sl^ ) f o r problems w i t h Neumann/Robbins ( D i r i c h l e t ) c o n d i t i o n s , r e s u l t i n g from the d i s c r e t i z a t i o n of the i n t e r i o r e q u a t i o n ( A . 1 a ) . In a d d i t i o n , the d i s c r e t i z a t i o n of d i f f e r e n t i a l boundary c o n d i t i o n s w i l l e f f e c t i v e l y s u p p l y an e q u a t i o n f o r the v a l u e of the f u n c t i o n a t any e x t r a g r i d p o i n t s which may have been i n t r o d u c e d as d e s c r i b e d above. These " e x t r a -boundary" f u n c t i o n v a l u e s w i l l t y p i c a l l y be r e f e r e n c e d by one i n t e r i o r d i f f e r e n c e e q u a t i o n and one boundary e q u a t i o n , and i t i s common p r a c t i c e t o e l i m i n a t e t h i s v a l u e from the two e q u a t i o n s , e f f e c t i v e l y i n c o r p o r a t i n g the boundary e q u a t i o n i n t o t h e i n t e r i o r e q u a t i o n . In any c a s e , a p r o p e r l y c o n s t r u c t e d d i f f e r e n c e scheme u s i n g a g r i d w i t h a t o t a l of n p o i n t s w i l l r e s u l t i n a system of n e q u a t i o n s f o r the n v a l u e s of t h e g r i d f u n c t i o n u . Because a t y p i c a l f i n i t e d i f f e r e n c e a p p r o x i m a t i o n i n v o l v e s o n l y the v a l u e of the f u n c t i o n a t the p o i n t of a p p l i c a t i o n and a few n e i g h b o r i n g f u n c t i o n v a l u e s , any g i v e n e q u a t i o n i n the system w i l l i n v o l v e o n l y a few of the unknowns. The next s e c t i o n s of t h i s appendix d e s c r i b e some of the methods f o r s o l v i n g such s p a r s e systems of e q u a t i o n s n u m e r i c a l l y . 1 44 A.3 Methods f o r s o l v i n g l i n e a r systems I f the o p e r a t o r s L and B i n (A.1) a r e b o t h l i n e a r , then the system of n e q u a t i o n s r e s u l t i n g from the d i s c r e t i z a t i o n of the complete boundary v a l u e problem w i l l a l s o be l i n e a r and may be w r i t t e n i n the form where A i s a n x n m a t r i x and u and b are n-component v e c t o r s . For the purpose of p r o v i d i n g rough comparisons of the e f f e c t i v e n e s s of vari.ous methods f o r s o l v i n g such l i n e a r systems, assume t h a t the number of g r i d p o i n t s N i n t h e x d i r e c t i o n i s about the same as the number of p o i n t s M i n the y d i r e c t i o n - t h a t i s N^M3>i*n. Where p o s s i b l e , methods w i l l be compared u s i n g 'order of N' e s t i m a t e s of two q u a n t i t i e s -o p e r a t i o n s r e q u i r e d t o s o l v e the system and computer memory needed t o p e r f o r m the s o l u t i o n . Here, an o p e r a t i o n w i l l t y p i c a l l y be a m u l t i p l i c a t i o n , d i v i s i o n , or a d d i t i o n , of two f l o a t i n g p o i n t numbers. Such a r i t h m e t i c o p e r a t i o n s u s u a l l y account f o r the m a j o r i t y of e x e c u t i o n time of any of the methods t o be d e s c r i b e d . Methods f o r s o l v i n g l a r g e , s p a r s e l i n e a r systems f a l l i n t o two g e n e r a l c l a s s e s - d i r e c t methods and i t e r a t i v e methods. These c l a s s e s a r e d i s c u s s e d s e p a r a t e l y i n the f o l l o w i n g two s u b - s e c t i o n s . 145 A. 3 .1 D i r e c t methods A d i r e c t method f o r s o l v i n g the system (A.17) u s u a l l y i n v o l v e s decomposing the m a t r i x A i n t o lower and upper t r i a n g u l a r f a c t o r s u s i n g some v a r i a n t of G a u s s i a n e l i m i n a t i o n . (The reader who i s u n f a m i l i a r w i t h G a u s s i a n e l i m i n a t i o n i s r e f e r r e d t o any e l e m e n t a r y t e x t i n l i n e a r a l g e b r a or n u m e r i c a l a n a l y s i s . ) (A.17) becomes i I . I I , , - \D ( A ' 1 8 ) where U A and L A a r e n x n upper t r i a n g u l a r and lower t r i a n g u l a r m a t r i c e s r e s p e c t i v e l y . Once the d e c o m p o s i t i o n has been a c c o m p l i s h e d , the s o l u t i o n v e c t o r i s d e t e r m i n e d by s o l v i n g s u c c e s s i v e l y the systems (A.19) arid The s o l u t i o n of such t r i a n g u l a r systems i s r e a d i l y a c c o m p l i s h e d and f o r l a r g e v a l u e s of n, r e p r e s e n t s o n l y a s m a l l f r a c t i o n of the work needed t o p e r f o r m the f a c t o r i z a t i o n . As a consequence, r e s e a r c h e f f o r t s have c o n c e n t r a t e d on ways of i m p r o v i n g the f a c t o r i z a t i o n p r o c e s s . The major problem e n c o u n t e r e d i n f a c t o r i n g a m a t r i x 1 46 a r i s i n g from a f i n i t e d i f f e r e n c e scheme i n v o l v e s the phenomenon of f i l l - i n - the upper and lower t r i a n g u l a r f a c t o r s w i l l i n g e n e r a l have non-zero elements which a r e z e r o i n the c o r r e s p o n d i n g p o s i t i o n s of the o r i g i n a l m a t r i x . S i n c e the o p e r a t i o n s needed t o p e r f o r m the f a c t o r i z a t i o n , as w e l l as the minimum memory s t o r a g e r e q u i r e d , depend on the number of non-z e r o s i n L A and , much work has been devoted t o the d e s i g n of a l g o r i t h m s which reduce f i l l - i n . The amount of f i l l s u f f e r e d by the m a t r i x A i n the c o u r s e of the f a c t o r i z a t i o n p r o c e s s depends on the may i n which the unknowns u| i=1,...,n a r e o r d e r e d . For example, i f the system t o be s o l v e d r e s u l t s from the d i s c r e t i z a t i o n of an e l l i p t i c problem u s i n g second-order c e n t r e d d i f f e r e n c e s , and the unknowns u " a r e numbered u s i n g a n a t u r a l o r d e r i n g such as the one d e p i c t e d i n F i g u r e 21 f o r a 10 x 10 g r i d , then i t can be shown t h a t a l t h o u g h t h e r e a r e o n l y 0 ( N 2 ) non-zeros i n A, t h e r e w i l l be 0 ( N 3 ) non-zeros i n each of the t r i a n g u l a r f a c t o r s L A and U\ , and the amount of work r e q u i r e d t o complete the f a c t o r i z a t i o n i s 0 ( N " ) . T h i s i s e s s e n t i a l l y a w o r s t - c a s e b e h a v i o u r and s e v e r a l methods f o r r e o r d e r i n g t h e unknowns t o improve t h i s performance have been d e v e l o p e d . Of t h e s e , the most s u c c e s s f u l f o r the type of d i f f e r e n c e system d e s c r i b e d above i s p r o b a b l y the n e s t e d d i s s e c t i o n method due t o George [ 1 7 ] , [ 1 8 ] , F i g u r e 22 shows a n e s t e d d i s s e c t i o n o r d e r i n g of a 10 x 10 g r i d . George has shown t h a t such an o r d e r i n g l e a d s t o 0 ( N 2 l o g N ) non-zeros i n the t r i a n g u l a r f a c t o r s and 0 ( N 3 ) o p e r a t i o n s t o compute the f a c t o r i z a t i o n , which r e p r e s e n t s a s i g n i f i c a n t improvement, f o r l a r g e N, over the n a t u r a l 147 1 2 11 12 21 22 31 32 41 42 51 52 61 62 71 72 81 82 91 92 3 4 13 14 23 24 33 34 43 44 53 54 63 64 73 74 83 84 93 94 5 6 15 16 25 26 35 36 45 46 55 56 65 66 75 76 85 86 95 96 7 8 17 18 27 28 37 38 47 48 57 58 67 68 77 78 87 88 97 98 9 10 19 20 29 30 39 40 49 50 59 60 69 70 79 80 89 90 99 100 N a t u r a l F i g u r e 21 O r d e r i n g o f a 1 0 x 1 0 G r i d 148 78 77 85 68 67 100 29 28 36 20 76 75 84 66 65 99 27 26 35 19 80 79 83 70 69 98 31 30 34 21 74 73 " 82 64 63 97 25 24 33 18 72 71 81 62 61 96 23 22 32 17 90 89 88 87 86 95 40 39 38 37 54 53 60 46 45 94 10 9 16 3 52 51 59 44 43 93 8 7 15 2 56 55 58 48 47 92 12 1 1 1 4 4 50 49 57 42 41 91 6 5 13 1 F i g u r e 22 N ested D i s s e c t i o n O r d e r i n g of a 10x10 G r i d 149 o r d e r i n g . I t has a l s o been shown [21] t h a t t h e s e o r d e r e s t i m a t e s a r e both o p t i m a l . I t s h o u l d be mentioned t h a t the o r d e r i n g of the unknowns i s sometimes i m p o r t a n t f o r an o t h e r r e a s o n . R e o r d e r i n g the components of the v e c t o r u i n (A.17) i m p l i e s t h a t the rows and columns o f the m a t r i x A must a l s o be permuted. However, f o r a g e n e r a l m a t r i x A, i t may be n e c e s s a r y t o permute the rows and/or columns of A as the e l i m i n a t i o n p r o c e d u r e p r o g r e s s e s t o m i n i m i z e n u m e r i c a l i n s t a b i l i t i e s which a r i s e i f the m a t r i x i s p o o r l y c o n d i t i o n e d . The c o n d i t i o n  number, c o n d ( A ) , of the m a t r i x , p r o v i d e s a measure of how " c l o s e " the m a t r i x i s t o b e i n g n u m e r i c a l l y s i n g u l a r . c < W ( M * H A i f l[ A""1 IIM \ 4 COr\d (A) i cu where II • l l M i n the above i s some m a t r i x norm, such as II A || M Stxp IIA x 1| with- 11 • II some v e c t o r norm. When cond(A) i s v e r y l a r g e , A i s s a i d t o be i l l - c o n d i t i o n e d . The need t o i n t e r c h a n g e the rows and/or columns of A i n the c o u r s e of the s o l u t i o n may l e a d t o a d i s r u p t i o n of the o r d e r i n g i n t e n d e d t o m i n i m i z e f i l l - i n and a compromise must u s u a l l y be e s t a b l i s h e d between the i n c r e a s e i n memory usage and e x e c u t i o n time due t o f i l l - i n , and the l o s s of a c c u r a c y due t o n u m e r i c a l i n s t a b i l i t y . F o r t u n a t e l y , m a t r i c e s a r i s i n g from the d i s c r e t i z a t i o n of e l l i p t i c systems are u s u a l l y w e l l - c o n d i t i o n e d so t h a t the problem of i n s t a b i l i t y does not a r i s e and an o r d e r i n g may be e s t a b l i s h e d 150 b e f o r e the f a c t o r i z a t i o n p r o c e s s commences. Another i m p o r t a n t c o n s i d e r a t i o n i n the d e s i g n of a l g o r i t h m s t o s o l v e a l a r g e s p a r s e system d i r e c t l y i s the manner i n which the m a t r i x and i t s f a c t o r s a r e a c t u a l l y s t o r e d i n the computer memory. I m p l i c i t i n the above d i s c u s s i o n of f i l l - i n i s the f a c t t h a t i t i s not f e a s i b l e t o s t o r e the e n t i r e m a t r i x A and i t s f a c t o r s i n t h e i r e n t i r e t y ^ I f n=l00, f o r example, which i s not u n r e a s o n a b l e , 100 m i l l i o n words of computer memory would be r e q u i r e d t o s t o r e A e x p l i c i t l y , the v a s t m a j o r i t y of which would c o n t a i n 0! Most a l g o r i t h m s employ sp a r s e m a t r i x s t o r a g e t e c h n i q u e s t o m i n i m i z e the number of z e r o elements which must be s t o r e d . Thorough d i s c u s s i o n s of many such schemes a r e found i n [ 1 8 ] . A.3.2 I t e r a t i v e methods ' In an i t e r a t i v e method f o r s o l v i n g ( A . 1 7 ) , s t a r t i n g w i t h some i n i t i a l e s t i m a t e of the s o l u t i o n u' o ) , a s e r i e s of i t e r a t e s u i s g e n e r a t e d such t h a t ^oo ~ ~ (A.21) Thus, i n c o n t r a s t w i t h d i r e c t methods w h i c h , n e g l e c t i n g the i n e x a c t n a t u r e of machine a r i t h m e t i c , y i e l d the e x a c t s o l u t i o n a f t e r a f i n i t e number of s t e p s , i t e r a t i v e methods r e q u i r e , i n p r i n c i p l e , an i n f i n i t e number of o p e r a t i o n s t o d e t e r m i n e u. In p r a c t i c e , the s o l u t i o n p r o c e s s i s g e n e r a l l y t e r m i n a t e d a f t e r a f i n i t e number of i t e r a t i o n s , when i t i s f e l t t h a t the e s t i m a t e 151 u t k ) i s " c l o s e enough" t o the s o l u t i o n u. A common way t o m o n i t o r the p r o g r e s s of an i t e r a t i v e scheme i s t o compute a t each i t e r a t i o n , the r e s i d u a l v e c t o r r °° r I f the i t e r a t i o n c o n v e r g e s , then c ° o (A.23) The i t e r a t i v e p r o c e d u r e may then be stopped when II r * II < (z (A.24) where II • II denotes some d i s c r e t e norm and € i s a convergence  parameter which i s t y p i c a l l y g i v e n a t the onset of the s o l u t i o n p r o c e s s . I t e r a t i v e methods a r e c h a r a c t e r i z e d by the manner i n which the new i t e r a t e i s c a l c u l a t e d from p r e v i o u s i t e r a t e s . In the g e n e r a l case u G ^ C u 0 0 , ^ " " V " , a"") * (A.25) where G i s some o p e r a t o r which, as the s u p e r s c r i p t i n d i c a t e s may change from i t e r a t i o n t o i t e r a t i o n , as may the v e c t o r c A p a r t i c u l a r c l a s s of methods, which i n c l u d e s a l l of the methods t o be d e s c r i b e d below, r e s u l t s when G i s l i n e a r , s t a t i o n a r y , and o p e r a t e s o n l y on the c u r r e n t s o l u t i o n 1 52 e s t i m a t e , and the v e c t o r c c l° i s c o n s t a n t . Then _ G O L C O ^ c ( A.26) and G may be r e p r e s e n t e d as a n x n m a t r i x and i s o f t e n c a l l e d the a m p l i f i c a t i o n m a t r i x . The e r r o r v e c t o r e of the k t h i t e r a t i o n i s d e f i n e d by eCK^ = u t C ' ^ - oc (A. 27) and i t can e a s i l y be shown t h a t e - Co e where G* i s the k t h power of the a m p l i f i c a t i o n m a t r i x . The i t e r a t i v e p rocedure w i l l converge i f JL^ II e c o l | * X;^ HGK -- 0 (A.28) A n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r the above t o be s a t i s f i e d i s t h a t the s p e c t r a l r a d i u s , p (G), of the a m p l i f i c a t i o n m a t r i x ( G) = max I X . C G ) I (A.29) where X t a r e the e i g e n v a l u e s of G, be l e s s than u n i t y . In a d d i t i o n , p (G) d e t e r m i n e s how q u i c k l y the i t e r a t i v e method c o n v e r g e s . 153 In g e n e r a l k.-*oc ^ £c*> \\ ' (A.30) and an a s y m p t o t i c r a t e of c o nvergence, R, may be d e f i n e d as f o l l o w s R - loc310 ( p - * ) (A.31 ) The r e c i p r o c a l of R y i e l d s the number of i t e r a t i o n s which must be performed t o reduce the e r r o r a s y m p t o t i c a l l y by a f a c t o r of t e n . The s t u d y of the a m p l i f i c a t i o n m a t r i x of a p a r t i c u l a r i t e r a t i v e method i s v e r y u s e f u l f o r e s t a b l i s h i n g t h e o r e t i c a l r e s u l t s c o n c e r n i n g the convergence p r o p e r t i e s of the method. However, f o r any but the v e r y s i m p l e s t i t e r a t i v e schemes, the e x p l i c i t form of G becomes q u i t e c o m p l i c a t e d , and w r i t i n g the i t e r a t i o n s i n the form of (A.26) tends t o make the methods l o o k more complex than t h e y a c t u a l l y a r e . An a l t e r n a t i v e approach i s t o d e s c r i b e the methods i n a manner which c l o s e l y p a r a l l e l s the way they a r e t y p i c a l l y implemented as computer programs. Let t h e n p o i n t s of the g r i d be numbered i n some f a s h i o n - u s i n g a n a t u r a l o r d e r i n g , f o r example. Denote the v a l u e of the unknown g r i d f u n c t i o n a t g r i d p o s i t i o n i by u;, and the elements of the m a t r i x A by a;j . A p a r t i c u l a r l y s i m p l e method, commonly c a l l e d the J a c o b i i t e r a t i o n , r e s u l t s from v i s i t i n g each g r i d p o i n t i n s u c c e s s i o n , c h a n g i n g the v a l u e of each 154 unknown so t h a t u s i n g the r e q u i r e d n e i g h b o r i n g v a l u e s from the p r e v i o u s i t e r a t i o n , the l o c a l d i f f e r e n c e e q u a t i o n s a r e s a t i s f i e d . That i s - ~ S a c j U j °° ' b l (A.32) Note t h a t t h e r e i s no need t o e x p l i c i t l y s t o r e the m a t r i x A, when t h i s method i s implemented on a computer. A l l t h a t i s r e q u i r e d i s s u f f i c i e n t i n f o r m a t i o n t o e v a l u a t e the d i f f e r e n c e e q u a t i o n s a t each p o i n t . T h i s f e a t u r e i s c h a r a c t e r i s t i c 'of most i t e r a t i v e methods and i s one of the p r i m a r y r e a s o n s t h a t v e r y l a r g e systems of d i f f e r e n c e e q u a t i o n s have t r a d i t i o n a l l y been s o l v e d by i t e r a t i v e , r a t h e r than d i r e c t methods. The b u l k of the s t o r a g e needed f o r the J a c o b i method i s used f o r m a i n t a i n i n g two v e c t o r s of l e n g t h n which c o n t a i n a t any , time the g r i d f u n c t i o n e s t i m a t e s of the c u r r e n t and p r e v i o u s i t e r a t i o n s . I f t h e J a c o b i method c o n v e r g e s , t h e n , i n g e n e r a l , u < k +° w i l l be a b e t t e r e s t i m a t e of u than u . T h i s s u g g e s t s t h a t an improvement on the J a c o b i i t e r a t i o n might r e s u l t by u s i n g newly c a l c u l a t e d q u a n t i t i e s whenever p o s s i b l e i n the c o u r s e of an i t e r a t i o n . Thus . a t i * & a i i u i t o ) * b : < A.33> a ; ; T h i s i s t h e G a u s s - S e i d e l (GS) i t e r a t i o n which has the advantage of o n l y r e q u i r i n g s t o r a g e f o r a s i n g l e v e c t o r of l e n g t h n t o m a i n t a i n the e s t i m a t e of the g r i d f u n c t i o n . 1 55 U n f o r t u n a t e l y , both of the above methods te n d t o have slow a s y m p t o t i c r a t e s of convergence. T y p i c a l l y , the number of i t e r a t i o n s r e q u i r e d t o reduce the e r r o r by an o r d e r of magnitude i s 0 ( N 2 ) . The number of c a l c u l a t i o n s needed t o p e r f o r m a s i n g l e i t e r a t i o n , or r e l a x a t i o n sweep, as i t i s o f t e n c a l l e d , i s 0 ( N 2 ) i f the number of o p e r a t i o n s r e q u i r e d t o e v a l u a t e each d i f f e r e n c e e q u a t i o n i s 0 { 1 ) . Thus, the t o t a l work n e c e s s a r y t o s o l v e the system by e i t h e r of the methods i s 0 ( N * ) , w h i l e the memory r e q u i r e d i s 0 ( N 2 ) . H i s t o r i c a l l y , r e s e a r c h e r s who used the G a u s s - S e i d e l method f o r s o l v i n g systems of d i f f e r e n c e e q u a t i o n s d i s c o v e r e d t h a t convergence c o u l d o f t e n be a c c e l e r a t e d by ' m o d i f y i n g the i t e r a t i o n so t h a t - ~ a r i > * n " ~ } u ^ ( A ' 3 4 ) where 0^^° i s d e t e r m i n e d by the r i g h t hand s i d e of (A . 3 3 ) , and co i s c a l l e d the r e l a x a t i o n parameter. I f 0 < " < 2, then the above d e f i n e s the well-known s u c c e s s i v e o v e r - r e l a x a t i o n (SOR) i t e r a t i o n which i n c o r p o r a t e s t h e G a u s s - S e i d e l i t e r a t i o n as a s p e c i a l case when co=1. The SOR method has been the s u b j e c t of a g r e a t d e a l of r e s e a r c h i n the p a s t and optimum v a l u e s of co have been d e t e r m i n e d a n a l y t i c a l l y f o r some s p e c i a l t y p e s of d i f f e r e n c e systems and domains. In th e s e c a s e s , the number of r e l a x a t i o n sweeps n e c e s s a r y f o r convergence may be O(N), l e a d i n g t o 0 ( N 3 ) o p e r a t i o n s f o r s o l u t i o n of the system, a g a i n w i t h 0 ( N 2 ) memory s t o r a g e . In g e n e r a l , however, the o p t i m a l 156 v a l u e of co , * j o p * , which depends on the g r i d s p a c i n g , the shape of the domain, and the d i f f e r e n c e scheme employed, must be d e t e r m i n e d e i t h e r t h r o u g h n u m e r i c a l experiment or by some s o r t of a d a p t i v e p r o c e d u r e which a t t e m p t s t o e s t i m a t e cooPt i n the c o u r s e of the s o l u t i o n p r o c e s s . Because the above methods s o l v e f o r one new f u n c t i o n v a l u e a t a t i m e , they a r e o f t e n r e f e r r e d t o as p o i n t - r e l a x a t i o n methods. Another c l a s s of r e l a x a t i o n methods i n v o l v e s t h e s i m u l t a n e o u s update of a group of unknowns. For example, assume t h a t the g r i d has been numbered u s i n g a n a t u r a l o r d e r i n g such as the one shown i n F i g u r e 21. L e t u*,tx = 1,...,M be the N-component v e c t o r s of unknowns c o r r e s p o n d i n g t o a l l g r i d p o i n t s h a v i n g the y c o o r d i n a t e y*. Then a l i n e - G a u s s - S e i d e l (LGS) method i n v o l v e s s o l v i n g e g u a t i o n (A.33) s i m u l t a n e o u s l y f o r a l l u, e u*f f o r oc = 1 ,. .. ,M. Thus one i t e r a t i o n of LGS r e q u i r e s the s o l u t i o n of M systems of N e q u a t i o n s . Each such system, however, i s u s u a l l y q u i t e easy t o s o l v e - g e n e r a l l y r e q u i r i n g o n l y 0(N) o p e r a t i o n s , so the work needed t o p e r f o r m a r e l a x a t i o n sweep i s s t i l l 0 ( N 2 ) . C l e a r l y , one might a l s o choose t o use l i n e s of unknowns h a v i n g c o n s t a n t x c o o r d i n a t e s , or even a l t e r n a t e the g r o u p i n g c h o i c e from i t e r a t i o n t o i t e r a t i o n . Line-SOR (LSOR) schemes may s i m i l a r l y be f o r m u l a t e d . As w i t h the p o i n t - i t e r a t i v e methods, t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s i n d i c a t e t h a t 0 ( N 2 ) LGS sweeps or at- l e a s t 0(N) LSOR sweeps a r e r e q u i r e d f o r convergence of a t y p i c a l f i n i t e d i f f e r e n c e problem, so t h e work and s t o r a g e e s t i m a t e s a r e of the same o r d e r s as t h e c o r r e s p o n d i n g p o i n t - w i s e schemes. However the t h e o r y which has 157 been e s t a b l i s h e d f o r l i n e - r e l a x a t i o n schemes, n o t a b l y LSOR, c o v e r s a more g e n e r a l c l a s s of f i n i t e d i f f e r e n c e problems than the r e s u l t s f o r p o i n t - r e l a x a t i o n methods, and i n p r a c t i c e , LSOR f r e q u e n t l y o u t p e r f o r m s SOR. A f i n a l , v e r y i m p o r t a n t p o i n t must be mentioned i n c o n n e c t i o n w i t h a l l of the above i t e r a t i v e schemes. In g e n e r a l , a l l of these r e l a x a t i o n methods w i l l converge i f and o n l y i f the system of d i f f e r e n c e e q u a t i o n s i s d e f i n i t e ; t h a t i s , i f X ^ A X > O or ~ } ~ (A.35) For most d i f f e r e n c e schemes r e s u l t i n g from the d i s c r e t i z a t i o n of a l i n e a r e l l i p t i c problem h a v i n g D i r i c h l e t or Neumann boundary c o n d i t i o n s , A w i l l be d e f i n i t e and r e l a x a t i o n methods may be a p p l i e d . Thorough t r e a t m e n t s of i t e r a t i v e t e c h n i q u e s f o r l a r g e l i n e a r systems a r e found i n [36] and [ 3 8 ] . 158 A.4 Methods f o r s o l v i n g n o n - 1 i n e a r systems I f e i t h e r of the o p e r a t o r s i n (A.1) i s n o n - l i n e a r , then the d i s c r e t i z a t i o n p r o c e s s g i v e s r i s e t o a system of non-l i n e a r a l g e b r a i c e q u a t i o n s which may be w r i t t e n as F-t ( u -y , u t j • • • , LA c ") = O or more s u c c i n c t l y F ( a ) -• o (A.36) ,—' These e q u a t i o n s must be s o l v e d i t e r a t i v e l y s i n c e t h e r e i s no e x i s t i n g a l g o r i t h m f o r the d i r e c t s o l u t i o n of g e n e r a l non-l i n e a r systems. As i n the case of l i n e a r i t e r a t i v e methods, the problem i s u s u a l l y c o n s i d e r e d s o l v e d when some convergence c r i t e r i o n has been met. A g a i n , a q u a n t i t y o f t e n m o n i t o r e d i s C\c) the norm of the r e s i d u a l v e c t o r , r r 1 5 V- ( a ) (A.37) The s o l u t i o n p r o c e s s may then be t e r m i n a t e d when 11 r ( K l || < e (A.38) where £ i s the convergence t o l e r a n c e as b e f o r e . A c o n s i d e r a b l e number of methods f o r the s o l u t i o n of non-159 l i n e a r systems have been d e v e l o p e d . Here, a t t e n t i o n w i l l be r e s t r i c t e d t o a few schemes which a l l i n v o l v e a g e n e r a l i z a t i o n t o n d i m e n s i o n s of Newton's method f o r the s o l u t i o n of a s i n g l e n o n - l i n e a r e q u a t i o n i n one unknown. To r e v i e w t h i s method, i f the e q u a t i o n t o be s o l v e d i s S (iC) - O (A.39) and x i s t h e c u r r e n t e s t i m a t e of the s o l u t i o n , then the new e s t i m a t e xCk"*"°is de t e r m i n e d by x. x ~ (A. 40) There a r e e s s e n t i a l l y two ways t o e x t e n d t h i s i t e r a t i o n t o n d i m e n s i o n s i n a s t r a i g h t f o r w a r d manner. The f i r s t method i n v o l v e s l i n e a r i z i n g the complete system of e q u a t i o n s a t each i t e r a t i o n . That i s , g i v e n an e s t i m a t e u , the new i t e r a t e i s g i v e n by U. - u. + o u ~ (A.41) where the v e c t o r Su c l c ' s a t i s f i e s the f o l l o w i n g l i n e a r system. F ' l W C ^ - ~ F ( o c c ^ ) (A.42) Here F' i s the J a c o b i a n m a t r i x of the system h a v i n g elements 160 At each i t e r a t i o n , t h e system (A.43) must then be s o l v e d , u s i n g , f o r example, any of the methods d e s c r i b e d i n the p r e v i o u s s e c t i o n f o r the s o l u t i o n of l i n e a r e q u a t i o n s . One of the methods used f o r comparison w i t h the m u l t i - g r i d method i n Chapter 5, employed the above i t e r a t i o n i n c o n j u n c t i o n w i t h a n e s t e d - d i s s e c t i o n d i r e c t method t o s o l v e the l i n e a r systems. The work r e q u i r e d t o s o l v e a n o n - l i n e a r system i n t h i s f a s h i o n depends c r u c i a l l y on the i n i t i a l e s t i m a t e . I f the e s t i m a t e i s good enough, then convergence may be a c h i e v e d w i t h j u s t a few Newton i t e r a t i o n s , so t h e o r d e r of t h e work e s t i m a t e may be the same as the e s t i m a t e f o r the method used f o r s o l v i n g the l i n e a r systems. The second way of employing Newton's method i n n d i m e n s i o n s i s t o s u c c e s s i v e l y a p p l y the i t e r a t i o n t o each i n d i v i d u a l e q u a t i o n , or a group of e q u a t i o n s c o r r e s p o n d i n g t o a l i n e of unknowns, i n the s p i r i t of the r e l a x a t i o n methods f o r s o l v i n g l i n e a r sytems. For example, the o n e - s t e p p o i n t - Gauss-Seidel-Newton i t e r a t i o n i s g i v e n by The term o n e - s t e p i s used s i n c e o n l y a s i n g l e Newton i t e r a t i o n i s performed b e f o r e moving on t o the next e q u a t i o n . ( I n c e r t a i n c a s e s , i t may be d e s i r a b l e t o p e r f o r m more than one s t e p ) . S i m i l a r l y , the o n e - s t e p point-SOR-Newton i t e r a t i o n i s 161 where, as b e f o r e , oJ i s the r e l a x a t i o n parameter, whose o p t i m a l v a l u e w i l l n o r m a l l y have t o be d e t e r m i n e d from n u m e r i c a l e x p e r i m e n t s . These i t e r a t i o n s may be m o d i f i e d so t h a t an e n t i r e l i n e of unknowns i s updated s i m u l t a n e o u s l y . U s i n g the n o t a t i o n of S e c t i o n A.3.2, a t y p i c a l line-SOR-Newton i t e r a t i o n i s g i v e n by Uot = U.A t co bu_* ; K~ 1, ••J<^\ (A.46) where Su? i s d e t e r m i n e d by P.! ( a t k ) ) S_u/° = -F« Cu"°) (A.47) V F; 6 F« ; (A. 48) ' c f C*M c)u; a = 0- ClO A g a i n , s o l u t i o n of the above l i n e a r systems i s u s u a l l y r e a d i l y a c c o m p l i s h e d . Both point-SOR-Newton and line-SOR-Newton methods were coded f o r comparison w i t h the m u l t i - g r i d method. The amount of work n e c e s s a r y t o d e t e r m i n e a s o l u t i o n by n o n - l i n e a r r e l a x a t i o n methods i s d i f f i c u l t t o e s t i m a t e i n g e n e r a l . I f the i n i t a l guess i s good, then the o r d e r e s t i m a t e s may be the same as the c o r r e s p o n d i n g l i n e a r r e l a x a t i o n methods. B e t t e r performance i s c e r t a i n l y not t o be e x p e c t e d . A v e r y good r e f e r e n c e f o r the s o l u t i o n of n o n - l i n e a r systems* i s due t o Ortega and R h e i n b o l d t [ 3 2 ] . 162 APPENDIX B I mplementation of the M u l t i - g r i d A l g o r i t h m B.1 G r i d o r g a n i z a t i o n f o r a d a p t i v e d i s c r e t i z a t i o n As s t a t e d i n Chapter 4, the m u l t i - g r i d program which i s l i s t e d i n t h i s appendix i s based on the a l g o r i t h m p r e s e n t e d i n pseudo-code form i n F i g u r e 5. However, the a c t u a l program was o d e s i g n e d t o accomodate more g e n e r a l p a t t e r n s of d i s c r e t i z a t i o n than the one used i n the pseudo-code a l g o r i t h m . In the a l g o r i t h m of Chapter 4, the u n i f o r m g r i d s used a l l c o v e r e d the e n t i r e domain of the problem, and each l e v e l of d i s c r e t i z a t i o n was a s s o c i a t e d w i t h a unique g r i d . For the purpose of a d a p t i v e d i s c r e t i z a t i o n , the program a l l o w s more than one g r i d t o be employed a t each l e v e l , and the g r i d s used may e x t e n d over sub-domains of the problem. F i g u r e 23a i l l u s t r a t e s an example of the type of d i s c r e t i z a t i o n the program was d e s i g n e d t o h a n d l e . A l l g r i d s used a r e r e q u i r e d t o be r e c t a n g u l a r w i t h u n i f o r m mesh s p a c i n g s i n b oth d i r e c t i o n s . The c o a r s e s t g r i d used i s assumed t o e x t e n d over the e n t i r e domain. A l l g r i d s , w i t h the e x c e p t i o n of the c o a r s e s t are r e q u i r e d t o have a unique f a t h e r g r i d which i s used f o r t h e purpose of c o a r s e g r i d c o r r e c t i o n s . The f a t h e r ' s domain must c o n t a i n the domains of a l l of i t s sons. The mesh s p a c i n g r a t i o s between a f a t h e r / s o n p a i r of g r i d s must be 2:1 or 1:1, w i t h the 2:1 (a) F i g u r e 23 Non-uniform D i s c r e t i z a t i o n and A s s o c i a t e d Tree S t r u c t u r e 164 r a t i o h o l d i n g f o r a t l e a s t one of the d i r e c t i o n s . In g e n e r a l , a s e t of g r i d s which may be used i n the program can be r e p r e s e n t e d by a t r e e s t r u c t u r e . F i g u r e 23b shows the t r e e s t r u c t u r e a s s o c i a t e d w i t h the g r i d s of F i g u r e 23a. The m o d i f i c a t i o n s t o the FAS a l g o r i t h m f o r such an o r g a n i z a t i o n of g r i d s a r e s t r a i g h t f o r w a r d . To s o l v e a problem a t l e v e l k, r e l a x a t i o n sweeps a r e a p p l i e d t o a l l g r i d s d e f i n e d on l e v e l k u n t i l convergence on a l l g r i d s i s slow. A c o a r s e g r i d c o r r e c t i o n i s performed by u p d a t i n g the systems of e q u a t i o n s f o r a l l g r i d s on l e v e l k-1 h a v i n g sons on l e v e l ' k. R e l a x a t i o n sweeps a r e then performed over a l l g r i d s a t l e v e l k-1, e t c . Note t h a t i n g e n e r a l , some of the c o a r s e g r i d s w i l l s e r v e d u a l purposes i n such a scheme; F i r s t t h ey w i l l a c t as c o r r e c t i n g g r i d s f o r t h e i r sons and s e c o n d l y , they may be the f i n e s t g r i d s used i n some r e g i o n s of the s o l u t i o n domain. The d a t a s t r u c t u r e which was d e s i g n e d t o implement t h i s t y p e of g r i d o r g a n i z a t i o n i s d e s c r i b e d i n the program documentation which f o l l o w s t he t e x t of t h i s a ppendix. Each g r i d i s l a b e l l e d by an i n t e g e r and v a r i o u s i n f o r m a t i o n d e s c r i b i n g the g r i d s i s s t o r e d i n s i n g l e d i m e n s i o n e d a r r a y s . P o i n t e r i n f o r m a t i o n t o f a c i l i t a t e a c c e s s t o a l l g r i d s a t a g i v e n l e v e l , the f a t h e r of a g i v e n g r i d , or the sons of a g i v e n g r i d , i s a l s o m a i n t a i n e d . F u n c t i o n s may be d e f i n e d on the p o i n t s of a g r i d or on the two s e t s of c o o r d i n a t e s which d e f i n e t he g r i d . A l l of th e s e f u n c t i o n s a r e s t o r e d i n a s i n g l e a r r a y , MEMORY. P o i n t e r s t o the s t a r t i n g l o c a t i o n s i n MEMORY f o r the v a r i o u s f u n c t i o n s a r e a l s o kept i n the g r i d d a t a s t r u c t u r e . The a r r a y MEMORY i s a l s o used throughout the 165 program t o p r o v i d e temporary working s t o r a g e . I t s h o u l d be n o t e d t h a t the m u l t i - g r i d runs d e s c r i b e d i n Chapter 5 d i d not demand as g e n e r a l a g r i d o r g a n i z a t i o n as the program was d e s i g n e d t o a l l o w . However, the a d d i t i o n a l e x e c u t i o n time which r e s u l t s from u s i n g the more g e n e r a l scheme i s t r u l y n e g l i g i b l e i n comparison t o the work i n v o l v e d i n r e l a x a t i o n sweeps, e t c . which must be performed by any i m p l e m e n t a t i o n of the FAS a l g o r i t h m . B.2 Summary of major m u l t i - g r i d r o u t i n e s The program l i s t e d below c o n t a i n s two h i g h l e v e l r o u t i n e s which e s s e n t i a l l y implement PROCEDURE MULTI_GRID of the pseudo-code a l g o r i t h m . The f i r s t of t h e s e r o u t i n e s i s a d r i v e r which s i m p l y i n p u t s some run parameters and then c a l l s the second r o u t i n e , MGMAIN. MGMAIN c a l l s i n t e r m e d i a t e l e v e l ' r o u t i n e s t o s o l v e the problem on s u c c e s s i v e l y f i n e r l e v e l s u n t i l some maximum l e v e l of d i s c r e t i z a t i o n has been used, or the e s t i m a t e d l o c a l t r u n c a t i o n e r r o r e s t i m a t e i s l e s s than the s u p p l i e d convergence c r i t e r i o n everywhere i n the d i s c r e t e domain. MGMAIN i s a l s o r e s p o n s i b l e f o r c a l l i n g the r o u t i n e ADAPT which d e t e r m i n e s where a new f i n e g r i d s h o u l d be i n t r o d u c e d . The second pseudo-code r o u t i n e PROCEDURE SOLVE_ON_GRID, i s implemented u s i n g f o u r r o u t i n e s : CYCLE, SOLVE, CGCST, and CGCFIN. The i m p l e m e n t a t i o n i s somewhat d i f f e r e n t than the a l g o r i t h m s i n c e " s t a n d a r d " FORTRAN does not p e r m i t r e c u r s i v e r o u t i n e s . CYCLE a t t e m p t s t o s o l v e the l e v e l k problem as 166 f o l l o w s . The r o u t i n e SOLVE i s c a l l e d , which i n t u r n c a l l s the r e l a x a t i o n r o u t i n e , LINRLX, t o a p p l y r e l a x a t i o n sweeps t o the l e v e l S. = k d i f f e r e n c e e q u a t i o n s u n t i l t he system i s s o l v e d , or slow convergence i s d e t e c t e d . I f A i s the c o a r s e s t l e v e l , a r o u t i n e which s o l v e s the system d i r e c t l y i s i n v o k e d . SOLVE r e t u r n s a f l a g which i n d i c a t e s whether the problem has been s o l v e d or convergence i s slow. In the l a t t e r c a s e , CYCLE c a l l s the r o u t i n e CGCST which s t a r t s the c o a r s e g r i d c o r r e c t i o n p r o c e s s . The r i g h t hand s i d e of the l e v e l d i f f e r e n c e e q u a t i o n s and the convergence c r i t e r i a f o r l e v e l s A-\ and p o s s i b l y £ a r e updated. CYCLE then c a l l s SOLVE on l e v e l A - ] . When the v a l u e of the f l a g t h a t SOLVE r e t u r n s i n d i c a t e s t h a t a problem has been s o l v e d , and the problem i s not on l e v e l k, a c o a r s e g r i d c o r r e c t i o n has been c o m p l e t e d . In t h i s c a s e , CYCLE i n v o k e s CGCFIN which updates the f i n e g r i d f u n c t i o n ; CYCLE then c a l l s SOLVE on l e v e l . When the l e v e l k problem has been s o l v e d , CYCLE r e t u r n s c o n t r o l t o MGMAIN. As a guard a g a i n s t program run-away, CYCLE has a parameter which l i m i t s the number of c o a r s e g r i d c o r r e c t i o n c y c l e s performed i n an attempt t o s o l v e the l e v e l k problem. B.3 R e l a x a t i o n r o u t i n e s As d e s c r i b e d i n Chapter 4, the d e t e r m i n a t i o n of a good r e l a x a t i o n scheme f o r a g i v e n d i s c r e t i z e d problem i s v i t a l f o r the p r o p e r o p e r a t i o n of any m u l t i - g r i d a l g o r i t h m . O r i g i n a l l y , a o n e - s t e p point-SOR-Newton r e l a x a t i o n r o u t i n e (see Appendix A, s e c t i o n A.4) was coded f o r use i n the a u t h o r ' s m u l t i - g r i d 167 program. However, t h i s scheme d i d not seem t o have a v e r y h i g h smoothing r a t e and r e s u l t e d i n slow o v e r a l l convergence. A line-SOR-Newton r o u t i n e , LINRLX, was then implemented. T h i s r o u t i n e i s c a p a b l e of u s i n g l i n e s of unknowns h a v i n g c o n s t a n t a n g u l a r or r a d i a l c o o r d i n a t e , and the l i n e s may be swept i n e i t h e r d i r e c t i o n - from low t o h i g h v a l u e s of g r i d c o o r d i n a t e or v i c e v e r s a . A f t e r some e x p e r i m e n t a t i o n , i t was found t h a t a scheme emp l o y i n g l i n e s of c o n s t a n t s swept from s=0 t o s=1 gave the best r e s u l t s . T h i s scheme was used i n a l l of the runs d e s c r i b e d i n Chapter 5 . R o u t i n e s t o smooth the n o n - D i r i c h l e t boundary e q u a t i o n s as d e s c r i b e d i n 4 . 4 were a l s o coded. As Brandt had c l a i m e d , i t was found t h a t two sweeps on a l l b o u n d a r i e s per i n t e r i o r r e l a x a t i o n sweep were s u f f i c i e n t t o m a i n t a i n p r o p e r convergence of the boundary e q u a t i o n s . B.4 C o n t r o l parameters Once the m u l t i - g r i d program had been w r i t t e n and debugged, some n u m e r i c a l e x p e r i m e n t a t i o n was performed t o determine s u i t a b l e v a l u e s f o r the t h r e e c o n t r o l parameters cx , , and S . The f o l l o w i n g v a l u e s were f i n a l l y chosen and used f o r a l l of the runs of Chapter 5 : ex = 0 . 3 5 , "7 = 0 . 7 0 , and 6 = 0 . 5 0 . In a d d i t i o n , the program was m o d i f i e d t o a c c e p t two a d d i t i o n a l p a r a m e t e r s , PSWP and QSWP which governed the minimum number of r e l a x a t i o n sweeps which were performed (on any l e v e l ) p r i o r t o and f o l l o w i n g a c o a r s e g r i d c o r r e c t i o n . These parameters were i n t r o d u c e d t o p r e v e n t the i n i t i a t i o n of a c o a r s e g r i d c o r r e c t i o n (as c o n t r o l l e d by the parameter ^ ) 168 b e f o r e the r e s i d u a l s had been s u f f i c i e n t l y smoothed. Without the a p p l i c a t i o n of a t l e a s t t h r e e or f o u r sweeps p r i o r t o a c o r r e c t i o n , the program sometimes e x h i b i t e d an o s c i l l a t o r y b e h a v i o u r - p e r f o r m i n g r e p e a t e d c o a r s e g r i d c o r r e c t i o n s which d i d not appear t o be c o n t r i b u t i n g v e r y much t o the s o l u t i o n of the f i n e g r i d e q u a t i o n s . 169 B.5 L i s t i n g of m u l t i - g r i d program c-c c c c-c c. c c c c c c c c c c c c c c c c c c c c c c c c c c c c D A T A S T R U C T U R E G R I D INTEGER GFSTRT(50,10), XFSTRT(50.10), YFSTRT(50,10) INTEGER BROTHR(50), FATHER(50), LHEAD(50), LLINK(50), S0N(50), NPT(50), NX(50), NY(50). XHIB(50), XLOB(50), YHIB(50). YL0B(50) REAL *8 AREA(50 ) , EPSI(50), ERR0R(50), NRMTAU(50), XINC(50), YINC(50) LOGICAL C0NV(50), SL0W(50) INTEGER CRSLEV, FINLEV, NGFUNC, NGRID, NXFUNC, NYFUNC INTEGER MEMPT INTEGER KSQR. R. RHS. RM, RP. TAU. W, WM, WP, U, OLDU INTEGER NULL / Z80808080 / COMMON / COMGST / LHEAD, LLINK, FATHER, BROTHR, SON, NX, NY, NPT, AREA, XINC. YINC, XFSTRT, YFSTRT, GFSTRT, XLOB, XHIB, Y LOB, YHIB, ERROR, CONV, SLOW, EPSI, NRMTAU, CRSLEV, FINLEV, NGRID, MEMPT COMMON / COMCON / R, RP, RM, W, WP, WM, U, OLDU, RHS, TAU, KSQR, NXFUNC, NYFUNC, NGFUNC 170 c  c C Pointer Information: C C LHEAD(L) > F i r s t g r i d on level L C LLINK(LHEAD(L)) > Next C LLINK(LLINK(LHEAD(L))) -> Etc. C C FATHER(G) > G's coarse g r i d correction g r i d C SON(G) > Finer g r i d contained by G C BROTHR(SON(G)) > Another g r i d contained by G C BROTHR(BROTHR(SON(G))) -> Etc. C C Fixed Grid Information: C C XINC(G) > Grid spacing in x d i r e c t i o n C YINC(G) > " " " y C NX(G) > Number of points in x d i r e c t i o n C NY (G) > " " " " y C NPT(G) > " " i n t e r i o r pts. in g r i d C AREA (G) > Area of gri d C XLOB(G) > Boundary fl a g s : C XHIB(G) / .EO. 0 -> D i r i c h l e t conditions C YLOB(G) / .NE. O -> non-D1richlet cond. C YHIB(G) / and contains number of smoothing C sweeps per i n t e r i o r relax, sweep C C Variable Grid Information: C C EPSI(G) > Current convergence c r i t e r i o n C ERROR(G) > Current dynamical residual norm C NRMTAU(G) > Norm of estimated local trunc. C error C CONV(G) > Flags solution on g r i d G C SLDW(G) > " slow convergence on G C C Function Pointers: ( s t a r t i n g locations 1n MEMORY) C C GFSTRT(G,I),I=1,NGFUNC -> Ptrs. to fens, of g r i d pts. C XFSTRT(G,I),I=1,NXFUNC -> x coords. C YFSTRT(G,I),1=1.NYFUNC -> " " " y coords. C C Miscellaneous: C C CRSLEV > Coarsest current level C FINLEV '-> Finest current level C NGRID > Current number of grids C MEMPT > Pointer to next ava i l a b l e C location in MEMORY C NGFUNC > # of fens, defined on g r i d pts. C NXFUNC > " " " " " x coords. C NYFUNC > " " " " " y " C NULL > Null pointer C c  c C T O P L E V E L R O U T I N E S C c  c  c C D R I V E R f o r m u l t i - g r i d and numerical I n t e g r a t i o n C r o u t i n e s . Note: Must be r e - c o m p i l e d to change amount of C memory a l l o c a t i o n (MEMORY). C C Run par a m e t e r s : C C p > Momentum C CVTAU > T r u n c a t i o n e r r o r convergence c r i t e r i o n C MAXLEV > Maximum number of l e v e l s of d i s c r e t 1 z 1 1 on C REFLEV > L e v e l a t which a d a p t i v e d i s c r e t i z a t i o n s t a r t s C MXYLEV > Maximum number of l e v e l s 1n y - d i r e c t l o n C LIST > .NE. 0 -> Enable program l i s t / t r a c e C .EQ. 0 -> D i s a b l e C PHIIN > .EQ. O -> C a l c u l a t e i n i t i a l c o a r s e g r i d e s t . C .NE. 0 -> Read " C C C R o u t i n e s c a l l e d : C C CDATE (System) C TIME (System) C MGMAIN C INTGRT C C C MTS l o g i c a l u n i t a s s i g n m e n t s : C C 2 -> (IN) I n i t i a l e s t i m a t e of c o a r s e g r i d f e n C 3 -> (OUT) F i n a l c a l c u l a t e d c o a r s e g r i d f e n C 4 -> (IN/OUT) F i n a l c a l c u l a t e d f i n e g r i d f e n s ' C 5 -> (OUT) L i s t i n g / program t r a c e C G -> (IN/OUT) User communication C 7 -> (IN) Run parameters C c  C C REAL*8 MEMORY(150000) C R E A L * 8 C0MMNT(1O) C REAL DATEAR(7 ) C REAL*8 CVTAU, DXTOL, DYTOL, P C INTEGER CONTRL, CPUT1. CPUT2. CPUTOT, I, LIST, LUNIT, * MAXCYC, MAXLEV, PHIIN, PHIOUT, RESOUT, USER, * REFLEV, MXYLEV C C DATA DXTOL / 1.0D-4 /, DYTOL / 1.0D-4 / 172 DATA MAXCYC / 4 /, PHIOUT / 3 /, * RESOUT / 4 /, LUNIT / 5 /, * USER / 6 /, CONTRL / 7 / C C C INPUT RUN PARAMETERS C READ(CONTRL,100) P, CVTAU. MAXLEV, REFLEV, MXYLEV, LIST, PHIIN 100 F0RMAT(F1O.3,E1O.4,5I4) C C ECHO INPUT, DATE, TIME, ETC. C CALL CDATE(DATEAR) WRITE(USER,110) 110 FORMAT(' ENTER COMMENT:') READ(USER,120) (COMMNT(I) , I = 1 , 10) 120 FORMAT(10A8 ) WRITE(LUNIT,130) (DATEAR(I) , 1 = 1 , 7 ) 130 FORMAT(1H-,7A4) WRITE(LUNIT,140) (COMMNT(I) , I = 1 , 10) 140 FORMAT(1H0,10A8) WRITE(LUNIT,150) P, MAXLEV 150 FORMAT('OMOMENTUM:',F20.3/' NUMBER OF LEVELS:',13) WRITE(LUNIT,155) MXYLEV, REFLEV 155 FORMAT(' MXYLEV: ',13,' REFLEV: '.13) WRITE(LUNIT,160) 160 FORMAT('— - - - - - - - - - M U L T I * ' G R I D D I N G B E G I N S ' / / ) C C MULTI-GRID C CALL TIME(O) CALL MGMAIN(MEMORY,150000,P,CVTAU,MAXLEV,REFLEV,MXYLEV,MAXCYC, * LI ST.LUNIT,PHI IN,PHIOUT,RESOUT,USER) CALL TIME(1,0,CPUT1) WRITE(LUNIT,170) 170 FORMATC-- - - - - - - - - - M U L T I * ' G R I D D I N G C O M P L E T E D'/) WRITE(LUNIT,180) CPUT1 180 F0RMAT('OMULTI-GRID CPU TIME:',T40,18,' MS.') C REWIND RESOUT C WRITE(LUNIT,190) 190 FORMAT( ' -- - - - - - - - - - N U M E R I C , * ' I N T E G R A T I O N * ' B E G I N S'//) C C NUMERIC INTEGRATION ('VOLUME' INTEGRAL) C CALL TIME(O) CALL INTGRT(MEMORY, 150000,DXTOL,DYTOL,RESOUT,LUNIT) CALL TIME(1.0.CPUT2) WRITE(LUNIT,200) 200 FORMATC-- - - - - - - - - - N U M E R I C , * ' I N T E G R A T I O N * ' C O M P L E T E D'/) CPUTOT = CPUT 1 + CPUT2 WRITE(LUNIT,210) CPUT2, CPUTOT 210 F0RMAT('OINTEGRATION CPU TIME:',T40.18,' MS.'/ * 'OTOTAL CPU TIME:',T40,18,' MS.') C STOP C C END SUBROUTINE MGMAIN M U L T I • - G R I D M A I N R O U T I N E C C Main r o u t i n e i n im p l e m e n t a t i o n of FAS m u l t i - g r i d a l g o r i t h m . C R e s p o n s i b l e f o r c o n t r o l l i n g o v e r a l l f l o w of a l g o r i t h m . C D l s c r e t l z e d boundary v a l u e problem 1s s o l v e d on a maximum C Of MAXLEV l e v e l s , or u n t i l the l o c a l t r u n c a t i o n e r r o r C e s t i m a t e a t some l e v e l i s u n i f o r m l y below CVTAU. R o u t i n e i s C r e s p o n s i b l e f o r i n i t i a l i z a t i o n o f the g r i d d a t a s t r u c t u r e , C g e n e r a t i o n of g r i d and c o o r d i n a t e f u n c t i o n s , d e f i n i t i o n C of a l 1 g r i d s , and output of f i n a l c a l c u l a t e d f u n c t i o n . C C Parameters: C C MEMORY(NMEM) C p C CVTAU C MAXLEV C REFLEV C C MXYLEV C C MAXCYC C LIST --C LUNIT C PHIIN ---C C C C PHIOUT C C RESOUT C C USER C C R o u t i n e s c a l l e d : C C INIGST C NEWGRD C DEFGRD C DIRSPC C PRVPHI C GUESS -> Main s t o r a g e a r r a y -> Momentum -> T r u n c a t i o n e r r o r convergence parameter -> Max. number of l e v e l s of d i s c e t 1 z a t 1 on -> Le v e l at which a d a p t i v e d i s c r e t i z a t i o n b e g i ns -> Max. number of l e v e l s of d i s c r e t i z a t i o n i n y d i r e c t i o n . -> Max. number of c o a r s e g r i d c o r r e c t i o n c y c l e s t o be used to s o l v e any system -> L i s t i n g / t r a c e f l a g -> L i s t i n g u n i t -> .EO. 0 -> c a l c u l a t e i n i t i a l e s t i m a t e of unknown on c o a r s e s t g r i d .NE. 0 -> r e a d I n i t i a l e s t i m a t e from l o g i c a l u n i t PHIIN -> L o g i c a l u n i t f o r output of f i n a l c a l c u l a t e d c o a r s e g r i d unknown -> L o g i c a l u n i t f o r output of c a l c u l a t e d f i n e g r i d unknown and o t h e r r e s u l t s •-> L o g i c a l u n i t f o r u s e r communication 174 C -CYCLE C ADAPT C INTERP C INJECT C 0UTRS1 C 0UTRS2 C C Major l o c a l v a r i a b l e s : C C Note: These v a r i a b l e s a r e I n i t i a l i z e d i n DATA C s t a t e m e n t s - r o u t i n e must be r e c o m p i l e d C 1f v a l u e s a r e to be changed. C C A > I n v e r s i o n r a d i u s ( s h o u l d remain C I n i t i a l i z e d t o 1.0D0 s i n c e some C o t h e r r o u t i n e s assume t h i s v a l u e ) C ALPHA > M u l t 1 - g r 1 d c o n t r o l parameters C DELTA / C ETA / C EPSIO > I n i t i a l c o a r s e g r i d convergence c r i t C XINCO, YINCO > Co a r s e g r i d (CG) x.y mesh s p a c i n g s C XMNO. YMNO > Coords, of lower l e f t c o r n e r of CG C XMXO, YMXO > Coords, of upper r i g h t c o r n e r of CG C XMNDOM, YMNDOM -> Coords, of lower l e f t c o r n e r of C d i s c r e t e domain C XMXDOM. YMXDOM -> Coords, of upper r i g h t c o r n e r of C d i s c r e t e domain C INILEV > Label f o r I n i t i a l l e v e l (CG) C MAXGRD > Maximum number of g r i d s t o be used C NXLO, NXHI, > Boundary f l a g s ( see d e s c r i p t i o n of C NYLO, NYHI / g r i d d a t a s t r u c t u r e ) C PSWP > Minimum number of pre-CGC sweeps C OSWP > " " " post-CGC C c  c c c c c-c c c c c c SUBROUTINE MGMAIN(MEMORY,NMEM,P,CVTAU,MAXLEV,REFLEV,MXYLEV, • MAXCYC.LIST,LUNIT,PHI IN,PHIOUT,RESOUT.USER) G R I D INTEGER INTEGER REAL*8 LOGICAL INTEGER INTEGER INTEGER D A T A S T R U C T U R E GFSTRT(50,10), XFSTRT(50,10). YFSTRT(50,10) BROTHR(50), FATHER(50), LHEAD(50), LLINK(50) SON(50), NPT(50), NX(50), NY(50), XHIB(50), XL0B(5O), YHIB(50), YL0B(50) AREA(50). EPSI(50), ERR0R(50) XINC(50), YINC(50) NRMTAU(50) C0NV(5O), SL0W(50) CRSLEV, FINLEV, NGFUNC, NGRID, NXFUNC, NYFUNC MEMPT KSQR, R, RHS, RM, RP, TAU, W, WM, WP, U, OLDU 175 INTEGER NULL / Z80808080 / C C-C C C C COMMON / COMGST / LHEAD, LLINK. FATHER, BROTHR, SON, NX NY, NPT, AREA, XINC. YINC, XFSTRT, YFSTRT, GFSTRT, XLOB, XHIB, YLOB, YHIB, ERROR, CONV, SLOW, EPSI, NRMTAU, CRSLEV, FINLEV, NGRID, MEMPT COMMON / COMCON / R, RP, RM, W, WP. WM, U, OLDU, RHS. TAU, KSQR, NXFUNC, NYFUNC, NGFUNC INTEGER NMEM REAL*8 MEMORY(NMEM) REAL*8 NORM REAL*8 A, ALPHA, DELTA, EPSIO, EPSILN, ETA. EXTRAP, P, XINCO, XINCL, XMNO, XMNDOM, XMXO, XMXDOM, YINCO, YINCL, YMNO, YMNDOM, YMXO, YMXDOM, MGNORM, AREAL, CVTAU, NWXMXD, XMX, XMXD INTEGER SWPCNT(50) INTEGER CGRID, GRID, INILEV, LEV, LIST, LUNIT, MAXCYC, MAXGRD, MAXLEV, NXHI, NXLO, NYHI, NYLO, OGRID, PHIIN, PHIOUT, PSWP, QSWP, RC, RESOUT, USER, CPUTIM, REFLEV, L1, L2, LEVM1, MXYLEV, NXLEV1 C C C C C C C C C C LOGICAL NOREF COMMON / COMSWP / SWPCNT DATA A / 1.OOODO /, * DELTA / 0.500DO /, * ETA / 0.700D0 /, * XINCO / 0.250D0 /, * XMNO / -0.250DO /, XMNDOM / 0.OOODO /, * YMNO / -0.250D0 /, * YMNDOM / 0.OOODO /, DATA INILEV / 1 /, MAXGRD * NXLO / 2 /, NXHI * NYLO / 2 /, NYHI * PSWP / 3 /, QSWP ALPHA EPSIO YINCO XMXO XMXDOM / INITIALIZE GRID STRUCTURE MEMPT = 1 CALL INIGST(INI LEV,MAXGRD) INITIALIZE SWEEP COUNTERS DO 10 I = 1 , MAXLEV SWPCNT(I ) = 0 / / / / YMXO / YMXDOM / / 10 /. 0.350D0 /, 5.0D-04 /, 0.250D0 /. 1.OOODO /, 1.OOODO /, 1.250D0 /, 1.OOODO / CONTINUE CALL NEWGRD(INI LEV,NULL,MAXGRD, RC) IF( RC .NE. O ) GO TO 130 DEFINE COARSEST GRID GRID = NGRID XINCL = XINCO YINCL = YINCO CALL DEFGRD(GRID,XMNO,XMXO,YMNO,YMXO,XMNDOM,XMXDOM, YMNDOM,YMXDOM.XINCL,YINCL,NXLO,NXHI,NYLO, NYHI.EPSIO,1.ODO) CALL INIGRD(MEMORY(GFSTRT(GRID,KSQR)),MEMORY(GFSTRT(GRID,RHS)), MEMORY(XFSTRT(GRID,R)),MEMORY(XFSTRT(GRID,RP)), MEMORY(XFSTRT(GRID,RM)),MEMORY(YFSTRT(GRID,W)), MEMORY(YFSTRT(GRID,WP)).MEMORY(YFSTRT(GRID,WM)), NX(GRID),NY(GRID),XMNO,YMNO, XINC(GRID),YINC(GRID),A,P) ALLOCATE MEMORY REQUIRED BY DIRECT SOLVER AND SET UP AS MUCH OF ACTUAL FINITE DIFFERENCE MATRIX AS POSSIBLE CALL DIRSPC(MEMORY,MEMPT,NX(GRID),NY(GRID), MEMORY(XFSTRT(GRID,RP)).MEMORY(XFSTRT(GRID,RM)) , MEMORY(YFSTRT(GRID,WP)).MEMORY(YFSTRT(GRID,WM)), XINC(GRID),YINC(GRID)) IF( PHIIN .EQ. 0 ) GO TO 20 READ IN INITIAL ESTIMATE OF PHI CALL PRVPHI(MEMORY(GFSTRT(GRID,U)),NX(GRID),NY(GRID), PHIIN,RC) IF( RC .NE. 0 ) GO TO 150 GO TO 30 CALCULATE INITIAL ESTIMATE OF PHI CALL GUESS(MEMORY(GFSTRT(GRID,U)).MEMORY(XFSTRT(GRID,R)). NX(GRID),NY(GRID)) CGRID = GRID DISABLE TAU EXTRAPOLATION EXTRAP = 1.ODO EPSILN = EPSIO LEV = INILEV L O O P . . . FOR EACH LEVEL ... WRITE(LUNIT,200) LEV, MEMPT F O R M A T ( ' - * * * * » L E V E L 14, ' * * * * * / / / / MEMORY ', 'USAGE:',18,' DOUBLE WORDS') ATTEMPT TO SOLVE LEVEL LEV PROBLEM CALL CYCLE(MEMORY,LEV,MAXCYC,ALPHA,DELTA.ETA,EXTRAP, PSWP,QSWP,LI ST,LUNIT,RC) WRITE(LUNIT,210) ERROR(GRID) FORMAT('OFINAL DYNAMIC RESIDUAL NORM: '.E14.5) IF( LEV .EO. MAXLEV ) GO TO 76 WAS PROBLEM SOLVED USING MAXCYC OR FEWER CGC'S IF( RC .NE. O ) GO TO 140 IF SO, PROCEED TO NEXT LEVEL LEV = LEV + 1 L1 = LEV - 1 CALL NEWGRD(LEV,GRID,MAXGRD,RC) IF( RC .NE. 0 ) GO TO 130 IF( LEV .GE. REFLEV ) GO TO 60 FULL-SIZE NEW GRID AREAL = XINCL = IF( LEV XMX XMXD GO TO 70 1 .ODO 0.5D0 * XINCL . LE. MXYLEV ) YINCL XMXO XMXDOM 0.5D0 * YINCL NEW GRID DETERMINED ADAPTIVELY L2 = LEV - 2 CALL ADAPT(NX(L2),NY(L2), MEMORY(GFSTRT(L2,TAU)), MEMORY(XFSTRT(L2,R)).ALPHA, CVTAU,AREA(L1 ) ,AREAL,NWXMXD, NOREF) IF( NOREF ) GO TO 74 WRITE(LUNIT,215) LEV, NWXMXD FORMAT(' OUTER BOUNDARY OF GRID ',14, ' IS '.E14.6) XINCL = 0.5D0 * XINCL IF( LEV .LE. MXYLEV ) YINCL = 0.5D0 * YINCL XMX = NWXMXD XMXD = NWXMXD EPSILN = ALPHA * (EPSKL1) * AREA(L1 ) ) / AREAL OGRID = GRID GRID = NGRID DEFINE AND INITIALIZE NEW GRID. INTERPOLATE INITIAL ESTIMATE OF UNKNOWN FUNCTION CALL DEFGRD(GRID,XMNO.XMX,YMNO,YMXO.XMNDOM, XMXD.YMNDOM,YMXDOM,XINCL,YINCL, NXLO.NXHI,NYLO,NYHI,EPSILN,AREAL) CALL INIGRD(MEMORY(GFSTRT(GRID,KSQR)), MEMORY(GFSTRT(GRID,RHS)), MEMORY(XFSTRT(GRID,R)), MEMORY(XFSTRT(GRID.RP)), MEMORY(XFSTRT(GRID,RM)), MEMORY(YFSTRT(GRID,W)), MEMORY(YFSTRT(GRID,WP) ) , MEMORY(YFSTRT(GRID,WM)), NX ( GR ID ) , NY ( GR ID ) ,XMNO, YMNO, XINC(GRID),YINC(GRID),A,P) CALL INTERP(MEMORY.OGRID,GFSTRT(OGRID,U),GRID, GFSTRT(GRID,U),4,LIST,LUNIT) GO TO 40 E N D L O O P NORMAL EXIT MAXLEV = LEV - 1 LEV = MAXLEV IF( LEV .EQ. O ) GO TO 160 NX LEV 1 = NX(LEV) - 1 IF( MEMORY(XFSTRT(LEV,R)+NXLEV1) .EQ. 1.ODO ) GO TO 90 LEV = LEV - 1 GO TO 77 GRID = LEV INJECT FINE GRID FUNCTIONS TO FINEST LEVEL WHICH COVERS ENTIRE DISCRETE DOMAIN LEV = MAXLEV IF( LEV .EQ. GRID ) GO TO 110 LEVM1 = LEV - 1 CALL INJECT(MEMORY(GFSTRT(LEVM1,U)),MEMORY(GFSTRT(LEV,U)), NX(LEVM1),NY(LEVM1),NX(LEV),NY(LEV) , MEMORY(XFSTRT(LEVM1,R)),MEM0RY(YFSTRT(LEVM1,W)), MEMORY(XFSTRT(LEV,R)),MEMORY(YFSTRT(LEV,W)), XINC(LEVM1),YINC(LEVM1),XINC(LEV),YINC(LEV), XL0B(LEVM1),XHIB(LEVM1),YLOB(LEVM1),YHIB(LEVM1), XLOB(LEV).XHIB(LEV),YLOB(LEV),YHIB(LEV) . 1 ) LEV = LEVM1 GO TO 100 DO 120 1 = 1 , MAXLEV WRITE(LUNIT,220) SWPCNT(I), I FORMAT(14,' SWEEPS ON LEVEL ',14) CONTINUE OUTPUT CALCULATED QUANTITIES CALL OUTRS1(MEMORY(GFSTRT(GRID,KSQR)).MEMORY(GFSTRT(GRID,U)) , MEMORY(XFSTRT(GRID,R)),MEMORY(YFSTRT(GRID,W)), NX(GRID),NY(GRID).RESOUT) CALL 0UTRS2(MEMORY(GFSTRT(CGRID,U)),NX(CGRID),NY(CGRID), PHIOUT) WRITE(LUNIT,230) MEMPT FORMAT('-MEMORY USAGE: ',16,' DOUBLE WORDS.') RETURN ABNORMAL EXITS WRITE(USER,240) FORMAT('-PROGRAM ABORTING FOLLOWING ERROR RETURN FROM S/R ' NEWGRD. ' ) RETURN WRITE(USER,250) MAXCYC, LEV FORMATC-NO CONVERGENCE AFTER ',14,' CYCLES ON LEVEL ',14 ' <CR> TO CONTINUE, 1 TO EXIT.') READ(USER,260) RC FORMAT(I 1 ) IF( RC .EQ. 0 ) GO TO 50 GO TO 76 WRrTE(USER,270) FORMAT('-PROGRAM ABORTING FOLLOWING ERROR RETURN FROM S/R ' PRVPHI.') WRITE(USER,280) FORMAT('-COULD NOT FIND S=1 ! ! ' ) RETURN 180 c ' c C I N T E R M E D I A T E L E V E L R O U T I N E S e C F O R P E R F O R M I N G C O A R S E G R I D C C C O R R E C T I O N C Y C L E A N D A D A P T I V E C C D I S C R E T I Z A T I O N C c  c  C C SUBROUTINE CYCLE C C-C C C C C C C C C C C C C C c c c c c c c c c c c c c c-c c SUBROUTINE CYCLE(MEMORY,L,MAXCYC,ALPHA,DELTA,ETA,EXTRAP, * PSWP.QSWP,LIST,UNIT,RC) C C G R I D D A T A S T R U C T U R E C INTEGER GFSTRT(50,10), XFSTRT(50,10), YFSTRT(50,10) C INTEGER BROTHR(50), FATHER(50), LHEAD(50), LLINK(50), * S0N(50), NPT(50), NX(50), NY(50), XHIB(50), * XL0B(50), YHIB(50), YL0B(5O) C REAL*8 AREA(50), EPSI(50), ERR0R(50), NRMTAU(50), Attempts to s o l v e l e v e l L problem by c a l l i n g SOLVE to pe r f o r m r e l a x a t 1 on or d i r e c t s o l u t i o n and by p e r f o r m i n g at most MAXCYC c o a r s e g r i d c o r r e c t i o n c y c l e s . P a rameters: MEMORY -> Main s t o r a g e a r r a y L  -> L e v e l of c u r r e n t problem MAXCYC -> Maximum number of c o a r s e g r i d c o r r e c t i o n c y c l e s ALPHA --> Alpha DELTA --> Del t a ETA -> E t a EXTRAP -> T a u - e x t r a p o l a t i o n parameter PSWP ---> Minimum number of Pre-CGC r e l a x a t i o n sweeps OSWP ---> Minimum number of Post-CGC r e l a x a t i o n sweeps LIST ---> L i s t i n g f l a g UNIT --•-> L i s t ing uni t RC -> Ret u r n code -> .EO. 0 -> problem s o l v e d .EO. 1 -> problem not s o l v e d Rout 1nes c a l 1 e d : SOLVE : CGCST CGCFIN 181 c c c c c c c c c-c c c c c c c 10 90 c c c c c c 100 X INC(50), YINC(50) LOGICAL C0NV(5O), SLOW(50) INTEGER CRSLEV, FINLEV, NGFUNC, NGRID, NXFUNC, NYFUNC INTEGER MEMPT INTEGER KSQR, R, RHS, RM, RP, TAU, W, WM, WP, U, OLDU INTEGER NULL / Z80808080 / COMMON / COMGST / LHEAD, LLINK, FATHER, BROTHR, SON, NX, NY, NPT, AREA, XINC, YINC, XFSTRT, YFSTRT, GFSTRT, XLOB, XHIB, YLOB, YHIB, ERROR, CONV, SLOW, EPSI, NRMTAU, CRSLEV, FINLEV, NGRID, MEMPT COMMON / COMCON / R, RP, RM, W, WP, WM, U, OLDU, RHS, TAU, KSQR, NXFUNC, NYFUNC, NGFUNC REAL*8 MEMORY(1) REAL*8 ALPHA, DELTA, ETA, EXTRAP INTEGER L, LEV, LIST, MAXCYC, NCYC, RC, UNIT INTEGER MINSWP, NSWP, PSWP, QSWP LOGICAL ALCONV NCYC = O LEV = L MINSWP = PSWP CALL SOLVE(MEMORY,LEV,ALCONV,ETA,3,MINSWP,NSWP,LI ST,UNIT) IF( LIST .GT. 0 ) WRITE(UNIT, 90) NSWP, LEV F0RMAT(1H ,14,' SWEEPS AT LEVEL ',13) CHECK FOR CONVERGENCE IF( ALCONV ) GO TO 20 IF( LEV .EO. L .AND. NCYC .EO. MAXCYC ) GO TO 40 INITIATE COARSE GRID CORRECTION IF( LIST .NE. 0 ) WRITE(UNIT,100) FORMATC STARTING COARSE GRID CORRECTION') CALL CGCST(MEMORY,LEV,ALPHA.DELTA,EXTRAP,LI ST,UNIT) IF( LEV .EO. L ) NCYC = NCYC + 1 LEV = LEV - 1 MINSWP = PSWP GO TO 10 HAS CONVERGENCE AT LEVEL L BEEN ACHIEVED ? 182 20 IF( LEV .EQ. L ) GO TO 30 C C IF NOT, THEN A COARSE GRID CORRECTION IS TO BE C COMPLETED C IF( LIST .NE. 0 ) WRITE(UNIT,110) 110 FORMAT(' COMPLETING COARSE GRID CORRECTION') CALL CGCFIN(MEMORY,LEV,LI ST,UNIT) LEV = LEV + 1 MINSWP = QSWP GO TO 10 C C CONVERGENCE ON LEVEL L IN MAXCYC OR FEWER CYCLES C 30 RC = O RETURN C C CONVERGENCE NOT ACHIEVED AFTER MAXCYC CYCLES C 40 RC = 1 RETURN C C END C--C C C C-C C C C C C C C C C C C C C C c c c c c c c c c c SUBROUTINE SOLVE C a l l s r e l a x a t i o n r o u t i n e to a p p l y minimum of MINSWP r e l a x a t i o n sweeps on a l l g r i d s at l e v e l L, or invokes d i r e c t s o l v e r i f L i s c o a r s e s t l e v e l . Returns when problem has been s o l v e d on l e v e l L, or i f convergence i s slow i n d i c a t i n g the need f o r a c o a r s e g r i d c o r r e c t i o n . P arameters: MEMORY L ALCONV ETA ---INIMOD MINSWP NSWP --LIST --UNIT --Main s t o r a g e a r r a y L e v e l t h a t problem i s b e i n g s o l v e d at •Returns .TRUE, i f problem i s s o l v e d on a l l g r i d s a t l e v e l L, .FALSE, i f not E t a I n i t i a l r e l a x a t i o n mode Minimum number of r e l a x a t i o n sweeps R e t u r n s number o f sweeps performed L i s t i n g f l a g L i s t i ng u n l t Rout 1nes c a l 1 e d : MAXO LINRLX DIRECT 183 c c c c c c c c-c c c c c c c c SUBROUTINE G R I D INTEGER INTEGER REAL*8 LOGICAL INTEGER INTEGER INTEGER INTEGER COMMON COMMON SOLVE(MEMORY,L,ALCONV,ETA,INIMOD,MINSWP,NSWP, LIST,UNIT) D A T A S T R U C T U R E GFSTRT(50,10), XFSTRT(50,10). YFSTRT(50,10) BROTHR(50), FATHER(50), LHEAD(50). LLINK(50), S0N(50), NPT(50), NX(50), NY(50), XHIB(50), XL0B(5O), YHIB(50), YL0B(50) AREA(50 ) , EPSI(50), ERR0R(50), NRMTAU(50), XINC(50), YINC(50) C0NV(50), SL0W(5O) CRSLEV, FINLEV, NGFUNC, NGRID, NXFUNC, NYFUNC MEMPT KSQR, R, RHS, RM. RP, TAU, W, WM, WP. U, OLDU NULL / Z80808080 / / COMGST / LHEAD, LLINK, FATHER, BROTHR, SON, NX, NY, NPT, AREA, XINC, YINC, XFSTRT, YFSTRT, GFSTRT, XLOB, XHIB, YLOB, YHIB, ERROR, CONV, SLOW, EPSI, NRMTAU, CRSLEV, FINLEV, NGRID, MEMPT / COMCON / R, RP, RM, W, WP, WM, U, OLDU, RHS, TAU, KSQR, NXFUNC, NYFUNC, NGFUNC REAL*8 MEMORY(1) REAL*8 DIRECT, LINRLX REAL*8 ERRGRD, DIREPS. ETA, INFNTY INTEGER NXTM0D(4), SWPCNT(50) INTEGER MAXO INTEGER GRID, INIMOD, L, LIST, MINSWP, MODE, NSWP, NTEMP, NXGRID, NYGRID, UNIT INTEGER DELPHI, JACOB, MNITER, MX ITER, NJ, NXW, NYW, RC, WORK LOGICAL ALSLOW, ALCONV COMMON / DIR / JACOB, NJ, WORK, NXW, NYW, DELPHI / COMSWP / SWPCNT DATA INFNTY / 1.0D50 /, MNITER / 3 /, MXITER / 10 /, NXTMOD / 2, 1, 3, 3/ WRITE(UNIT,2000) (EPSI(I ) ,I=1,L) F0RMAT(8E14.5) GRID = LHEAD(L) NSWP = O MODE = INIMOD IF( GRID .EQ. NULL ) GO TO 20 SLOW(GRID) = .FALSE. CONV(GRID) = .FALSE. ERROR(GRID ) = INFNTY GRID = LLINK(GRID) GO TO 10 ALSLOW = .TRUE. ALCONV = .TRUE. GRID = LHEAD(L) IF( GRID .EQ. NULL ) GO TO 50 IF( NSWP .LT. MINSWP ) GO TO 32 IF( SLOW(GRID) .AND. L .NE. CRSLEV ) GO TO 40 IF( CONV(GRID) ) GO TO 40 NXGRID = NX(GRID) NYGRID = NY(GRID) IF( L .EQ. CRSLEV ) GO TO 33 PERFORM RELAXATION SWEEP NTEMP = MAXO(NXGRID,NYGRID) ERRGRD = LINRLX(MEMORY(GFSTRT(GRID,U ) ) , MEMORY(GFSTRT(GRID,KSQR)), MEMORY(GFSTRT(GRID,RHS)), MEMORY(XFSTRT(GRID.R)), MEMORY(XFSTRT(GRID,RP) ) , MEMORY(XFSTRT(GRID,RM)), MEMORY(YFSTRT(GRID.W)), MEMORY(YFSTRT(GRID,WP ) ) , MEMORY(YFSTRT(GRID,WM)), X INC(GRID),YINC(GRID).NXGRID,NYGRID, XLOB(GRID) ,XHIB(GRID),YLOB(GRID),YHIB(GRID) MEMORY(MEMPT),MEMORY(MEMPT+NTEMP ) , MEMORY(MEMPT+2*NTEMP) .MEMORY(MEMPT+3*NTEMP) MODE,LIST,UNIT) SWPCNT(L) = SWPCNT(L) + 1 GO TO 34 SOLVE DIRECTLY DIREPS = EPSI(GRID) ERRGRD = DIRECT(MEMORY(GFSTRT(GRID,U)), MEMORY(GFSTRT(GRID,KSQR)), MEMORY(GFSTRT(GRID,RHS)), MEMORY(MEMPT), MEMORY(XFSTRT(GRID,R)), 185 c c 1000 c c c c c 34 MEMORY(XFSTRT(GRID,RP)), MEMORY(XFSTRT(GRID.RM)), MEMORY(YFSTRT(GRID,W)), MEMORY(YFSTRT(GRID.WP)), MEMORY(YFSTRT(GRID,WM)) , MEMORY(JACOB),MEMORY(WORK),MEMORY(DELPHI), XINC(GRID),YINC(GRID).NXGRID,NYGRID,NJ.NXW.NYW, XLOB(GRID),XHIB(GRID),YLOB(GRID),YHIB(GRID), DIREPS,MNITER.MXITER.LIST,UNIT,RC) SWPCNT(L) = SWPCNT(L) + 1 IF( RC .NE. O ) WRITE(UNIT,1000) FORMAT('-ABORT PROGRAM: FAILURE IN DIRECT') EXIT RETURN IF( ERRGRD .LE. EPSI(GRID) ) CONV(GRID) SLOW(GRID) = .FALSE. IF( (ERRGRD / ERROR(GRID)) .GT. ETA ) SLOW(GRID) = .TRUE. ERROR(GRID) = ERRGRD .TRUE. 40 ALSLOW = ALSLOW .AND. SLOW(GRID) ALCONV = ALCONV .AND. CONV(GRID) GRID = LLINK(GRID) GO TO 30 50 IF( ALCONV ) RETURN NSWP = NSWP + 1 MODE = NXTMOD(MODE) IF( L .EQ. CRSLEV .OR. .OR. .NOT.ALSLOW NSWP .LT. MINSWP ) GO TO 20 RETURN END <j C C SUBROUTINE CGCST C c  C C S t a r t s c o a r s e g r i d c o r r e c t i o n of l e v e l LEV on l e v e l LEV-1. C Updates r 1ght-hand-s 1de of l e v e l LEV-1 e q u a t i o n s , c a l c u l a t e s C r e l a t i v e t r u n c a t i o n e r r o r e s t i m a t e s , and updates convergence C c r i t e r i a f o r l e v e l LEV-1 and p o s s i b l y l e v e l LEV. C C Parameters: C C MEMORY -> Main s t o r a g e a r r a y 186 c c c c c c c c c c c c c c c c c-c c c c-c c c c c c c c c-c c c c LEV > L e v e l b e i n g c o r r e c t e d ALPHA --> Al p h a DELTA --> D e l t a EXTRAP -> T a u - e x t r a p o l a t i o n parameter LIST > L i s t i n g f l a g UNIT > L i s t i n g u n i t R o u t i n e s c a l l e d : COPY INIT OPERAT INJECT CLCTAU NEWRHS SUBROUTINE CGCST(MEMORY,LEV,ALPHA,DELTA.EXTRAP,LI ST,UNIT) G R I D D A T A S T R U C T U R E INTEGER GFSTRT(50,10), XFSTRT(50,10), YFSTRT(50,10) INTEGER BROTHR(50), FATHER(50), LHEAD(50), LLINK(50), • S0N(50), NPT(50), NX(50), NY(50), XHIB(50), • XL0B(5O), YHIB(50), YL0B(50) REAL*8 AREA(50), EPSI(50), ERR0R(5O), NRMTAU(50), • XINC(50), YINC(50) LOGICAL C0NV(50), SL0W(5O) INTEGER CRSLEV, FINLEV, NGFUNC, NGRID, NXFUNC, NYFUNC INTEGER MEMPT INTEGER KSQR, R, RHS, RM, RP, TAU, W, WM, WP, U, OLDU INTEGER NULL / Z80808080 / COMMON / COMGST / LHEAD, LLINK, FATHER, BROTHR, SON, * NX, NY, NPT, AREA, XINC, YINC, * XFSTRT, YFSTRT, GFSTRT. XLOB, XHIB, * YLOB, YHIB. ERROR, CONV, SLOW, EPSI, * NRMTAU, CRSLEV, FINLEV, NGRID, MEMPT COMMON / COMCON / R, RP, RM, W, WP, WM, U, OLDU, RHS, * TAU, KSQR, NXFUNC, NYFUNC, NGFUNC REAL*8 MEMORY(1 ) REAL*8 CLCTAU REAL*8 ALPHA, BETA, EXTRAP, MAXERR, NOTDEF, TAREA INTEGER DAD, DADLEV, FSTBRN, GRID, LEV, LIST, NXD, 18 NYD, TAULEV, TEMP 1, TEMP2, UNIT DATA NOTDEF / Z8O8O8O8O8O8O8O8O / OMEMPT = MEMPT DADLEV = LEV - 1 DAD = LHEAD(DADLEV) FOR EACH GRID ON LEVEL LEV - 1 IF( DAD .EQ. NULL ) GO TO 50 MEMPT = OMEMPT FSTBRN = SON(DAD) IF GRID HAS ANY SONS IF( FSTBRN .EO. NULL ) GO TO 40 THEN CREATE COPY OF CURRENT GRID FUNCTION AND ALLOCATE STORAGE NXD = NX(DAD) NYD = NY(DAD) TAREA = O.ODO MAXERR = O.ODO GRID = FSTBRN CALL COPY(MEMORY(GFSTRT(DAD,U)), MEMORY(GFSTRT(DAD,OLDU)),NXD,NYD) TEMPI = MEMPT MEMPT = MEMPT + NPT(DAD) CALL INIT(MEMORY(TEMPI),NXD,NYD,NOTDEF) TEMP2 = MEMPT MEMPT = MEMPT + NPT(DAD) CALL INIT(MEMORY(TEMP2),NXD,NYD,O.ODO) FOR EACH SON, APPLY DIFFERENCE OPERATOR, INJECT INTO TEMP 1, AND INJECT (GRID.U) INTO (DAD,OLDU) IF( GRID .EO. NULL ) GO TO 30 CALL OPERAT(MEMORY(GFSTRT(GRID,U)), MEMORY(GFSTRT(GRID,KSQR)),MEMORY(MEMPT), MEMORY(MEMPT) , MEMORY(XFSTRT(GRID,R)).MEMORY(XFSTRT(GRID,RP)), MEMORY(XFSTRT(GRID,RM)).MEMORY(YFSTRT(GRID,W)), MEMORY(YFSTRT(GRID.WP)).MEMORY(YFSTRT(GRID.WM)) X INC(GRID),YINC(GRID),NX(GRID),NY(GRID), XLOB(GRID),XHIB(GRID),YL0B(GRID),YHIB(GRID), O.ODO,LIST,UNIT) CALL INJECT (MEMORY (TEMP.1 ), MEMORY (MEMPT ), NXD , NYD , NX( GRID ) , NY ( GRID ) , MEMORY ( XFSTRT (DAD , R ) ) , MEMORY(YFSTRT(DAD,W)),MEMORY(XFSTRT(GRID,R)). MEMORY(YFSTRT(GRID,W)).XINC(DAD),YINC(DAD), XINC(GRID),YINC(GRID),XLOB(DAD),XHIB(DAD), YLOB(DAD),YHIB(DAD),XLOB(GRID),XHIB(GRID), YLOB(GRID),YHIB(GRID),1) CALL INJECT(MEMORY(GFSTRT(DAD,OLDU)), MEMORY(GFSTRT(GRID.U)),NXD, NYD,NX(GRID),NY(GRID).MEMORY(XFSTRT(OAD.R)), 188 * MEMORY(YFSTRT(DAD,W)),MEMORY(XFSTRT(GRID.R)), * MEMORY(YFSTRT(GRID.W)),XINC(DAD).YINC(DAD), + X INC(GRID),YINC(GRID),XLOB(DAD),XHIB(DAD), * YLOB(DAD),YHIB(DAD),XLOB(GRID).XHIB(GRID), * YLOB(GRID),YHIB(GRID),0) CALL INJECT(MEMORY(TEMP2), * MEMORY(GFSTRT(GRID,RHS)),NXD, * NYD,NX(GRID),NY(GRID).MEMORY(XFSTRT(DAD,R)), * MEMORY(YFSTRT(DAD,W)),MEMORY(XFSTRT(GRID,R)), * MEMORY(YFSTRT(GRID,W)).XINC(DAD),YINC(DAD), * XINC(GRID),YINC(GRID),XLOB(DAD),XHIB(DAD), * YLOB(DAD),YHIB(DAD),XLOB(GRID),XHIB(GRID), * YLOB(GRID),YHIB(GRID),1) TAREA = TAREA + AREA(GRID) IF( ERROR(GRID) .GT. MAXERR ) * MAXERR = ERROR(GRID) GRID = BROTHR(GRID ) GO TO 20 C C CALCULATE LOCAL TRUNCATION ERROR AND UPDATE C (DAD,RHS) C 30 CALL OPERAT(MEMORY(GFSTRT(DAD,OLDU)), * MEMORY(GFSTRT(DAD,KSQR)),MEMORY(MEMPT), * MEMORY(MEMPT), * MEMORY(XFSTRT(DAD,R)),MEMORY(XFSTRT(DAD,RP)), • * MEMORY(XFSTRT(DAD,RM)),MEMORY(YFSTRT(DAD,W)), * MEMORY(YFSTRT(DAD,WP)).MEMORY(YFSTRT(DAD.WM)), * XINC(DAD),YINC(DAD),NXD,NYD, * XLOB(DAD),XHIB(DAD).YLOB(DAD),YHIB(DAD), * O.OOO,LIST,UNIT) NRMTAU(DAD) = CLCTAU(MEMORY(GFSTRT(DAD,TAU)), * MEMORY(MEMPT),MEMORY(TEMP 1),MEMORY(XFSTRT(DAD,R)) * MEMORY(YFSTRT(DAD,W)),NXD,NYD) CALL COPY(MEMORY(GFSTRT(DAD,OLDU)), * MEMORY(GFSTRT(DAD,U)),NXD,NYD) CALL NEWRHS(MEMORY(GFSTRT(DAD.RHS)), * MEMORY(GFSTRT(DAD.TAU)).MEMORY(TEMP2),NXD.NYD, * DADLEV,FINLEV,EXTRAP) C C UPDATE CONVERGENCE CRITERIA C EPSI(DAD) = DELTA * MAXERR * * (TAREA / AREA(DAD)) 32 WRITE(6,1000) DADLEV, EPSI(DAD) 1000 FORMAT('OCONVERGENCE CRITERIA ON LEVEL ',13, * ' I S NOW ',E15.5) IF( DADLEV .NE. (FINLEV - 1) ) GO TO 40 TGRID = SON(DAD) 35 IF( TGRID .EQ. NULL ) GO TO 40 EPSI(TGRID) = ALPHA * NRMTAU(DAD) * * (AREA(DAD ) / AREA(TGRID ) ) TGRID = BROTHR(TGRID) GO TO 35 C 40 DAD = LLINK(DAD) C GO TO 10 C 50 MEMPT = OMEMPT 189 RETURN END C C C C C C C C C C C C C C C C C C C C C C C C C C c c-c SUBROUTINE CGCFIN F in ishes coarse g r i d c o r r e c t i o n on level is in te rpo la ted to level LEV+1 and f i n e updated appropr ia te l y . LEV. gr i d Correct ion funct ions are Parameters: MEMORY -> Main storage array LEV > Level that c o r r e c t i o n has LIST > L i s t i n g f l a g UNIT > L i s t i n g unit been ca lcu la ted on Routines c a l l e d : ADDSUB INIT INTERP SUBROUTINE CGCFIN(MEMORY,LEV,LI ST,UNIT) G R I D D A T A S T R U C T U R E INTEGER GF STRT(50, 10), XFSTRT(50, 10), YFSTRT(50, 10) INTEGER BROTHR(50), FATHER(50), LHEAD(50), LLINK(50), * SON(50), NPT(50), NX(50), NY(50). XHIB(50), * XL0B(50), YHIB(50), YL0B(50) REAL*8 AREA(50), EPSI(50), ERR0R(50), NRMTAU(50), * XINC(50), YINC(50) LOGICAL C0NV(50), SL0W(50) INTEGER CRSLEV. FINLEV, NGFUNC, NGRID, NXFUNC, NYFUNC INTEGER MEMPT INTEGER KSQR, R, RHS, RM, RP, TAU, W, WM, WP, U, OLDU INTEGER NULL / Z80808080 / COMMON / COMGST / LHEAD, LLINK, FATHER, BROTHR, SON, * NX, NY, NPT, AREA, XINC, YINC, * XFSTRT, YFSTRT, GFSTRT, XLOB, XHIB, * YLOB, YHIB, ERROR, CONV, SLOW, EPSI 190 NRMTAU, CRSLEV, FINLEV, NGRID, MEMPT COMMON / COMCON / R, RP, RM, W, WP, WM, U, OLDU, RHS, TAU, KSQR, NXFUNC, NYFUNC, NGFUNC REAL*8 MEMORY(1) INTEGER DAD, FSTBRN, GRID, LEV, LIST, NXG, NYG, UNIT C C C C DAD = LHEAD(LEV) C C FOR EACH GRID ON LEVEL LEV C 10 IF( DAD .EO. NULL ) GO TO 40 C C IF GRID HAS SONS C FSTBRN = SON(DAD) IF( FSTBRN .EO. NULL ) GO TO 30 C C CALUCLATE (DAD,U) - (DAD.OLDU) C CALL ADDSUB(MEMORY(GFSTRT(DAD.U) ), * MEMORY(GFSTRT(DAD,OLDU)),MEMORY(GFSTRT(DAD,OLDU)) * NX(DAD),NY(DAD),- 1.ODO) C C FOR EACH GRID WHICH IS A SON OF DAD. UPDATE GRID C FUNCTION (GRID.U) C GRID = FSTBRN C 20 IF( GRID .EO. NULL ) GO TO 30 NXG = NX(GRID) NYG = NY(GRID) CALL INIT(MEMORY(MEMPT ),NXG,NYG,O.ODO) CALL INTERP(MEMORY,DAD,GFSTRT(DAD,OLDU).GRID, * MEMPT,2,LIST,UNIT) CALL ADDSUB(MEMORY(GFSTRT(GRID,U)), * MEMORY(MEMPT),MEMORY(GFSTRT(GRID,U)),NXG. * NYG.1.ODO) GRID = BROTHR(GRID) GO TO 20 C 30 DAD = LLINK(DAD) GO TO 10 C 40 RETURN C C END C C SUBROUTINE ADAPT C 191 c C Scans t runcat ion er ror estimate contained in TAUL2 to C decide where outer boundary of new f i n e g r i d should be. C (If new g r i d is at level L, t runcat ion er ror estimate is C for level L-2 r e l a t i v e to L-1) C C Parameters: C C TAUL2(NXL2,NYL2) -> Truncation error estimate C XL2(NXL2) > X coordinates of level L-2 C ALPHA > Alpha C CVTAU > Truncation er ror convergence c r i t e r i o n C AREAL1 > No longer used - dummy parameter C AREAL > " C NWXMXD > Coordinate of new outer boundary C NOREF > Returns .TRUE, i f t runcat ion error C uniformly below CVTAU C c  C c SUBROUTINE ADAPT(NXL2,NYL2,TAUL2,XL2,ALPHA,CVTAU,AREAL1 , * AREAL,NWXMXD,NOREF) C C C C C C C c c c c c c 10 c INTEGER NXL2, NYL2 REAL*8 TAUL2(NXL2,NY L2) REAL*8 XL2(NXL2) REAL*8 DABS REAL*8 ALPHA, AREAL, AREAL 1 , CVSM, CVTAU, ETAULJ, NOCVSM, NOTDEF, NWXMXD, SUMTAU, TAUIJ, TOTSM INTEGER I, IMXNW, J , NXL2M1, NYL2M1. NYL2M2 LOGICAL NOREF DATA NOTDEF / Z8080808080808080 / CVSM NOCVSM NXL2M1 NYL2M1 NYL2M2 NOREF IMXNW = O.ODO = O.ODO = NXL2 -= NYL2 -= NYL2 -= .TRUE. = NXL2 1 1 2 I = NXL2M1 IF( I . LT. 2 ) GO TO 60 SUMTAU = O.ODO DO 20 J = 2 , NYL2M1 TAUIJ = TAUL2(I,J) IF( TAUIJ .EQ. NOTDEF ) GO TO 50 SUMTAU = SUMTAU + DABS(TAUIJ) 20 CONTINUE C ETAULJ = ALPHA * SUMTAU / NYL2M2 IF( ETAULJ .LE . CVTAU .AND. NOREF ) GO TO 40 C C REFINEMENT NECESSARY IN THIS REGION C IF( .NOT. NOREF ) GO TO 30 NOREF = .FALSE. IF( I .NE. NXL2M1 ) IMXNW = I 30 NOCVSM = NOCVSM + ETAULJ GO TO 50 C C NO REFINEMENT NEEDED YET C 40 CVSM = CVSM + ETAULJ C 50 1 = 1 - 1 C GO TO 10 -C 60 TOTSM = CVSM + NOCVSM NWAREA = XL2(IMXNW) NWXMXD = XL2(IMXNW) IF( IMXNW .EO. 2 ) NOREF = .TRUE. C RETURN C C END c  C C FUNCTION CLCTAU C c  C C Ca lcu la tes r e l a t i v e t runcat ion er ror estimate from funct ions C LIU and ILU and places in array TAU. Returns norm of TAU. C Value of NOTDEF is p laced in TAU anywhere where t runcat ion C e r ro r Is undef ined. C C Parameters: C C TAU(NX,NY) - -> Re la t i ve t runcat ion er ror estimate C LIU(NX.NY) - -> Residuals of Injected f i n e g r i d fen C ILU(NX.NY) - -> Injected f i n e g r i d res idua ls C X, Y > Gr id coordinates (cur rent ly unused) C C Routines c a l l e d : C C DABS C c  c c DOUBLE PRECISION FUNCTION CLCTAU(TAU,LIU,ILU,X,Y,NX,NY) 193 c c c c c c c c c c 10 20 30 40 50 C C C C INTEGER NX, NY REAL*8 ILU(NX,NY), LIU(NX,NY), TAU(NX,NY) REAL*8 X(1) , Y(1) REAL*8 DABS REAL*8 ILUXY, NOTDEF, NRMTAU, TAUXY INTEGER INRM, IX, IY, NX 1, NY 1 DATA NOTDEF / Z8080808080808080 / NRMTAU = DO 30 IY O.ODO = 1 , NY DO 20 IX = 1 , NX ILUXY = ILU(IX.IY) IF( ILUXY .EO. NOTDEF ) GO TO 10 TAUXY = LIU(IX.IY) - ILUXY TAU(IX.IY) = TAUXY GO TO 20 TAU(IX.IY) = NOTDEF CONTINUE » CONTINUE NX 1 = NX NY 1 = NY DO 50 IX = 2 NX 1 DO 40 IY = 2 , NY1 NRMTAU = NRMTAU + DABS(TAU(IX,IY ) ) CONTINUE CONTINUE CLCTAU = NRMTAU / ((NX 1 - 1) * (NY 1 - 1)) RETURN END C-C C C C-C C C C C C SUBROUTINE NEWRHS Ca lcu la tes new r i g h t - h a n d - s i d e of d i f fe rence equations from t runcat ion er ror est imate, TAU and Injected r i g h t - h a n d - s i d e , INJRHS of f i n e r level d i f f e r e n c e equations. Parameters: 194 c C RHS(NX,NY) > New rhs of d i f fe rence equations C TAU(NX,NY) > Re lat i ve local trunc. er ror estimate C INJRHS(NX,NY) -> Injected rhs of f i n e r level equations C L > Level C LASTL > Current f i n e s t level C EXTRAP > Tau -ext rapo lat ion parameter C c  C c c c c c c c c c SUBROUTINE NEWRHS(RHS,TAU,INJRHS.NX,NY,L,LASTL,EXTRAP) INTEGER NX, NY REAL*8 INJRHS(NX,NY), RHS(NX,NY), TAU(NX.NY) REAL*8 EXTRAP, NOTDEF, TAUEXT, TAUXY INTEGER IX, IY, L, LASTL DATA NOTDEF / Z8O8O8O8O8O8O8O8O / IF( L .EQ. (LASTL - 1) ) GO TO 10 TAUEXT = 1.ODO GO TO 20 10 TAUEXT = EXTRAP C 20 DO 40 IY = 1 , NY C DO 30 IX = 1 , NX TAUXY = TAU(IX.IY) IF( TAUXY .EQ. NOTDEF ) GO TO 25 RHS(IX.IY) = TAUEXT * TAUXY + INJRHS(IX,IY) GO TO 30 25 TAU(IX,IY) = O.ODO 30 CONTINUE C 40 CONTINUE C C C RETURN END 195 c  c C R E L A X A T I O N A N D R E S I D U A L C C C A L C U L A T I N G R O U T I N E S C c  c  c C FUNCTION LINRLX C c  c C Line re laxa t ion rout ine . Performs one l ine re laxat ion C sweep of unknown funct ion PHI. If MODE .GT. 2, then l ines C of constant rad ia l coordinate are used; If MODE .LE . 2, C l i nes of constant angular coordinate are used. If MODE is C odd, forward sweeping d i r e c t i o n is used; If MODE 1s even C reverse sweeping d i r e c t i o n Is used. Uses external rout ine C TRISLV (MTS l i b r a r y funct ion) to solve t r id iagonal systems. C Invokes appropr iate rout ines to smooth boundary d i f fe rence C equat ions. Returns norm of i n t e r i o r dynamic r e s i d u a l s . C C Parameters: C C PHI(NS.NW) - -> Unknown funct ion C KSQR(NS,NW) -> E x t r i n s i c curvature squared C RHS(NS.NW) - -> Right-hand-s1de of d i f fe rence system C S, S2, S4 > Radial coordinate funct ions C W, WP, WM > Angular C DELS > Radial mesh spacing C DELW > Angular C XLOB, XHIB, -> Boundary f l a g s C YLOB, YHIB / C LOWER > Lower diagonal of t r1 -d iagonal system C DI AG > Main C UPPER > Upper C RESRHS > R1ght-hand-side of C Also used for update of PHI C MODE > Se lects l i n e grouping/sweeping d i r e c t i o n C LIST > L i s t i n g f l a g C UNIT > L i s t i n g unit C C Routines c a l l e d : C C NRMACC C NRMCLC C TRISLV C RXLOB C RYB C c  C c DOUBLE PRECISION FUNCTION LINRLX(PHI,KSQR,RHS, * S,S2,S4,W,WP,WM,DELS, * DELW,NS.NW,XLOB,XHIB,YLOB,YHIB,LOWER, * DIAG,UPPER,RESRHS,MODE,LI ST,UNIT) INTEGER NS, NW REAL*8 KSOR(NS.NW), PHI(NS,NW), RHS(NS.NW) REAL*8 DIAG(1), LOWER(1), RESRHS(1), S(1) , S2(1), S4(1), UPPERO), W(1), WM(1), WP(1) REAL*8 NRMACC, NRMCLC REAL*8 C1, C1A, C2, C3, D1, D2, DELS, DELSM2, DELW, DELWM2, KPHIM7, KSQRIJ, PHIIJ, PHIM7, RES, S2I, S4I, WMJ, WPJ INTEGER I, IFIN, 1ST, ISTEP, ITRI, J , JFIN, JST, JSTEP, LIST, MODE, NPT, NSM1, NTRI, NWM1, UNIT, XBST, XBFIN, XHIB, XLOB, YBST, YBFIN, YHIB, YLOB LINRLX = O.ODO NSM1 = NS - 1 NWM1 = NW - 1 NPT = (NS - 2) * (NW - 2) DELSM2 = 1.0DO / (DELS ** 2) DELWM2 = 1.0DO / (DELW ** 2) INTERIOR RELAXATION IF( MODE .GT. 2 ) GO TO €0 LINES OF CONSTANT ANGULAR COORDINATE NTRI = NS - 2 IF( MODE .EO. 2 ) GO TO 10 FORWARD SWEEPING DIRECTION JST = 2 JFIN = NWM1 JSTEP = 1 GO TO 20 REVERSE SWEEPING DIRECTION JST = NWM1 JFIN = 2 JSTEP = -1 J = JST START LOOP WPJ = WP(J) WMJ = WM(J) ITRI = 1 DO 40 I = 2 , NSM1 KSORIJ = KSOR(I,J) PHIIJ = PHI( I ,J ) PHIM7 = 1.000 / PHIIJ ** 7 KPH1M7 = KSQRIJ * PHIM7 C1 = S4(I ) * DELSM2 S2I = S2(I) C2 = S2I * DELWM2 * WPJ C3 = S2I * DELWM2 * WMJ D1 = C1 * (PHI(I+1,J) - 2.ODO * PHIIJ +• PHI ( I -1 , J ) ) D2 = C2 * (PHI(I,J+1) - PHIIJ) + C3 * ( P H K I . J - 1 ) - PHIIJ) RES = D1 + D2 + 0.125DO * KPHIM7 - RHS(I.J) LINRLX = NRMACC(LINRLX,RES,O.ODO,O.ODO) SET UP TRIDIAGONAL SYSTEM DIAG(ITRI) = -(2.ODO * C1 + C2 + C3 + 0.875D0 * KPHIM7 / PHIIJ) UPPER(ITRI) = C1 LOWER(ITRI ) = C1 RESRHS(ITRI) = RES ITRI = ITRI + 1 CONTINUE SOLVE TRIDIAGONAL SYSTEM CALL TRISLV(NTRI,LOWER , DI AG, UPPER , RESRHS , 0,8.300) ITRI = 1 UPDATE PHI DO 50 1 = 2 , NSM1 PHI(I .J ) = PHI(I .J) - RESRHS(ITRI) ITRI = ITRI + 1 CONTINUE END LOOP IF( J .EO. JFIN ) GO TO 120 J = J + JSTEP GO TO 30 LINES OF CONSTANT RADIAL COORDINATE NTRI = NW - 2 IF( MODE .EO. 4 ) GO TO 70 FORWARD SWEEPING DIRECTION 1ST = 2 IFIN = NSM1 ISTEP = 1 GO TO 80 REVERSE SWEEPING DIRECTION 1ST = NSM1 IFIN = 2 ISTEP = -1 •I = 1ST START LOOP S2I = S2(I) S4I = S4(I) C1 = S4I * DELSM2 C1A = S2I * DELWM2 ITRI = 1 DO 100 J = 2 , NWM1 KSQRIJ = KSQR(I.J) PHIIJ = PHI(I .J) PHIM7 = 1.0D0 / PHIIJ ** 7 KPHIM7 = KSQRIJ * PHIM7 C2 = C1A * WP(J) C3 = C1A * WM(J) 01 = C1 * (PHI(I+1,J) - 2.ODO * PHIIJ + PHI ( I -1 , J ) ) D2 = C2 * (PHI(I,J+1) - PHIIJ) + C3 * (PHI( I , J -1) - PHIIJ) RES = D1 + D2 + 0.125D0 * KPHIM7 - RHS(I.J) LINRLX = NRMACC(LINRLX,RES,O.ODO,O.ODO) SET UP TRIDIAGONAL SYSTEM DIAG(ITRI) = -(2.ODO * C1 + C2 + C3 + 0.875DO * KPHIM7 / PHIIJ) UPPER(ITRI) = C2 LOWER(ITRI) = C3 RESRHS(ITRI) = RES ITRI = ITRI + 1 CONTINUE SOLVE TRIDIAGONAL SYSTEM CALL TRISLV(NTRI,LOWER,DIAG,UPPER,RESRHS,0,8.300) ITRI = 1 UPDATE PHI DO 110 J = 2 , NWM1 PHI(I .d) = PHI( I .J ) - RESRHS(ITRI) ITRI = ITRI + 1 CONTINUE END LOOP IF( I .EQ. IFIN ) GO TO 120 1 = 1 + ISTEP GO TO 90 LINRLX = NRMCLC(LINRLX,NPT) IF( LIST .NE. 0 ) WRITE(UNIT,200) LINRLX FORMAT(' INTERIOR RESIDUAL:' ,TGO,E12.4) BOUNDARY RELAXATIONS XBST = 1 199 XBFIN = NS - 1 YBST = 1 YBFIN = NW IF( XLOB .NE. O ) XBST = 2 IF( Y LOB .NE. O ) YBST = 2 IF( YHIB .NE. O ) YBFIN = NW C IF( XLOB .NE. 0 ) RES = RXLOB(PHI.RHS,NS,NW,XLOB,1,DELS , * YBST,YBFIN,0.9DO,3,LIST,UNIT) IF( YLOB .NE. O ) RES = RYB(PHI,RHS,NS,NW,YLOB,1,DELW, * XBST,XBFIN,0.5DO,3.LIST.UNIT) IF( YHIB .NE. 0 ) RES = RYB(PHI,RHS,NS,NW,YHIB,NW,DELW, * XBST,XBFIN,0.5DO,3,LIST,UNIT) C C RETURN C C ABNORMAL EXITS C 300 WRITE(UNIT,310) 310 F0RMAT('OS/R LINRLX: ERROR IN EXT. ROUTINE TRISLV) RETURN C C END c  c C FUNCTION RXLOB C c  C C Performs NSWP smoothing sweeps of inner x boundary C d i f f e r e n c e equations. Returns norm of r e s i d u a l s . C Sweeping d i r e c t i o n c o n t r o l l e d by value of MODE. C C Parameters: See FUNCTION LINRLX C c  C c DOUBLE PRECISION FUNCTION RXLOB(PHI,RHS,NX,NY,NSWP,BX,DELX, * YBST,YBFIN,ETA,MODE,LIST,UNIT) C C INTEGER NX, NY C REAL*8 PHI(NX.NY), RHS(NX.NY) C REAL*8 DABS C REAL*8 DELXM2, DELX, ETA, RFINNW, RPOS, RPOSM, RPOSNW. * RPOSP, RST, RSTNW, RSTP, RTOT G INTEGER START(3) C INTEGER BX, BX1, BX2, IFIN, 1ST, LIST, MODE, N, NSWP, POS, * ST, STEP, SWEEP. UNIT, YBFIN, YBST C N = YBFIN - YBST + 1 DELXM2 = -0.5D0 / DELX BX1 = BX + 1 BX2 = BX + 2 1ST = YBST + 1 IFIN = YBFIN - 1 GO TO( 10, 20, 30 ), MODE ST = YBST STEP = 1 GO TO 40 ST = YBFIN STEP = -1 GO TO 40 ST = YBST STEP = 1 START(1) = YBST START(3) = YBFIN SWEEP = 0 RTOT = O.ODO RST = DELXM2 * (PHI(BX.ST) - PHI(BX2,ST)) + 0.5D0 * PHI(BX1,ST) - RHS(BX.ST) RSTP = DELXM2 * (PHI(BX,ST+STEP) - PHI(BX2,ST+STEP)) + 0.5D0 * PHI(BX1,ST+STEP) - RHS(BX,ST+STEP) RSTNW = ETA * RST RTOT = RTOT + DABS(RSTNW) PHI(BX.ST) = PHI(BX.ST) + (RSTNW - RST) / DELXM2 POS = ST + STEP RPOS = RSTP RPOSM = RSTNW DO 60 I = 1ST , IFIN RPOSP = DELXM2 * (PHI(BX,POS+STEP) - PHI(BX2,POS+STEP)) 0.5D0 * PHI(BX1,POS+STEP) - RHS(BX,POS+STEP) RPOSNW = 0.5D0 * (RPOSM + RPOSP) RTOT = RTOT + DABS(RPOSNW) PHI(BX.POS) = PHI(BX.POS) + (RPOSNW - RPOS) / DELXM2 RPOSM = RPOSNW RPOS = RPOSP POS = POS + STEP CONTINUE RFINNW = ETA * RPOS RTOT = RTOT + DABS(RFINNW) PHI(BX,POS) = PHI(BX.POS) + (RFINNW - RPOS) / DELXM2 RXLOB = RTOT / N IF( LIST .EQ. O ) GO TO 65 WRITE(UNIT,100) RXLOB - FORMATC INNER BOUNDARY RES IDUAL: ' ,T60,E12.4) SWEEP = SWEEP + 1 IF( SWEEP .GE. NSWP ) GO TO 70 C IF( MODE .NE. 3 ) GO TO 50 ST = START(STEP+2) STEP = -STEP GO TO 50 C 70 RETURN C C END c-C C C C-C C C C C C C C C-C C C C C C c c c FUNCTION RYB Performs NSWP smoothing sweeps of y boundary d i f f e r e n c e equat ions. Returns norm of res idua ls , Boundary swept c o n t r o l l e d by value of BY. Sweeping d i r e c t i o n c o n t r o l l e d by value of MODE. Parameters: See FUNCTION LINRLX DOUBLE PRECISION FUNCTION RYB(PHI,RHS,NX,NY,NSWP,BY,DELY * XBST,XBFIN,ETA,MODE,LIST,UNIT) INTEGER REAL*8 REAL*8 REAL*8 INTEGER INTEGER NX, NY PHI(NX.NY), RHS(NX,NY) DABS DELYM2, DELY, ETA, RPOSP, RST, RSTNW, START(3) RFINNW, RPOS, RSTP, RTOT RPOSM, RPOSNW, BY, BY2, IFIN, 1ST, LIST, MODE, N, NSWP, POS, ST STEP, SWEEP, UNIT, XBFIN, XBST N = XBFIN - XBST + 1 DELYM2 = 0.5D0 / DELY IF( BY .EQ. NY ) GO TO 3 BY2 = 3 DELYM2 = -DELYM2 GO TO 5 BY2 = NY - 2 1ST = XBST + 1 IFIN = XBFIN - 1 GO T0( 10, 20, 30 ), MODE 202 c 10 ST = XBST STEP = 1 GO TO 40 C 20 ST = XBFIN STEP = -1 GO TO 40 C 30 ST = XBST STEP = 1 START(1) = XBST START(3) = XBFIN C 40 SWEEP = O C 50 RTOT = O.ODO RST = DELYM2 * (PHI(ST.BY) - PHI(ST,BY2)) - RHS(ST,BY) RSTP = DELYM2 * (PHI(ST+STEP,BY) - PHI(ST+STEP,BY2)) -* RHS(ST+STEP,BY) RSTNW = ETA * RST RTOT = RTOT + DABS(RSTNW) PHI(ST.BY) = PHI(ST.BY) + (RSTNW - RST) / DELYM2 POS = ST + STEP RPOS = RSTP RPOSM = RSTNW C DO 60 I = 1ST , IFIN RPOSP = DELYM2 * (PHI(POS+STEP,BY) - PHI(POS+STEP,BY2)) * RHS(POS+STEP,BY) RPOSNW = 0.5D0 * (RPOSM + RPOSP) RTOT = RTOT + DABS(RPOSNW) PHI(POS.BY) = PHI(POS.BY) + (RPOSNW - RPOS) / DELYM2 RPOSM = RPOSNW RPOS = RPOSP POS = POS + STEP 60 CONTINUE C RFINNW = ETA * RPOS RTOT = RTOT + DABS(RFINNW) PHI(POS.BY) = PHI(POS.BY) + (RFINNW - RPOS) / DELYM2 C RYB = RTOT / N IF( LIST .EO. 0 ) GO TO 65 IF( BY .EO. NY ) GO TO 62 WRITE(UNIT,100) RYB 100 FORMAT(' BOTTOM BOUNDARY RESIDUAL: ' ,T60,E12.4) GO TO 65 62 WRITE(UNIT,110) RYB 110 FORMAT(' TOP BOUNDARY RESIDUAL: ' ,T60,E12.4) C 65 SWEEP = SWEEP + 1 C IF( SWEEP .GE. NSWP ) GO TO 70 C IF( MODE .NE. 3 ) GO TO 50 ST = START(STEP+2) STEP = -STEP GO TO 50 C 203 70 RETURN END C--C C C C-C C C C C C C C C C C C C C C C C C C c-c c c c c c c c c c-c c c c c c c SUBROUTINE OPERAT Appl ies i n t e r i o r d i f fe rence equations to unknown funct ion PHI, p l a c i n g res idua ls in array RESID. (R ight -hand-s ide of d i f f e r e n c e equations may be subtracted by se t t ing RHSSW to 1.0DO - e lse should be O.ODO) C a l l s appropriate rout ines to c a l c u l a t e boundary r e s i d u a l s , i f necessary. Parameters: PHI(NS.NW) - -> Unknown funct ion KSQR(NS.NW) -> E x t r i n s i c curvature squared RHS(NS.NW) - -> R ight -hand-s ide of d i f fe rence system RESID(NS,NW)-> Residual array S, S2, S4 > Radial coordinate funct ions W, WP, WM > Angular " " DELS > Radial mesh spacing DELW > Angular " XLOB, XHIB, -> Boundary f l a g s YLOB, YHIB / RESRHS > Enables/disables subt ract ion of RHS LIST > L i s t i n g f l a g UNIT > L i s t i n g unit Rout i nes ca l1ed : VALURC OPXLOB OPYB SUBROUTINE OPERAT(PHI,KSQR,RHS,RESID,S,S2,S4,W,WP,WM,DELS, * DELW,NS,NW,XLOB,XHIB,YLOB,YHIB,RHSSW, * LIST,UNIT) INTEGER NS, NW REAL*8 KSQR(NS.NW), PHI(NS.NW). RHS(NS,NW), RESID(NS.NW) REAL*8 S(1) , S2(1), S4(1), W(1), WM(1) . WP(1) REAL*8 C1, C2, C3, D1. 02, DELS, DELSM2. DELW, DELWM2, PHIIJ, RHSSW, S2I, WMJ, WPd INTEGER I, J , LIST, NSM1, NWM1, UNIT, XHIB, XLOB, YHIB, YLOB C 204 IF( LIST .GT. 0 ) WRITE(UNIT,200) NS, NW 200 FORMAT( ' ENTERING OPERAT - ',14,' X ',14,' GRID') C NSM1 = NS - 1 NWM1 = NW - 1 DELSM2 = 1.0D0 / (DELS ** 2) DELWM2 = 1.0D0 / (DELW ** 2) C DO 20 0 = 2 , NWM1 WPJ = WP(J) WMJ = WM(J). C DO 10 I = 2 , NSM1 PHIIJ = P H I ( I . J ) C1 = S4(I ) * DELSM2 S2I = S 2 ( I ) C2 = S2I * DELWM2 * WPJ C3 = S2I * DELWM2 * WMJ D1 = C1 * (PHI(I+1,J) - 2.ODO * PHIIJ + P H I ( I - 1 , J ) ) D2 = C2 * ( P H K I . J + 1 ) - PHIIJ) + * C3 * ( P H K I . J - 1 ) - PHIIJ) RESID(I.J) = D1 + D2 + 0.125DO * KSOR(I.J) / P H I I J * - RHS(I.J) * RHSSW 10 CONTINUE C 20 CONTINUE C IF( XLOB .NE. O ) GO TO 30 CALL VALURC(RESID,NS,NW,1.0,0.ODO) GO TO 40 30 CALL OPXLOB(PHI,RHS,RESID.DELS,NS.NW,RHSSW) 40 CALL VALURC(RESID,NS,NW,NS,0,O.ODO) IF( YLOB .NE. 0 ) GO TO 50 CALL VALURC(RESID,NS.NW,1,1.O.ODO) GO TO GO 50 CALL OPYB(PHI,RHS,RESID,DELW,NS.NW,1.RHSSW) 60 IF( YHIB .NE. 0 ) GO TO 70 CALL VALURC(RESID,NS,NW,NW,1,O.ODO) GO TO 80 70 CALL OPYB(PHI,RHS,RESID,DELW,NS.NW.NW,RHSSW) C 80 RETURN C C END c  c C SUBROUTINE OPXLOB C c  C C A p p l i e s i n n e r x boundary d i f f e r e n c e e q u a t i o n s to unknown C f u n c t i o n PHI C C Parameters: See SUBROUTINE OPERAT C c  205 c c SUBROUTINE OPXLOB(PHI,RHS,RESID,DELS,NS,NW,RHSSW) C C INTEGER NS, NW C REAL*8 PHI(NS.NW), RHS(NS.NW), RESID(NS.NW) C REAL*8 DELS, DELS2M, RHSSW C INTEGER J C C DELS2M = 0.5D0 / DELS C DO 10 J = 1 , NW RESID(1,J) = DELS2M * (PHI(3,d) - PHI(1,d)) + * 0.5DO * PHI(2,J) - RHSO.d) * RHSSW 10 CONTINUE C RETURN C C END c  c C SUBROUTINE OPYB C c  C C A p p l i e s y boundary d i f f e r e n c e e q u a t i o n s t o unknown f u n c t i o n C PHI C C P a r a m e t e r s : see SUBROUTINE OPERAT C c  c c SUBROUTINE OPYB(PHI,RHS,RESID,DELW,NR,NW,BY,RHSSW) C INTEGER NR, NW C REAL*8 PHI(NR.NW), RHS(NR.NW), RESID(NR.NW) C REAL*8 DELW, DELW2, RHSSW, SIGN C INTEGER BY, BY2, I C C DELW2 = 0.5D0 / DELW IF( BY .EO. 1 ) GO TO 10 BY2 = NW - 2 SIGN = 1.ODO GO TO 20 10 BY2 = 3 SIGN = -1.ODO 20 CONTINUE 2 0 6 c DO 30 1 = 1 , NR RESID(I.BY) = SIGN * (PHI(I,BY) - PHI(I,BY2)) * DELW2 -* RHS(I.BY) * RHSSW 30 CONTINUE C RETURN C C END 207 c  c C D I R E C T S O L U T I O N R O U T I N E S C c  C c c SUBROUTINE DIRECT c c Routine for so l v ing d i f f e r e n c e equations using c Newton-direct method 1. Performs minimum of MNITER Newton c i t e r a t i o n s , u n t i l res idual norm is less than EPSI or c MXITER i t e r a t i o n s have been completed. C a l l s external c CHLSKY to solve l inear system. c c Parameters: c c PHI(NX,NY) - - -> Unknown funct ion c KSQR(NX,NY) - -> E x t r i n s i c curvature squared c RHS(NX,NY ) - - -> Right -hand-s ide of d i f fe rence system c RESID(NX,NY) -> Residual array c S, S2, S4 - - - -> Radial coordinate funct ions c W, WP, WM -> Angular " " c JACOB(NJ.NJ) -> Jacobian array c WORK(NXW.NYW) -> Work array for Inversion of JACOB c DELPHI -> Function update array c DELS Radial mesh spacing c DELW Angular " c XLOB, XHIB, - -> Boundary f l a g s c YLOB, YHIB / c EPSI Convergence c r i t e r i o n c MNITER Minimum number of Newton i te ra t ions c MXITER Maximum c LIST L i s t i n g f l a g c UNIT L i s t i n g unit c RC .EQ. 0 -> So lut ion successful c .NE. 0 -> No convergence a f te r MXITER c i t e r a t i o n s c c Rout 1nes ca l1ed : c c OPERAT c FDIAG c c CHLSKY c c DOUBLE PRECISION FUNCTION DIRECT(PHI,KSQR.RHS,RESID, * S S2 S4,W,WP,WM, * JACOB,WORK,DELPHI,DELS,DELW,NX,NY,NJ, * NXW.NYW,XLOB,XHIB,YLOB,YHIB,EPSI.MNITER, * MXITER,LIST,UNIT,RC) C C INTEGER NJ, NX, NXW, NY, NYW REAL*8 DABS REAL*8 JACOB(NJ.NJ), KSQR(NX.NY), PHI(NX.NY), RESID(NX,NY), RHS(NX,NY), WORK(NXW,NYW) REAL*8 DELPHI(1 ) , S(1), S 2(1). S4(1). W(1), WM(1), WP(1) REAL*8 DELS, DELW. EPSI, RES, RTOT INTEGER FX, FY, I, ITER, IX, J . LIST, MNITER, MX ITER, NINT, NXM1, NXM2, NYM1, POS, RC, UNIT, XHIB, XLOB, YHIB, YLOB NXM1 = NX - 1 NXM2 = NX - 2 NYM1 = NY - 1 NINT = (NY - 2) * NXM2 ITER = O MAIN LOOP . . . CALCULATE RESIDUALS AND PLACE IN WORK ARRAY CALL OPERAT(PHI,KSQR,RHS,RESID,S,S2,S4,W.WP,WM,DELS, DELW,NX,NY,XLOB,XHIB,YLOB,YHIB, 1.ODO,LI ST, UNIT) FX = NXM1 POS = 1 RTOT = O.ODO DO 30 IX = 1 , NXM2 DO 20 FY = 1 , NY RES = RESID(FX,FY ) WORK(POS.NYW) = RES IF( FY .EO. 1 .OR. FY .EO. NY ) GO TO 25 RTOT = RTOT + DABS(RES) POS = POS + 1 CONTINUE FX = FX - 1 CONTINUE DO 40 FY = 2 , NYM1 WORK(POS.NYW) = RESID(1 ,FY) POS = POS + 1 CONTINUE CALCULATE RESIDUAL NORM AND EXIT IF CONVERGED DIRECT = RTOT / NINT IF( LIST .NE. O ) WRITE(UNIT,1000) ITER, DIRECT FORMATC ITERATION: ' , 1 3 , T 6 0 , E12 . 4 ) IF( DIRECT .LE . EPSI .AND. ITER .GE. MNITER ) GO TO 100 FILL OFF-DIAGONAL ELEMENTS OF WORK ARRAY DO 60 I = 1 . NJ DO 50 J = 1 . NJ WORK(I.J) = JACOB(I.J) CONTINUE CONTINUE DEFINE DIAGONAL ELEMENTS CALL FDI AG(WORK,NXW,NYW,PHI,KSOR.NX,NY.S2,S4,WP,WM, DELS,DELW) SOLVE SYSTEM OF LINEAR EQUATIONS CALL CHLSKY(WORK,NXW,NYW,DELPHI) UPDATE PHI FX = NXM1 POS = 1 DO 80 IX = 1 , NXM2 DO 70 FY = 1 , NY PHI(FX,FY) = PHI(FX.FY) - DELPHI(POS) POS = POS + 1 CONTINUE FX = FX - 1 CONTINUE DO 90 FY = 2 , NYM1 PHI(1,FY) = PHIO .FY) - DELPHI(POS) POS = POS + 1 CONTINUE NEWTON ITERATION COMPLETE . . . ENOUGH DONE ??? ITER = ITER + 1 IF( ITER .GE. MXITER ) GO TO 110 . . . END MAIN LOOP GO TO 10 RC = O RETURN RC = 1 RETURN 210 C SUBROUTINE FOFFD C c  C C D e f i n e s o f f - d i a g o n a l elements of J a c o b i a n a r r a y JACOB f o r C s o l u t i o n of d i f f e r e n c e system by d i r e c t means. C C Parameters: C C JAC0B(N,N) > J a c o b i a n a r r a y C S2, S4 > R a d i a l c o o r d i n a t e f u n c t i o n s C WP, WM > Angular C DELS > R a d i a l mesh s p a c i n g C DELW > Angular " C c . c c c c c c c c SUBROUTINE FOF FD(JACOB,N,NX,NY,S2,S4,WP.WM,DELS,DELW) INTEGER N REAL*8 JACOB(N.N) REAL*8 S 2 ( 1 ) . S 4 ( 1 ) , WM(1), WP(1) REAL*8 C1 FX, C2FX, DELS, DELS2M, DELSM1, DELSM2, * DELW, DELW2M, DELWM1, DELWM2 INTEGER IX, IY, FX, FY, NX, NXM1, NXM2, NY, NYM1, * NYM2, NYNYM1 NXM1 = NX - 1 NXM2 = NX - 2 NYM1 = NY - 1 NYM2 = NY - 2 NYNYM1 = NY + NYM1 DELSM1 = 1.ODO / DELS DELSM2 = DELSM1 ** 2 DELS2M = 0.5DO * DELSM1 DELWM1 = 1.ODO / DELW DELWM2 = DELWM1 ** 2 DELW2M = 0.5DO * DELWM1 C C ZERO JACOBIAN ARRAY DO 10 I = 1 , N DO 10 J = 1 , N JACOB(I,J) = O.ODO 10 CONTINUE FX = NXM1 POS = 1 DO 90 IX = 1 , NXM2 FY = 1 C1 FX = DELSM2 * S4(FX) 'C2FX = DELWM2 * S2(FX) DO 80 IY = 1 , NY IF( IY .EO. 1 ) GO TO 50 IF( IY .EO. NY ) GO TO 60 INTERIOR POINTS IF( IX .EO. 1 ) GO TO 20 IF( IX .EO. NXM2 ) GO TO 30 CENTRAL INTERIOR POINTS JAC0B(PDS,P0S+1 ) = C2FX * WP(FY) JAC0B(P0S,P0S-1 ) = C2FX * WM(FY) JACOE(POS,POS+NY) = C1 FX JACOB(POS,POS-NY) = C1 FX GO TO 40 RIGHT INTERIOR POINTS JAC0B(P0S,P0S+1 ) = C2FX * WP(FY) JAC0B(P0S,P0S-1 ) = C2FX * WM(FY) JACOB(POS,POS+NY) = C1 FX GO TO 40 LEFT INTERIOR POINTS JACOB(POS,POS+1 ) = C2FX * WP(FY) JAC0B(P0S,P0S-1 ) = C2FX * WM(FY) JAC0B(P0S.P0S+NYM1 ) = C 1 FX JACOB(POS.POS-NY) = C1 FX GO TO 70 LOWER BOUNDARY POINTS JAC0B(P0S,P0S+2) = DELW2M GO TO 70 UPPER BOUNDARY POINTS JAC0B(P0S,P0S-2) = -DELW2M POS = POS + 1 FY = FY + 1 CONTINUE FX = FX - 1 CONTINUE LEFT BOUNDARY POINTS FX = 1 FY = 2 212 DO 100 IY = 1 , NYM2 JAC0B(P0S,P0S-NYM1) = 0.5D0 JAC0B(POS,POS-NYNYM1) = DELS2M FY = FY + 1 POS = POS + 1 100 CONTINUE C RETURN C C END c  C C SUBROUTINE FDIAG C c  c C Defines diagonal elements of Jacobian array JACOB for C s o l u t i o n of d i f f e r e n c e system by d i r e c t means. C C Parameters: C C JACOB(NXJ.NYJ) -> Jacobian array C PHI(NX.NY) > Unknown funct ion C KSQR(NX.NY) > E x t r i n s i c curvature squared C S2, S4 > Radial coordinate funct ions C WP, WM > Angular " C DELS > Radial mesh spacing C DELW > Angular " C c  C C SUBROUTINE FDI AG( JACOB , NXJ , NY J , PHI , KSQR , NX , NY , S2', S4 , WP , WM, • DELS,DELW) INTEGER NX, NXJ, NY, NYJ REAL*8 JACOB(NXJ.NYJ), PHI(NX.NY), KSOR(NX.NY) REAL*8 S2(1), S4(1), WM(1), WP(1) REAL*8 C1 FX, C2FX, DELS, DELS2M, DELSM1, DELSM2, DELW, DELW2M, DELWM1, DELWM2 INTEGER IX, IY, FX, FY, NXM1, NXM2, NYM1, NYM2 NXM1 = NX - 1 NXM2 = NX - 2 NYM1 = NY - 1 NYM2 = NY - 2 DELSM1 = 1.ODO / DELS DELSM2 = DELSM1 ** 2 DELS2M = 0.5D0 * DELSM1 DELWM1 = 1.ODO / DELW DELWM2 = DELWM1 ** 2 213 DELW2M = 0.5D0 * DELWM1 C FX = NXM1 PDS = 1 C DO 50 IX = 1 , NXM2 FY = 1 C1 FX = DELSM2 * S4(FX) C2FX = DELWM2 * S2(FX) C DO 40 IY = 1 , NY IF( IY .EO. 1 ) GO TO 10 IF( IY .EO. NY ) GO TO 20 C C INTERIOR POINTS C C c c JACOB(POS,POS) = -(2.ODO * C1 FX + C2FX * (WP(FY) + WM(FY)) + 0.875D0 * KSQR(FX.FY) / (PHI(FX.FY) ** 8)) GO TO 30 C C LOWER BOUNDARY POINTS C 10 JACOB(POS,POS) = -DELW2M C GO TO 30 C C UPPER BOUNDARY POINTS C 20 JACOB(POS.POS) = DELW2M C C 30 POS = POS + 1 FY = FY + 1 40 CONTINUE C FX = FX - 1 50 CONTINUE C C LEFT BOUNDARY POINTS C FX = 1 FY = 2 C DO 60 IY = 1 , NYM2 JACOB(POS,POS) = -DELS2M FY = FY + 1 POS = POS + 1 60 CONTINUE RETURN END c  C C SUBROUTINE DIRSPC 214 c c  c C A l l o c a t e s space for d i r e c t so lu t ion of coarse g r i d C d i f f e r e n c e equations and c a l l s FOFFD to def ine ( s t a t i c ) C o f f - d i a g o n a l Jacobian matrix elements. C C Parameters: C C MEMORY -> Main storage array C MEMPT - -> Current a v a i l a b l e locat ion in MEMORY C NX, NY -> Dimensions of coarse g r i d C S2, S4 -> Radial coordinate funct ions C WP, WM -> Angular " C DELS > Radial mesh spacing C DELW > Angular " " C C Routines c a l l e d : C C FOFFD C c  C c c c c c c c c SUBROUTINE DIRSPC(MEMORY,MEMPT,NX.NY.S2 , S4,WP,WM,DELS,DELW) REAL*8 MEMORY(1), S2(1), S4(1), WM(1), WP(1) REAL*8 DELS, DELW INTEGER DELPHI, JACOB, MEMPT, NJ, NX, NXW, NY, NYW, WORK COMMON / DIR / JACOB, NJ, WORK, NXW, NYW, DELPHI NJ = (NX - 2) * NY + (NY - 2) NXW = NJ NYW = NXW + 1 JACOB = MEMPT MEMPT = MEMPT + NJ ** 2 WORK = MEMPT MEMPT = MEMPT + NXW * NYW DELPHI = MEMPT MEMPT = MEMPT + NJ C C DEFINE OFF-DAIGONAL JACOBIAN ELEMENTS C CALL FOFFD(MEMORY(JACOB),NJ,NX,NY,S2,S4,WP,WM,DELS,DELW) RETURN END 215 c  c C I N T E R P O L A T I O N A N D R E S T R I C T I O N C C R O U T I N E S C c  c  C C SUBROUTINE INTERP C c  C C C a l l s e i the r LININT to perform l inear Interpolat ion or C CUBINT to do cubic in te rpo la t ion of coarse g r i d funct ion C to f i n e g r i d . C o a r s e - t o - f i n e x, y mesh spacing r a t i o s are C assumed to be e i the r 2:1 or 1:1. Coarse g r i d domain is C assumed to conta in f i n e g r i d domain, and a l l i n t e r i o r C coarse g r i d l i nes must be f i n e g r i d l i n e s . C C Parameters: C C MEMORY -> Main storage array C CGRID - -> Coarse g r i d number C CFPT > Pointer to s t a r t of coarse g r i d fen in MEMORY C FGRID - -> Fine g r i d number C FFPT > Pointer to s ta r t of f i n e g r i d fen in MEMORY C ORDER - -> .EO. 4 -> Perform cubic in te rpo la t ion C .NE. 4 -> 1inear C C Routines c a l l e d : C C BIG C DFLOAT C SMALL C TRNCAT C CEIL C LININT C CUBINT C c  C C SUBROUTINE INTERP(MEMORY,CGRID,CFPT,FGRID.FFPT,ORDER, * LIST,UNIT) C C G R I D D A T A S T R U C T U R E C INTEGER GFSTRT(50,10), XFSTRT(50,10), YFSTRT(50,10) C INTEGER BR0THR(5O). FATHER(50), LHEAD(50), LLINK(50), * S0N(50), NPT(50). NX(50), NY(50). XHIB(50). * XL0B(50), YHIB(50), YL0B(50) C REAL*8 AREA(50), EPSI(50), ERR0R(5O), NRMTAU(50), * XINC(50), YINC(50) C LOGICAL C0NV(50), SL0W(50) 216 c c c c c c c c-c c c c c INTEGER INTEGER INTEGER INTEGER COMMON COMMON CRSLEV, FINLEV, NGFUNC, NGRID. NXFUNC, NYFUNC MEMPT KSQR, R. RHS. RM, RP, TAU, W, WM, WP, U, OLDU NULL / 280808080 / / COMGST / LHEAD, LLINK, FATHER, BROTHR, SON, NX, NY, NPT, AREA, XINC, YINC, XFSTRT, YFSTRT, GFSTRT, XLOB. XHIB, YLOB, YHIB, ERROR, CONV, SLOW, EPSI, NRMTAU, CRSLEV, FINLEV, NGRID, MEMPT / COMCON / R, RP, RM, TAU, KSQR, W, WP, WM, U, OLDU, RHS, NXFUNC, NYFUNC, NGFUNC C C C C REAL*8 MEMORY(1) REAL*8 BIG, DFLOAT, SMALL, TRNCAT REAL*8 CFX1, CFX2, CFY1, CFY2, CXO, CXINC, CYO, CYINC, FXO, FXINC, FYO, FYINC INTEGER CEIL INTEGER CFPT, CGRID, CIXFIN, CIXST, CIYFIN, CIYST, CNX, CNY. FFPT, FGRID, FIXFIN, FIXST, FIYFIN, FIYST, FNX, FNY. FTXST, FTYST, LIST. NCX, NCY, ORDER, TNX, TNY, UNIT, XBRST, XSTEP, YBRST, YSTEP CNX = NX(CGRID) CNY = NY(CGRID) FNX = NX(FGRID) FNY = NY(FGRID ) CXINC = XINC(CGRID) CYINC = YINC(CGRID) FXINC = XINC(FGRID) FYINC = YINC(FGRID) CXO = MEMORY(XFSTRT(CGRID,R)) CYO = MEMORY(YFSTRT(CGRID,W)) FXO = MEMORY(XFSTRT(FGRID,R)) FYO = MEMORY(YFSTRT(FGRID.W)) DETERMINE RELATIONSHIP OF FINE GRID BOUNDARY W.R.T COARSE GRID BOUNDARY CFX1 = (FXO - CXO) / CXINC CFY1 = (FYO - CYO) / CYINC CFX2 = (FXO + (FNX - 1) * FXINC - CXO) / CXINC CFY2 = (FYO + (FNY - 1) * FYINC - CYO) / CYINC CIXST = BIG(1.ODO,TRNCAT(CFX1) + 1.0D0) CIYST = BIG(1.ODO,TRNCAT(CFY1) + 1.0D0) CIXFIN = SMALL(DFLOAT(CNX),DFL0AT(CEIL(CFX2)) + 1.0D0) CIYFIN = SMALL(DFLOAT(CNY),DFL0AT(CEIL(CFY2)) + 1.0D0) 217 NCX" = CIXFIN - CIXST + 1 NCY = CIYFIN - CIYST + 1 c IF( CXINC . EO. FXINC ) GO TO 10 XSTEP = 1 GO TO 20 10 XSTEP = 2 20 IF( CYINC .EO. FY INC ) GO TO 30 YSTEP = 1 GO TO 40 30 YSTEP = 2 c 40 IF( ORDER .EO. 4 .AND NCX .GT * GO TO 70 c 1 10 .AND. NCY .GT. 3 ) C C LINEAR INTERPOLATION C IF( CFX1 .LT. O.ODO ) GO TO 50 FIXST = 1 FIXFIN = FNX XBRST = 0 IF( CFX1 .NE. TRNCAT(CFX1 ) ) XBRST = 1 GO TO 60 50 FIXST = 2 FIXFIN = FNX XBRST = O 60 FIYST = 1 FIYFIN = FNY YBRST = O IF( CFY1 .NE. TRNCAT(CFY1 ) ) YBRST = 1 C CALL LININT(MEMORY(CFPT),MEMORY(FFPT),CNX,CNY, * FNX,FNY,CIXST.CIYST,FIXST,FIXFIN, * ' FIYST,FIYFIN,XBRST,YBRST,XSTEP,YSTEP) C GO TO 140 C C CUBIC INTERPOLATION C 70 IF( XSTEP .EO. 2 ) GO TO 80 TNX = 2 * NCX - 1 GO TO 90 80 TNX = CNX 90 IF( YSTEP .EO. 2 ) GO TO 100 TNY = 2 * NCY - 1 GO TO 110 10O TNY = CNY IF( CFX1 .LT. O.ODO ) GO TO 120 FIXST = 1 FIXFIN = FNX FTXST = 1 IF( CFX1 .NE. TRNCAT(CFX1 ) ) FTXST = 2 GO TO 130 120 FIXST = 2 FIXFIN = .FNX FTXST = 1 130 FIYST = 1 FIYFIN = FNY FTYST = 1 IF( CFY1 .NE. TRNCAT(CFY1 ) ) FTYST = 2 C 140 C C CALL CUBINT(MEMORY(CFPT).MEMORY(FFPT).MEMORY(MEMPT), CNX,CNY,FNX,FNY,TNX,TNY,CIXST.CIXFIN, CIYST,CIYFIN.FIXST.FIXFIN.F1YST.FIYFIN, FTXST,FTYST,XSTEP,YSTEP) RETURN END C-C C C C-C C C C C C c c c c c c c c c c c c c c c-c c SUBROUTINE- LININT Performs l inear Interpolat ion of coarse g r i d funct ion COARSE to f i n e g r i d funct ion FINE. See subroutine INTERP for assumptions regarding g r i d spacings e t c . Parameters: COARSE(CNX,CNY) FINE(FNX,FNY) -CIXST, CIYST - -FIXST, FIYST - -FIXFIN, FIYFIN XBRST, YBRST - -Coarse g r i d funct ion Fine g r i d funct ion S ta r t ing Indices for in te rpo la t ion Indices for f i r s t in terpolated value Indices for las t in terpolated value Flags ind ica t ing p o s i t i o n i n g of f ine g r i d w . r . t coarse g r i d XSTEP, YSTEP > Coarse - to - f1ne mesh spacing ra t ios -> •-> •-> •-> --> --> Rout 1nes ca l1ed : MOD C C C SUBROUTINE LININT(COARSE,FINE,CNX,CNY,FNX,FNY,CIXST, • , CIYST,FIXST,FIXFIN,FIYST,FIYFIN, * XBRST, YBRST,XSTEP,YSTEP) INTEGER MOD INTEGER CNX, CNY, FNX, FNY REAL*8 COARSE (CNX, CNY") , F INE ( FNX . FNY ) INTEGER CIX, CIXST, CIY, CIYST. FIX, FIXFIN, FIXST, FIY, FIYFIN, FIYST, XBR, XBRST, XSTEP, YBR, YBRST, YSTEP CIY = CIYST YBR = YBRST DO 60 FIY = FIYST , FIYFIN CIX = CIXST XBR = XBRST DO 50 FIX = FIXST , FIXFIN IF( XBR .EO. 1 ) GO TO 20 IF( YBR .EO. 1 ) GO TO 10 FINE(FIX.FIY) = COARSE(CIX,CIY) GO TO 40 FINE(FIX.FIY) = 0.5D0 * (COARSE(CIX,CIY ) + C0ARSE(CIX,CIY+1 )) GO TO 40 IF( YBR .EO. 1 ) GO TO 30 FINE(FIX.FIY) = 0.5D0 * (COARSE(CIX,CIY) + C0ARSE(CIX+1,CIY)) GO TO 40 FINE(FIX.FIY) = 0.25D0 * (COARSE(CIX,CIY) + C0ARSE(CIX+1 ,CIY) + COARSE(CIX,CIY+1) + C0ARSE(CIX+1.CIY+1)) XBR = M0D(XBR + XSTEP,2) IF( XBR .EO. O .OR. XSTEP .EO. 2 ) CIX = CIX + 1 CONTINUE YBR = MOD(YBR + YSTEP,2) IF( YBR .EO. 0 .OR. YSTEP .EO. 2 ) CIY = CIY + 1 60 CONTINUE C RETURN C C END c  c C SUBROUTINE CUBINT C c  C C Performs cubic Interpolat ion of coarse g r i d funct ion C. COARSE to f i n e g r i d funct ion FINE. See subroutine INTERP C for assumptions regarding g r i d spacings e t c . Interpolat ion 1s C . f i r s t performed Into array TEMP which corresponds to a f i n e C g r i d which covers the same region as the coarse g r i d . C Appropr iate values are then t ransfer red to FINE. C C Parameters: C C COARSE(CNX.CNY) -> Coarse g r i d funct ion C FINE(FNX,FNY) > Fine g r i d funct ion C TEMP(TNX,TNY) > Temporary work array C CIXST, CIYST > S ta r t ing indices for in te rpo la t ion C FIXST, FIYST > Indices for f i r s t Interpolated value C FIXFIN, FIYFIN --> Indices for last interpolated value C FTXST, FTYST > S t a r t i n g indices for t ransfer of C Interpolated values from TEMP to FINE C XSTEP, YSTEP > Coarse - to - f1ne mesh spacing ra t ios C C Routines c a l l e d : 10 20 30 40 50 220 c c c c c-c c c c c c c c c c c c MOD CUBIC 1 SUBROUTINE CUBINT(COARSE,FINE,TEMP,CNX,CNY,FNX,FNY, * TNX,TNY,CIXST,CIXFIN,CIYST,CIYFIN, * FIXST,FIXFIN,FIYST,FIYFIN.FTXST, * FTYST,XSTEP,YSTEP) 10 15 20 30 INTEGER CNX, CNY, FNX, FNY, TNX, TNY REAL*8 COARSE(CNX.CNY), FINE(FNX,FNY), TEMP(TNX,TNY) REAL*8 CUBIC 1 INTEGER MOD INTEGER CIX, CIXFIN, CIXST, CIY, CIYFIN. CIYST. CXINT, FIX, FIXFIN, FIXST, FIY, FIYFIN, FIYST, FTXST, FTYST, IINT, NXINT, NX INT 1, NY INT, NY INT 1, T1, T2, TIX, TIY, TYINT, XBR, XSTEP, YBR, YSTEP T1 = CIXFIN - CIXST T2 = CIYFIN - CIYST NXINT = O NY INT = 0 IF( XSTEP .EO. 1 IF( YSTEP .EO. 1 NX INT 1 = NXINT -NYINT1 = NY INT -) NXINT = T1 ) NY INT = T2 1 1 INTERPOLATE IN X DIRECTION TIY 1 DO 40 CIY = CIYST , CIYFIN CIX = CIXST CXINT = CIXST XBR = 0 IINT = 1 DO 30 TIX = 1 , TNX IF( XBR .EO. 1 ) GO TO 10 TEMP(TIX.TIY) = COARSE(CIX.CIY) CIX = CIX + 1 GO TO 20 TEMP(TIX.TIY) = CUBICKCOARSE,CNX,CNY,CXINT, CIY,O,IINT,NXINT) .OR .IINT .GE. NX INT 1 ) GO TO 15 1 .LE . 1 IF( IINT CXINT = CXINT + IINT = IINT + 1 XBR = MOD(XBR + XSTEP,2) CONTINUE TIY TIY + (3 - YSTEP) 221 40 CONTINUE C IF( YSTEP .EO. 2 ) GO TO 80 C C INTERPOLATE IN Y DIRECTION C DO 70 TIX = 1 , TNX YBR = O TYINT = 1 11NT = 1 C DO 60 TIY = 1 , TNY IF( YBR .EO. O ) GO TO 50 TEMP(TIX,TIY) = CUBIC1(TEMP,TNX,TNY,TIX, * TYINT, 1,11NT,NY INT) IF( IINT .LE . 1 .OR. IINT .GE. NY INT 1 * GO TO 45 TYINT = TYINT + 2 45 IINT = IINT + 1 50 YBR = MOD(YBR + YSTEP,2) 60 CONTINUE C 70 CONTINUE C 80 TIY = FTYST C C TRANSFER VALUES FROM TEMP TO FINE C DO 100 FIY = FIYST , FIYFIN TIX = FTXST C DO 90 FIX = FIXST , FIXFIN FINE(FIX.FIY) = TEMP(TIX,TIY) TIX = TIX + 1 90 CONTINUE C TIY = TIY + 1 100 CONTINUE C RETURN END <j C C FUNCTION CUBIC 1 C c  C C Helper rout ine for SUBROUTINE CUBINT. Determines which C d i r e c t i o n i n t e r p o l a t i o n is being performed 1n, c a l l s CUBIC C to produce Interpolated va lue. C C Parameters: C C F(NX,NY ) - -> Function values used for in te rpo la t ion C I, J > Indices of in te rpo la t ion point (IP) C ROWCOL > .EO. 0 -> In terpo lat ion In x d i r e c t i o n 222 c c c c c c c c c c-c c c c c c c c c c 10 20 30 40 50 .NE. o -> " " y I I N T > p o s i t i o n of IP 1n row or column MINT > Number of values to be Interpolated 1n row or column Rout 1nes cal1ed CUBIC DOUBLE PRECISION FUNCTION CUBIC1(F.NX,NY,I.J,ROWCOL,I INT,NINT) INTEGER NX, NY REAL*8 F(NX,NY) REAL*8 CUBIC REAL*8 FO, F1, F2, F3 , FIVE, ONE, THREE INTEGER I, IINT, J , NINT, ROWCOL DATA ONE / 1.ODO / , THREE / 3.ODO / , FIVE / 5.ODO / IF( ROWCOL .EO. 1 ) GO TO 10 FO = F ( I , J ) F1 = F(I+1, J ) F2 = F(I+2,J) F3 = F(I+3,J) GO TO 20 FO = F ( I , J ) F1 = F(I ,J+2) F2 = F(I ,J+4) F3 = F(I,J+6) IF( IINT .EO. 1 ) GO TO 40 IF( IINT .EO. NINT ) GO TO 30 CUBIC 1 = CUBIC(THREE,FO,F1,F2,F3) GO TO 50 CUBIC 1 = CUBIC(FIVE,F0,F1,F2.F3) GO TO 50 CUBIC 1 = CUBIC(0NE,F0,F1,F2,F3) RETURN C C END c  C C FUNCTION CUBIC C c  C Helper rout ine for FUNCTION CUBIC 1. Returns interpolated C value from FO, F1, F2, F3. N ( 1 , 3 , 5 ) s p e c i f i e s p o s i t i o n i n g C. of in te rpo la ted value with respaect to FO. C C Parameters: As above C C-C C DOUBLE PRECISION FUNCTION CUBIC(N,FO,F1,F2,F3) REAL*8 FO, F1, F2, F3. N, T1, T2. T3 T1 = N * 0.5DO T2 = T1 * ( N - 2.ODO ) * 0.25DO T3 = T2 * ( N - 4.ODO ) / 6.ODO CUBIC = FO + T1 * ( F1 - FO ) + T2 * ( F2 - 2.ODO * F1 + FO ) + T3 * ( F3 - 3.ODO * F2 + 3.ODO * F1 - FO ) RETURN END c--C C C C - -C C C C C C C C c c c c c c c c c c c c c c c c c c c-c c SUBROUTINE INJECT R e s t r i c t s f i n e g r i d funct ion FINE to coarse g r i d ^ t i o n COARSE by i n j e c t i n g appropr iate i n t e r i o r values ^nd poss ib l y ex t rapo la t ing boundary values^ See rout ine INTERP for assumed r e l a t i o n between coarse and f i n e g r i d s . Parameters: COARSE(CNX,CNY) -> FINE(FNX,FNY) > CXO, CYO > FXO, FYO > CXINC, CYINC > FXINC, FYINC > CXLOB, CXHIB, > CYLOB, CYHIB / FXLOB, FXHIB, > FYLOB, FYHIB / MODE > Routines c a l l e d : Coarse g r i d (CG) funct ion Fine g r i d (FG) funct ion Coords, of lower l e f t corner of CG n II II II II " FG x,y mesh spacings of CG H H II 11 F G CG boundary f lags FG boundary f l a g s .EO. 0 -> extrapolate boundary values .NE. O -> in ject boundary values CEIL TRNCAT SUBROUTINE INJECT(COARSE,FINE.CNX,CNY,FNX,FNY,CXO, 224 10 20 30 40 50 GO 70 80 C C C INTEGER REAL*8 REAL*8 REAL*8 INTEGER INTEGER INTEGER CYO,FXO,FYO,CXINC,CYINC,FXINC.FY INC, CXLOES, CXHIB , CYLOB , CYHIB , FXLOB , FXHIB , FYLOB.FYHIB,MODE) CNX, CNY, FNX, FNY COARSE(CNX.CNY), FINE(FNX,FNY) TRNCAT CFX, CFY, CXO, CXINC, CYO, CYINC, FXO, FXINC, FYO, FY INC CEIL CXHIB, CXLOB, CYHIB, FYHIB, FYLOB CYLOB, FXHIB, FXLOB, CIX, CIXFIN, CIXST, CIY, CIYFIN, CIYST, FIX, FIXST, FIY, FIYST, FNXM1, FNXM2, FNXM3, FNYM1, FNYM2, FNYM3, MODE, NIX, NIY, XSTEP, YSTEP CFX = (FXO • - CXO) / CXINC CFY = (FYO • - CYO) / CYINC IF( CFX .NE . TRNCAT(CFX) ) CIXST = CFX + 1.ODO FIXST = 1 GO TO 20 CIXST = CFX + 1.5D0 FIXST = 2 IF( CFY .NE . TRNCAT(CFY) ) CIYST = CFY + 1.ODO FIYST = 1 GO TO 40 CIYST = CFY + 1.5D0 FIYST = 2 NIX = CEIL(((FNX + 1 NIY = CEIL(((FNY + 1 CIXFIN = CIXST + NIX CIYFIN = CIYST + NIY FIXST) * FXINC) / CXINC) FIYST) * FY INC) / CYINC) 1 1 FXINC .EO XSTEP = 2 TO GO XSTEP = 1 FY INC .EO YSTEP = 2 TO 80 YSTEP = 1 FIX = FIXST INTERIOR INJECTION DO 100 CIX = CIXST FIY = FIYST CIXFIN DO 90 CIY = CIYST , CIYFIN COARSE(CIX.CIY) = FINE(FIX.FIY) FIY = FIY + YSTEP CONTINUE FIX = FIX + XSTEP CONTINUE BOUNDARY EXTRAPOLATION IF( ( CYHIB .EO. O .AND. FYHIB .EO. 0 ) .OR. YSTEP .EO. 1 ) GO TO 120 EXTRAPOLATE "UPPER" BOUNDARY FIX = FIXST FNYM1 = FNY - 1 FNYM2 = FNY - 2 FNYM3 = FNY - 3 IF( MODE .NE. 0 ) GO TO 112 DO 110 CIX = CIXST , CIXFIN COARSE(CIX.CNY) = 3.5D0 * FINE(FIX,FNY) -4.5D0 * FINE(FIX,FNYM1) 2.5DO * FINE(FIX,FNYM2 ) 0.5D0 * FINE(FIX,FNYM3) FIX = FIX + XSTEP CONTINUE GO TO 120 DO 115 CIX = CIXST , CIXFIN COARSE(CIX,CNY) = FINE(FIX,FNY) FIX = FIX + XSTEP CONTINUE IF( ( CYLOB .EO. 0 .AND. FYLOB .EO. O ) .OR. YSTEP .EO. 1 ) GO TO 140 EXTRAPOLATE "LOWER" BOUNDARY FIX = FIXST IF( MODE .NE. 0 ) GO TO 132 DO 130 CIX = CIXST , CIXFIN COARSE(CIX, 1 ) = 3.5DO * FINE(FIX.I) -4.5D0 * FINE(FIX,2) + 2.5D0 * FINE(FIX,3) -0.5D0 * FINE(FIX,4) FIX = FIX + XSTEP CONTINUE GO TO 140 DO 135 CIX = CIXST , CIXFIN COARSE(CIX,1 ) = FINE(FIX,1) FIX = FIX + XSTEP CONTINUE IF( ( CXHIB .EO. 0 .AND. FXHIB .EO. 0 ) .OR. XSTEP .EO. 1 ) GO TO 160 EXTRAPOLATE "OUTER" BOUNDARY FIY = FIYST 226 150 C 152 155 C 160 C C C FNXM1 = FNX - 1 FNXM2 = FNX - 2 FNXM3 = FNX - 3 IF( MODE .NE. O ) GO TO 152 DO 150 CIY = CIYST , CIYFIN COARSE(CNX.CIY) 3 . 5D0 4 . 5D0 2.5D0 0.5D0 FINE(FNX,FIY) -FINE(FNXM1,FIY) + FINE(FNXM2,FIY) -FINE(FNXM3,FIY) FIY = FIY CONTINUE GO TO 160 YSTEP DO 155 CIY = CIYST , CIYFIN COARSE(CNX.CIY) = FINE(FNX,FIY) FIY = FIY + YSTEP CONTINUE IF( ( CXLOB .EQ. O .AND. FXLOB .EO. 0 ) .OR. XSTEP .EO. 1 ) GO TO 180 EXTRAPOLATE "INNER" BOUNDARY FIY = FIYST IF( MODE .NE. 0 ) GO TO 172 DO 170 CIY CIYST CIYFIN 170 C 172 175 C 180 C C COARSE(1,CIY) = 3.5D0 4 . 5D0 2.5DO 0.5D0 FIY = FIY + YSTEP CONTINUE GO TO 180 DO 175 CIY = CIYST , CIYFIN COARSE(1,CIY) = FINE(1,FIY) FIY = FIY + YSTEP CONTINUE FINE(1,FIY) -FINE(2,FIY) + FINE(3,FIY) -FINE(4,FIY) RETURN END 227 c-c c c c c c c c-R O U T I N E S F O R G R I D D E F I N I T I O N A N D I N I T I A L I Z A T I O N ; F U N C T I O N D E F I N I T I O N c-C C C C-C C C C C C C C C c-C C C C-C SUBROUTINE INIGST I n i t i a l i z e s g r i d s t ructure Parameters: INILEV -> I n i t i a l level (usual ly 1 i f f i r s t level to be used w i l l be coarsest level throughout run) MAXGRD -> Maximum number of g r ids to be used in run C C C C C C SUBROUTINE G R I D INTEGER INTEGER REAL*8 * LOGICAL INTEGER INTEGER INTEGER INTEGER COMMON COMMON INIGST(INI LEV,MAXGRD) D A T A S T R U C T U R E GFSTRT(50,10), XFSTRT(50,10), YFSTRT(50,10) BROTHR(50), FATHER(50), LHEAD(50), LLINK(50), SON(50), NPT(50), NX(50), NY(50), XHIB(50), XL0B(50), YHIB(50), YL0B(50) AREA(50), EPSI(50). ERR0R(5O), NRMTAU(50), XINC(50), YINC(50) C0NV(50), SL0W(5O) CRSLEV, FINLEV, NGFUNC, NGRID, NXFUNC. NYFUNC MEMPT KSQR, R, RHS, RM, RP, TAU, W, WM, WP, U, OLDU NULL / Z80808080 / / COMGST / LHEAD, LLINK, FATHER, BROTHR, SON, NX, NY, NPT, AREA, XINC, YINC, XFSTRT, YFSTRT, GFSTRT, XLOB, XHIB, YLOB, YHIB, ERROR, CONV, SLOW, EPSI, NRMTAU, CRSLEV, FINLEV, NGRID, MEMPT / COMCON / R, RP, RM, W, WP, WM, U. OLDU, RHS, TAU, KSQR, NXFUNC, NYFUNC, NGFUNC INTEGER I, INILEV, MAXGRD INITIALIZE TREE STRUCTURE DO 10 I = 1 , MAXGRD LHEAD(I) = NULL LLINK(I) = NULL FATHER(I) = NULL BROTHR(I ) = NULL SON(I) = NULL 10 CONTINUE INITIALIZE FUNCTION "POINTERS" MEMPT = 1 NGRID = O CRSLEV = INILEV FINLEV = INILEV NXFUNC = 3 NYFUNC = 3 NGFUNC = 5 R = 1 RP = 2 RM = 3 W = 1 WP = 2 WM = 3 U = 1 OLDU = 2 RHS = 3 TAU = 4 KSQR = 5 C RETURN C C END C-C C C C-C C C C C C C C C-C C SUBROUTINE NEWGRD Attempts to insert new nul l g r i d Into g r i d is at level LEV and has father DAD. MAXGRD number of g r ids which may be def ined . RC zero value If i nse r t ion attempt 1s Parameters: As above s t ruc tu re . Gr id is the maximum returns with a non-unsuccessful . SUBROUTINE NEWGRD(LEV,DAD,MAXGRD,RC) 229 c c c c c c c c c c c c. c c c 10 G R I D D A T A S T R U C T U R E INTEGER GFSTRT( 50, 10), XFSTRT(50, 10), YFSTRT(50, 10) INTEGER BROTHR(50), FATHER(50), LHEAD(50), LLINK(50), S0N(50). NPT(50), NX(50), NY(50), XHIB(50), XL0B(5O), YHIB(50), YL0B(50) REAL *8 AREA(50), EPSI(50), ERR0R(50), NRMTAU(50), XINC(50), YINC(50) LOGICAL C0NV(50), SL0W(50) INTEGER CRSLEV, FINLEV, NGFUNC. NGRID, NXFUNC, NYFUNC INTEGER MEMPT INTEGER KSQR, R, RHS, RM, RP, TAU, W, WM, WP, U, OLDU INTEGER NULL / Z80808080 / COMMON / COMGST / LHEAD, LLINK, FATHER, BROTHR, SON, NX, NY, NPT, AREA, XINC, YINC, XFSTRT, YFSTRT, GFSTRT, XLOB, XHIB, YLOB, YHIB, ERROR, CONV. SLOW, EPSI, NRMTAU, CRSLEV, FINLEV, NGRID, MEMPT COMMON / COMCON / R. RP, RM, W, WP, WM. U. OLDU, RHS, TAU, KSQR, NXFUNC, NYFUNC, NGFUNC INTEGER CRSLV1, DAD, FINLV1, LEV, MAXGRD, NGRID1, PSON, RC NGRID1 = NGRID + 1 IF( NGRID1 .GT. MAXGRD ) GO TO 60 DO GRIDS ALREADY EXIST AT THIS LEVEL ? IF( LHEAD(LEV) .EQ. NULL ) GO TO 10 IF SO, THEN A FATHER MUST BE SPECIFIED IF( DAD .EO. NULL ) GO TO 30 INSERT NEW GRID INTO STRUCTURE, UPDATE POINTERS PSON = SON(DAD) SON(DAD) = NGRID1 FATHER(NGRID1) = DAD BROTHR(NGRID 1) = PSON LLINK(NGRID1) = LHEAD(LEV) LHEAD(LEV) = NGRID 1 GO TO 20 CHECK FOR INVALID LEVEL SPECIFICATION CRSLV1 = CRSLEV - 1 230 FINLV1 = FINLEV + 1 IF( LEV .LT. CRSLV1 ) GO TO 40 IF( LEV .EO. CRSLV1 ) CRSLEV = CRSLV1 IF( LEV .GT. FINLV1 ) GO TO 50 IF( LEV .EO. FINLV1 ) FINLEV = FINLV1 C C INSERT NEW GRID, UPDATE POINTERS C LHEAD(LEV) = NGRID1 LLINK(NGRID1 ) = NULL FATHER(NGRID 1 ) = DAD IF( DAD .NE. NULL ) SON(DAD) = NGRID1 SON(NGRIDI) = LHEAD(LEV + 1) BR0THR(NGRID1) = NULL C 20 NGRID = NGRID1 C C NORMAL RETURN C RC = O RETURN C C ATTEMPT TO INSERT GRID AT EXISTING LEVEL WITH NULL FATHER C 30 RC = 1 RETURN C C SPECIFIED INSERTION LEVEL TOO COARSE C 40 RC = 2 RETURN C C SPECIFIED INSERTION LEVEL TOO FINE C 50 RC = 3 RETURN C C NO MORE SPACE AVAILABLE IN GRID STRUCTURE C GO RC = 4 ETURN END c  c C SUBROUTINE INIGRD C c  C C I n i t i a l i z e s coordinate and g r i d funct ions . C C Parameters: C C KSOR(NS.NW) - -> E x t r i n s i c curvature squared C RHS(NS.NW) > R1ght-hand-slde of g r i d equations C S, S2, S4 > Radial coordinate funct ions C W, WP, WM > Angular 231 C SMNGRD > Minimum rad ia l coordinate of g r id C WMNGRD > " angular C DELS > Radial mesh spacing C DELW > Angular mesh spacing C A > a (dummy parameter - assumed to be 1 .ODO) C P > Momentum C C Routines c a l l e d : C C DSIN C INIT C DEFKSQ C c  C c SUBROUTINE INIGRD(KSQR,RHS,S,S2,S4,W,WP,WM,NS,NW,SMNGRD, * WMNGRD,DELS,DELW,A,P) INTEGER NS, NW REAL*8 KSOR(NS.NW), RHS(NS.NW) S(1) , S2(1), S4(1), W(1), WM(1), WP(1) DSIN REAL*8 REAL*8 REAL*8 A, DELS, DELW, DELW2, DSWJM1, P, SI, SIM1, SMNGRD, Wd, WMNGRD INTEGER I, d, NSM1 DELW2 = 0.5DO * DELW S(1) = SMNGRD W(1) = -DELW W(2) = O.ODO WP(2) = 2.ODO WM(2) = 2.ODO 10 DO 10 I = 2 , NS SI = S( I -1 ) + DELS S(I) = SI SIM1 = 1.ODO - SI S2(I) = SIM1 ** 2 S4(I) = SIM1 ** 4 CONTINUE 20 DO 20 d = 3 , NW Wd = W(d-1) + DELW DSWdMI = 1.0D0 / DSIN(Wd) W(J) = Wd WP(d) = DSWdMI * DSIN(Wd + DELW2) WM(d) = DSWdMI * DSIN(Wd - DELW2) CONTINUE CALL INIT(RHS,NS.NW,O.ODO) CALL DEFKSQ(KSQR,NS,NW,S,W,A,P) 2 3 2 RETURN C C END c  C C SUBROUTINE DEFGRD C c  C C I n i t i a l i z e s sca la r g r i d quant i t i es such as area, mesh C s p a d n g s , convergence c r i t e r i o n , e t c . Determines boundary C types and i n i t i a l i z e s boundary f l a g s . A l loca tes space for C g r i d and coordinate funct ions . C C Parameters: C C GRID > Gr id number C XMNGRD, XMXGRD --> x coords of g r i d boundaries C YMNGRD, YMXGRD --> y C XMNDOM, XMXDOM - -> x " " domain C YMNDOM, YMXDOM --> y C NWXINC, NWYINC --> x,y mesh spadngs C NXLO, NXHI, > Boundary f l a g s C NYLO, NYHI / C EPSILN > I n i t i a l convergence c r i t e r i o n C NWAREA > Area of g r i d C C Routines c a l l e d : C C DABS C c  c c SUBROUTINE DEFGRD(GRID,XMNGRD,XMXGRD,YMNGRD,YMXGRD,XMNDOM, * XMXDOM,YMNDOM,YMXDOM,NWXINC.NWYINC, * NXLO,NXHI,NYLO,NYHI,EPSILN,NWAREA) C C G R I D D A T A S T R U C T U R E C INTEGER GFSTRT(50,10), XFSTRT(50,10), YFSTRT(50,10) C INTEGER BROTHR(50), FATHER(50), LHEAD(50), LLINK(50), * S0N(5O), NPT(50), NX(50), NY(50). XHIB(50), * XL0B(5O), YHIB(50), YL0B(50) C REAL*8 AREA(50), EPSI(50), ERR0R(5O), NRMTAU(50), • .* XINC(50), YINC(50) LOGICAL C0NV(50), SL0W(5O) INTEGER CRSLEV, FINLEV, NGFUNC, NGRID, NXFUNC, NYFUNC INTEGER MEMPT INTEGER KSQR, R, RHS, RM, RP, TAU, W, WM. WP, U, OLDU 233 c c c c c 10 20 30 40 50 60 70 80 INTEGER NULL / Z80808080 / COMMON / COMGST / LHEAD, LLINK, FATHER, BROTHR, SON, NX, NY, NPT, AREA, XINC, YINC, XFSTRT, YFSTRT, GFSTRT, XLOB, XHIB, YLOB, YHIB, ERROR, CONV, SLOW, EPSI, NRMTAU, CRSLEV, FINLEV, NGRID, MEMPT COMMON / COMCON / R, RP, RM, W, WP, WM, U, OLDU, RHS, TAU, KSQR, NXFUNC, NYFUNC, NGFUNC REAL*8 REAL*8 REAL*8 INTEGER DELXMN DELXMX DELYMN DELYMX DABS DELXMN, DELXMX, DELYMN, DELYMX, EPSILN, NWXINC, NWYINC, XL, XMNDOM, XMNGRD, XMXDOM, XMXGRD, YL, YMNDOM, YMNGRD, YMXDOM, YMXGRD, NWAREA PIBY2 / Z411921FB54442D18 / IGFUNC, IXFUNC, IYFUNC, GRID, NPOINT, NXGRID, NXH1, NXLO, NYGRID, NYHI, NYLO XMNGRD XMXDOM YMNGRD YMXDOM XMNDOM XMXGRD YMNDOM YMXGRD ESTABLISH THE 4 BOUNDARY TYPES DIRICHLET: 0, NEUMANN, ROBBIN: > 0 IF( DELXMN .GE. O.ODO) GO TO 10 XLOB(GRID) = NXLO IF( DABS(DELXMN) .GT. NWXINC ) XMNGRD = XMNGRD + NWXINC GO TO 20 XLOB(GRID ) = O IF( DELXMX .GE. O.ODO ) GO TO 30 XHIB(GRID) = NXHI IF( DABS(DELXMX) .GT. NWXINC ) XMXGRD = XMXGRD - NWXINC GO TO 40 XHIB(GRID) = 0 IF( DELYMN .GE. O.ODO) GO TO 50 YLOB(GRID) = NYLO IF( DABS(DELYMN) .GT. NWYINC ) YMNGRD = YMNGRD + NWYINC GO TO 60 YLOB(GRID) = O IF( DELYMX .GE. O.ODO) GO TO 70 YHIB(GRID) = NYHI IF( DABS(DELYMX) .GT. NWYINC ) YMXGRD = YMXGRD - NWYINC GO TO 80 YHIB(GRID) = O XL = XMXGRD - XMNGRD YL = YMXGRD - YMNGRD NXGRID = XL / NWXINC + 1 NYGRID = YL / NWYINC + 1 XINC(GRID) = NWXINC 234 YINC(GRID) = PIBY2 / (NYGRID - 3) NX(GRID) = NXGRID NY(GRID) = NYGRID NPOINT = NXGRID * NYGRID NPT(GRID) = NPOINT AREA(GRID) = NWAREA EPSI(GRID) = EPSILN C C ALLOCATE SPACE FOR COORDINATE AND GRID FUNCTIONS C DO 90 IXFUNC = 1 , NXFUNC XFSTRT(GRID,IXFUNC) = MEMPT MEMPT = MEMPT + NXGRID 90 CONTINUE C DO 100 IYFUNC = 1 , NYFUNC YFSTRT(GRID,IYFUNC) = MEMPT MEMPT = MEMPT + NYGRID 100 CONTINUE C DO 110 IGFUNC = 1 , NGFUNC GFSTRT(GRID,IGFUNC) = MEMPT MEMPT = MEMPT + NPOINT 110 CONTINUE C RETURN C C END C-C C C C-C C C C C C C C-C C C C C C C C c SUBROUTINE GUESS Ca lcu la tes i n i t i a l estimate- of funct ion PHI(NS.NW). Radial coordinate array S is suppl ied as parameter. Cur rent ly uniformly i n i t i a l i z e s PHI to 1.0D0 Parameters: As above SUBROUTINE GUESS(PHI,S,NS,NW) INTEGER NS, NW REAL*8 PHI(NS.NW) REAL*8 S(1) INTEGER IS, JW DO 20 IS = 1 , NS 235 "DO 10 JW = 1 , NW PHI(IS,JW) = 1.ODO 10 CONTINUE C 20 CONTINUE C RETURN C C END c  C C SUBROUTINE DEFKSO C c  C C Defines " e x t r i n s i c curvature squared" (H) for exact ly C so lub le model problem. C C Parameters: C C KSQR(NS.NW) -> E x t r i n s i c curvature squared C S > Radial coordinates C W > Angular coordinates C A > Inversion radius C P > Linear momentum C c  C c c c c c c c c c c c SUBROUTINE DEFKSO(KSQR,NS,NW,S,W,A,P) INTEGER NS, NW REAL+8 KSQR(NS.NW) REAL*8 S(1) , W(1) REAL*8 A, ARIS2, P, P26, RIS, T1, T2, WJW2 INTEGER IS, JW, NSM1 C NSM1 = NS - 1 P26 = 6.ODO * P ** 2 DO 20 JW = 1 , NW DO 10 IS = 1 , NSM1 RIS = 1.0D0 / (1.0D0 - S(IS)) ARIS2 = (A / RIS) ** 2 T1 = (1.ODO - ARIS2) ** 2 KSQR(IS.vJW) = P26 * T1 / RIS ** 4 10 CONTINUE KSQR(NS.JW) = P26 236 20 CONTINUE C RETURN C C END c : c C SUBROUTINE DEFKSQ C Defines " e x t r i n s i c curvature squared" (H) for boosted black ho les . Parameters: KSQR(NS.NW) - > E x t r i n s i c curvature squared S •-> Radial coordinates W •-> Angular coordinates A - > Inversion radius P •-> Linear momentum Routines c a l 1 e d : DCOS C C SUBROUTINE DEFKSQ(KSQR,NS,NW,S,W,A,P) C C C C C C INTEGER NS, NW REAL*8 KSOR(NS.NW) REAL*8 S O ) . WO) REAL*8 DCOS REAL*8 A, ARIS2, P. P245, RIS, T1, T2, WJW2 INTEGER IS, JW, NSM1 NSM1 = NS - 1 P245 = 4. 5D0 * P ** 2 DO 20 JW = 1 , NW WJW2 = •  DCOS(W(JW)) ** 2 c DO 10 IS = 1 , NSM1 RIS = 1.0D0 / O.ODO - S(IS)) ARIS2 = (A / RIS) ** 2 T1 = 1.000 +. ( -4 . ODO + ARIS2) * ARIS2 T2 = O.ODO + ARIS2) ** 2 + 2.ODO * WJW2 * T1 KSOR(IS.JW) = P245 * T2 / RIS ** 4 237 10 CONTINUE C KSQR(NS.dW) = P245 * (1.ODO + 2.ODO * WJW2) C 20 CONTINUE C C C C C C C C C C RETURN END c  c C SUBROUTINE DEFKSO C c  c C Defines " e x t r i n s i c curvature squared" (H) for spinning C black ho les . C C Parameters: C C KSOR(NS.NW) -> E x t r i n s i c curvature squared C S > Radial coordinates C W > Angular coordinates C A > Inversion radius (unused) C d > Angular momentum C C Rout i nes c a l 1 e d : C C DSIN C c  C c SUBROUTINE DEFKSQ(KSQR,NS,NW,S,W,A,d) INTEGER NS, NW REAL*8 KSQR(NS,NW) REAL*8 S O ) , W(1) REAL*8 DSIN REAL*8 A, d. d18, d18Wd, RISM6, WdW2M1 INTEGER IS, dW, NSM1 NSM1 = NS - 1 d18 = 18.ODO * d ** 2 DO 20 dW = 1 , NW WdW2M1 = DSIN(W(dW)) ** 2 d18WJ = d18 * WdW2M1 DO 10 IS = 1 , NSM1 238 RISM6 = (1.0D0 - S(IS)) ** 6 KSQR(IS.JW) = J18WJ * RISM6 10 CONTINUE C KSQR(NS,JW) = O.ODO C 20 CONTINUE C RETURN C C END 239 c  c C U T I L I T Y R O U T I N E S F O R A R R A Y C C O P E R A T I O N S C c  c  c C SUBROUTINE ADDSUB C c  C C Adds {subtracts} en t r ies of F2 to {from} F1, p lac ing resu l t s C 1n FRES C C Parameters: C C F1(FNX,FNY) - -> As above C F2 (FNX,FNY) / C FRES(FNX,FNY) / C SIGN > .EO. +1.0DO -> Add C .EO. -1.0DO -> Subtract C c  c c SUBROUTINE ADDSUB(F1,F2,FRES,FNX,FNY,SIGN) C INTEGER FNX, FNY C REAL*8 F1(FNX,FNY), F2(FNX,FNY), FRES(FNX,FNY) C REAL*8 SIGN C INTEGER IX, IY C C DO 20 IX = 1 , FNX C DO 10 IY = 1 , FNY FRES(IX.IY) = F K I X . I Y ) + SIGN * F2(IX,IY) 10 CONTINUE C 20 CONTINUE C RETURN END c  C C SUBROUTINE COPY C c  C 2 4 0 C Copies array FROM(NX.NY) to array TO(NX.NY) C C Parameters: As above C c  C c SUBROUTINE COPY(FROM,TO,NX,NY) C INTEGER NX, NY C REAL*8 FROM(NX,NY), TO(NX.NY) C INTEGER IX, IY C C DO 20 IY = 1 , NY DO 10 IX = 1 , NX TO(IX,IY) = FROM(IX.IY) 10 CONTINUE 20 CONTINUE C RETURN C C END c  C C SUBROUTINE INIT C c  C C Places VALUE in a l l en t r ies of ARRAY(NX,NY) C C Parameters: As above C c  c c SUBROUTINE INIT(ARRAY,NX,NY,VALUE) C INTEGER NX, NY C REAL*8 ARRAY(NX,NY) C REAL*8 VALUE C INTEGER IX, IY C C DO 20 IY = 1 , NY DO 10 IX = 1 , NX ARRAY(IX,IY ) = VALUE 10 CONTINUE 20 CONTINUE C RETURN C 241 c END c  c C SUBROUTINE VALURC C c  c C Places VALUE in a l l e n t r i e s of a row or column of C ARRAY(NX,NY) C C C Parameters: As above and C C IRC > Row/column index C ROWCOL > .EO. O -> F i l l column < C .NE. 0 -> F i l l row C c  C c SUBROUTINE VALURC(ARRAY,NX,NY,IRC,ROWCOL,VALUE) C INTEGER NX, NY C REAL*8 ARRAY(NX,NY) C REAL*8 VALUE C INTEGER IRC, IX, IY, ROWCOL C C IF( ROWCOL ..NE. 0 ) GO TO 20 C DO 10 IY = 1 , NY ARRAY(IRC,IY ) = VALUE 10 CONTINUE C GO TO 40 C 20 DO 30 IX = 1 , NX ARRAY(IX,IRC) = VALUE 30 CONTINUE C 40 RETURN C C END 242 c-c c c c-c C Reads X(NX), Y(NY), KSQR(NX.NY), PHI(NX,NY) unformatted C from log ica l uni t FILE. C C Parameters: As above C c  C C I N P U T / O U T P U T R O U T I N E S SUBROUTINE INDATA c c c c c c SUBROUTINE INDATA(X,Y,KSQR,PHI,NX,NY.FILE) INTEGER NX, NY REAL*8 KSQR(NX,NY), PHI(NX.NY), X(NX), Y(NY) INTEGER . IX, IY, FILE READ(FILE) (X(IX) , IX = 1 , NX) READ(FILE) (Y(IY) , IY = 1 , NY) READ(FILE) ((KSQR(IX,IY) , IY = 1 , NY) , IX = 1 , NX) READ(FILE) ((PHI(IX.IY) , IY = 1 , NY) , IX = 1 . NX) RETURN END c  C C SUBROUTINE PRVPHI C c  c C Reads PHI(NX.NY) unformatted from log ica l unit FILE. Return C code RC set to 1 1f s i ze mismatch detected. C C Parameters: As above C c  c c SUBROUTINE PRVPHI(PHI , NX , NY ,.F I LE , RC) C C INTEGER NX, NY 243 c REAL*8 PHI(NX.NY) C INTEGER IX, IY, FILE, GNX, GNY, RC C C RC = 0 READ(FILE) GNX, GNY IF( GNX .NE. NX .OR. GNY .NE. NY ) GO TO 10 READ(FILE) ((PHI(IX.IY) , IY = 1 , NY) , IX = 1 , NX) C RETURN C 10 RC = 1 C RETURN C C END c  C C SUBROUTINE 0UTRS1 C c  c C Writes KSOR(NX.NY), PHI(NX,NY), X(NX), Y(NY) unformatted C on log i ca l uni t FILE. C C Parameters: As above C c  c c SUBROUTINE OUTRS1(KSQR,PHI,X,Y,NX,NY,FILE) INTEGER NX, NY REAL*8 KSQR(NX,NY), PHI(NX.NY), X(NX), Y(NY) INTEGER IX, IXFIN, IXST, IY, IYFIN, IYST, FILE, * OUTNX, OUTNY IXST = 2 IXFIN = NX OUTNX = IXFIN - IXST + 1 IYST = 2 IYFIN = NY - 1 OUTNY = IYFIN - IYST + 1 WRITE(FILE) OUTNX, OUTNY WRITE(FILE) (X(IX) , IX = WRITE(FILE) (Y(IY) . IY = WRITE(FlLE ) ((KSQR(IX,IY) WRITE(FILE) ((PHI(IX.IY) IXST , IXFIN) IYST , IYFIN) , IY = IYST , IYFIN) IX = IXST , IXFIN) , IY = IYST , IYFIN) IX = IXST , IXFIN) C 244 RETURN C C END c  c C SUBROUTINE 0UTRS2 C c  c C Writes PHI(NX.NY) unformatted on log ica l unit FILE. C C Parameters: As above C c  C c SUBROUTINE 0UTRS2(PHI,NX,NY.FILE) INTEGER NX, NY REAL*8 PHI(NX.NY) INTEGER IX, IY, FILE WRITE(FILE) NX, NY WRITE(FILE) ((PHI(IX,IY) , IY = 1 , NY) , IX = 1 . NX) RETURN END c c c c M I S C E L L A N E O U S R O U T I N E S c  C C FUNCTION BIG C c  c C Returns maximum of X and Y. C c  c c DOUBLE PRECISION FUNCTION BIG(X.Y) C REAL*8 X, Y BIG = X IF( Y .GT. X ) BIG = Y RETURN END c  c C FUNCTION SMALL C c  C C Returns minimum of X and Y. C c : C C DOUBLE PRECISION FUNCTION SMALL(X,Y) C REAL*8 X, Y SMALL = X IF( Y .LT. X ) SMALL = Y RETURN END 246 t, c C FUNCTION CEIL C c  C C Returns smal lest integer .GE. X C c  c c INTEGER FUNCTION CEIL(X) C REAL*8 X, X1 X1 = X + 0.99999DO CEIL = X1 RETURN END c  C C FUNCTION TRNCAT C c  C C Returns largest integral double p r e c i s i o n number .LE . X C C Routines c a l l e d : C C DFLOAT C c  c c DOUBLE PRECISION FUNCTION TRNCAT(X) C REAL*8 DFLOAT, X C INTEGER IX IX = X TRNCAT = DFLOAT(IX) RETURN END 

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