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Analysis of a Galerkin-Characteristic algorithm for the potential vorticity-stream function equations Bermejo, Rodolfo 1990

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ANALYSIS OF A GALERKIN-CHARACTERISTIC ALGORITHM FOR THE POTENTIAL VORTICITY-STREAM FUNCTION EQUATIONS By RODOLFO BERMEJO Dipl. Naval Architecture Universidad P o l i t e c n i c a de Madrid ,1977 M.Sc. The Univ e r s i t y of B r i t i s h Columbia,1986 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Mathematics and Inst.of Applied Mathematics) We accept t h i s thesis conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January 1990 © Rodolfo Bermejo, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada DE-6 (2788) ABSTRACT I n t h i s t h e s i s we d e v e l o p and a n a l y z e a G a l e r k i n - C h a r a c t e r i s t i c method t o i n t e g r a t e t h e p o t e n t i a l v o r t i c i t y e q u a t i o n s o f a b a r o c l i n i c o c e a n . The method p r o p o s e d i s a two s t a g e i n d u c t i v e a l g o r i t h m . I n t h e f i r s t s t a g e t h e m a t e r i a l d e r i v a t i v e o f t h e p o t e n t i a l v o r t i c i t y i s a p p r o x i m a t e d by c o m b i n i n g G a l e r k i n - C h a r a c t e r i s t i c and P a r t i c l e methods . T h i s y i e l d a c o m p u t a t i o n a l l y e f f i c i e n t a l g o r i t h m f o r t h i s s t a g e . S u c h an a l g o r i t h m c o n s i s t s o f u p d a t i n g t h e d e p e n d e n t v a r i a b l e a t t h e g r i d p o i n t s by c u b i c s p l i n e i n t e r p o l a t i o n a t t h e f o o t o f t h e c h a r a c t e r i s t i c c u r v e s o f t h e a d v e c t i v e component o f t h e e q u a t i o n s . The a l g o r i t h m i s u n c o n d i t i o n a l l y s t a b l e and c o n s e r v a t i v e f o r A t = 0 ( h ) . The e r r o r a n a l y s i s w i t h 2 r e s p e c t t o L - no rm shows t h a t t he a l g o r i t h m c o n v e r g e s w i t h o r d e r 0 ( h ) ; howeve r , i n t h e maximum norm i t i s p r o v e d t h a t f o r s u f f i c i e n t l y smooth f u n c t i o n s t h e f o o t o f t h e c h a r a c t e r i s t i c 4 c u r v e s a r e s u p e r c o n v e r g e n t p o i n t s o f o r d e r 0 ( h / A t ) . The s e c o n d s t a g e o f t h e a l g o r i t h m i s a p r o j e c t i o n o f t h e L a g r a n g i a n r e p r e s e n t a t i o n o f t h e f l o w o n t o t h e C a r t e s i a n s p a c e — t i m e E u l a r i a n r e p r e s e n t a t i o n c o o r d i n a t e d w i t h C r a n k — N i c h o l s o n F i n i t e E l e m e n t s . The e r r o r a n a l y s i s f o r t h i s 2 s t a g e w i t h r e s p e c t t o L —norm shows t h a t t h e a p p r o x i m a t i o n 2 component o f t h e g l o b a l e r r o r i s 0 ( h ) f o r t h e f r e e - s l i p b o u n d a r y c o n d i t i o n , and 0 ( h ) f o r t h e n o — s l i p b o u n d a r y c o n d i t i o n . The se e s t i m a t e s r e p r e s e n t an imp rovement w i t h r e s p e c t t o o t h e r e s t i m a t e s f o r t h e v o r t i c i t y p r e v i o u s l y i i r e p o r t e d i n t h e l i t e r a t u r e . The e v o l u t i o n a r y component o f t h e 2 g l o b a l e r r o r i s e q u a l t o K ( A t + h ) , where K i s a c o n s t a n t t h a t depend s on t h e d e r i v a t i v e s o f t h e a d v e c t i v e q u a n t i t y a l o n g t h e C h a r a c t e r i s t i c . S i n c e t h e p o t e n t i a l v o r t i c i t y i s a q u a s i - c o n s e r v a t i v e q u a n t i t i y , one c an c o n c l u d e t h a t K i s i n g e n e r a l s m a l l . N u m e r i c a l e x p e r i m e n t s i l l u s t r a t e o u r t h e o r e t i c a l r e s u l t s f o r b o t h s t a g e s o f t h e method. i i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i v LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENTS v i i i CHAPTER I. INTRODUCTION 1 CHAPTER II. THE GOVERNING EQUATIONS AND THEIR NUMERICAL DISCRETIZATION 8 § 1 . G o v e r n i n g E q u a t i o n s 7 § 2 . N o t a t i o n 14 § 3 . 1 . The Weak S o l u t i o n F o r m u l a t i o n 18 § 3 . 2 . The F i n i t e E l e m e n t A p p r o x i m a t i o n 29 CHAPTER III. A GALERKIN-CHARACTERISTIC ALGORITHM FOR THE HYPERBOLIC STAGE 38 § 1 . P r e l i m i n a r i e s 39 §2 . D e s c r i p t i o n o f t h e A l g o r i t h m 44 2 . 1 . The c o n t i n u o u s P r o b l e m 44 2 . 2 . The D i s c r e t e P r o b l e m 47 § 3 . P r o p e r t i e s 58 §4. E r r o r A n a l y s i s 60 § 5 . N u m e r i c a l E x p e r i m e n t s 70 i v CHAPTER IV. ERROR ANALYSIS OF THE POTENTIAL VORTICITY AND BAROTROPIC-BAROCLINIC MODE EQUATIONS 81 § 1 . Preliminaries 82 §2. Error Analysis of the Potential Vorticity Equation 86 §3. Error Analysis of the Barotropic and Baroclinic Modes..102 §4. Numerical Experiments 108 Appendix to Chapter IV 109 CHAPTER V. CONCLUSIONS 126 REFERENCES 130 APPENDIX 135 v LIST OF TABLES T a b l e Page 1. T ime E v o l u t i o n o f t h e Cone 73 2. T ime E v o l u t i o n o f t h e S l o t t e d C y l i n d e r 73 3. L o g a r i t h m o f r .m . s . E r r o r s 112 4. L o g a r i t h m o f r . m . s . E r r o r s 112 5. l o g a r i t h m o f r .m . s . E r r o r s 112 v i LIST OF FIGURES Figure Page 1. The Two Layer Model 9 2. Regularization of the Particle Point (X^.t) that wi l l occupied the grid Point x at t + At 74 3a. Rotating Cone after 0 Revolutions 75 3b. Rotating Cone after 6 Revolutions ..76 4a. Slotted Cylinder after 0 Revolutions 77 4b. Slotted Cylinder after 1/8 of a Revolution 78 4c. Slotted Cylinder after 1 Revolution 79 4d. Slotted Cylinder after 6 Revolutions 80 5a. Exact Stream Function Solution at Re = 1000 113 5b. Calculated Stream Function Solution at Re = 1000 114 5c. Pointwise Error distribution of the Stream—Function....115 6a. Exact Vorticity Solution at Re = 1000 116 6b. Calculated Vorticity Solution at Re = 1000 117 6c. Poitnwise error distribution of the Vorticity 118 v i i ACKNOWLEDGEMENTS I w o u l d l i k e t o a cknow ledge , w i t h g r a t i t u d e , t h e f o l l o w i n g p e o p l e ; W i l l i a m H s i e h , my s u p e r v i s o r , f o r a l l o w i n g me t h e f r e e d o m t o roam f a r and w i d e i n s e a r c h o f an u l t i m a t e scheme; t h e members o f my s u p e r v i s o r y c o m m i t t e e , U r i A s c h e r , M i k e Fo reman and B r i a n Seymour f o r t h e i r a d v i c e t o i m p r o v e t h e t e x t o f t h e t h e s i s ; Andrew S t a n i f o r t h , whose s u p p o r t and g u i d a n c e were c r u c i a l i n t h e p r o c e s s o f my r e s e a r c h ; and J o h n Heywood, who t a u g h t me t h e b e a u t y o f d o i n g n u m e r i c a l a n a l y s i s i n t h e f r a m e w o r k o f f i n i t e e l e m e n t s . On a more p e r s o n a l n o t e , my t h a n k s t o M a r t i n and A l e x , whose c o m p a n i o n s h i p and c o m p e t i t i v e n e s s we re v e r y h e l p f u l d u r i n g t h e w r i t i n g o f t h i s t h e s i s . F i n a l l y , t o my f a m i l y , who made i t a l l p o s s i b l e , i n p a r t i c u l a r t o M a r i a J o s e , t o whom t h i s t h e s i s i s d e d i c a t e d . v i i i CHAPTER I I N T R O D U C T I O N D u r i n g t h e p a s t 25 y e a r s t h e r e ha s b e e n a s t e a d y i n c r e a s e i n t h e u s e o f n u m e r i c a l methods t o s o l v e t h e e q u a t i o n s w h i c h g o v e r n t h e o c e a n d y n a m i c s . A c t i v e i n t e r a c t i o n among o b s e r v a t i o n a l e v i d e n c e s , n u m e r i c s , c ompu te r t e c h n o l o g y and s c i e n t i f i c r e a s o n i n g h a s p r o d u c e d a h i e r a r c h y o f m o d e l s t o s t u d y d i f f e r e n t a s p e c t s o f t h e o c e a n d y n a m i c s . T h i s h i e r a r c h y s pan s f r o m mode l s w h i c h u se t h e t h r e e d i m e n s i o n a l N a v i e r - S t o k e s e q u a t i o n s a l o n g w i t h mass, h e a t and s t a t e e q u a t i o n s down t o t h e q u a s i - g e o s t r o p h i c mode l s (QGM). The l a t t e r o n l y r e t a i n t h e p h y s i c s r e l a t e d t o a r a n g e o f s p a c e - t i m e s c a l e s s i g n i f i c a n t t o m i d - l a t i t u d e o c e a n d y n a m i c s . Though s i m p l e , QGM have c o n t r i b u t e d s i g n i f i c a n t l y t o o u r u n d e r s t a n d i n g o f t h e o c e a n d y n a m i c s d u r i n g t h e p a s t d e c a d e . F rom a n u m e r i c a l v i e w p o i n t , one o f t h e r e a s o n s o f s u c h a s u c c e s s i s t h a t t h e p h y s i c a l a s s u m p t i o n s , w h i c h l e a d t o t h e f o r m u l a t i o n o f t h e QG e q u a t i o n s , a l l o w t h e u s e o f r e a s o n a b l y l a r g e t i m e s t e p s w i t h a g r i d s p a c i n g s m a l l enough t o m i n i m a l l y r e s o l v e some o f t h e i m p o r t a n t f e a t u r e s o f t h e m i d - l a t i t u d e o c e a n c i r c u l a t i o n . Thus, one i s a b l e t o p e r f o r m c a l c u l a t i o n s f o r v e r y l o n g p e r i o d s o f t i m e i n o r d e r t o s t u d y t h e t i m e e v o l u t i o n o f t h e phenomena. Mos t o f t h e p r e s e n t mode l s i n t h e o c e a n m o d e l i n g commun i ty 1 u s e s t a n d a r d f i n i t e d i f f e r e n c e s , b o t h i n t i m e and i n s p a c e . The t i m e d i s c r e t i z a t i o n i s c a r r i e d o u t b y t h e so c a l l e d l e a p - f r o g scheme f o r most o f t h e te rms o f t h e e q u a t i o n s , w i t h t h e p o s s i b l e e x c e p t i o n o f t h e v i s c o u s t e r m s w h i c h a r e l a g g e d by one t i m e s t e p . The ma in p u r p o s e o f u s i n g t h i s t i m e d i s c r e t i z a t i o n f o r t h e v i s c o u s t e r m s i s t o k e e p t h e e x p l i c i t n e s s o f t h e scheme w i t h o u t s i g n i f i c a n t l y a l t e r i n g t h e CFL c o n d i t i o n ; however , by d o i n g so t h e t i m e e r r o r i s o f o r d e r O ( A t ) , i n s t e a d o f (A t ) a s t h e l e a p - f r o g scheme w o u l d s u g g e s t . The s p a t i a l e r r o r i n QGM w i t h f r e e - s l i p b o u n d a r y c o n d i t i o n s i s o f o r d e r o f 0 ( h ) , where h s t a n d s f o r a g e n e r i c g r i d s p a c i n g i n t e r v a l . A s we w i l l s ee i n C h a p t e r I I , t h e m a t h e m a t i c a l s t r u c t u r e o f t h e QG e q u a t i o n s c o n s i s t s o f a s e m i - l i n e a r p a r a b o l i c e q u a t i o n - t r a n s p o r t - d i f f u s i o n e q u a t i o n - f o r t h e p o t e n t i a l v o r t i c i t y q c o u p l e d w i t h an e l l i p t i c e q u a t i o n f o r t h e s t r e a m f u n c t i o n </». I t i s w e l l known t h a t t h e n u m e r i c a l s o l u t i o n o f t h e t r a n s p o r t - d i f f u s i o n e q u a t i o n po se s a c h a l l e n g i n g p r o b l e m t o t h e n u m e r i c a l a n a l y s t . T h i s i s due t o t h e f a c t t h a t s u c h an e q u a t i o n i s a c o m b i n a t i o n o f h y p e r b o l i c t e r m s - t h e a d v e c t i v e o p e r a t o r - and d i s s i p a t i v e te rms - t h e L a p l a c i a n o p e r a t o r . T h e r e f o r e , w h e t h e r t h e g l o b a l s t r u c t u r e o f t h e e q u a t i o n i s more h y p e r b o l i c o r p a r a b o l i c depends on t h e r e l a t i v e m a g n i t u d e o f one t e r m a g a i n s t t h e o t h e r . F u r t h e r m o r e , i n o c e a n c i r c u l a t i o n p r o b l e m s t h e g l o b a l s t r u c t u r e o f t h e e q u a t i o n h a s a l s o a g e o g r a p h i c a l d i s t r i b u t i o n o r c h a r a c t e r . T h e r e a r e , f o r 2 i n s t a n c e , n a r r o w a r e a s where t h e c o n v e c t i v e t e rms become t h e d o m i n a n t one s , b u t w i t h t h e d i f f u s i v e t e r m s s t i l l p l a y i n g a s i g n i f i c a n t r o l e . I n c o n t r a s t , i n b r o a d a r e a s o f t h e s p a t i a l doma in , away f r o m t h e s o l i d b o u n d a r i e s , t h e p a r a m e t e r i z e d d i s s i p a t i v e t e r m s s h o u l d be v e r y s m a l l . The se c o n s i d e r a t i o n s p o i n t o u t t h e n u m e r i c a l d i f f i c u l t i e s f o u n d i n d e s i g n i n g an e f f i c i e n t t r a n s p o r t - d i f f u s i o n a l g o r i t h m . And s o , i t i s p e r f e c t l y u n d e r s t a n d a b l e why s u c h a l g o r i t h m s f o r m a v e r y a c t i v e r e s e a r c h a r e a i n c o m p u t a t i o n a l f l u i d d y n a m i c s . They a p p e a r u n d e r s e v e r a l v e r s i o n s ( c f . [ 5 ] , [ 8 ] , [ 1 3 ] , [ 1 9 ] , [ 2 1 ] , [ 2 2 ] , [ 2 8 ] , [ 3 1 ] ) ) w i t h t h e ma i n o b j e c t i v e o f g e n e r a t i n g s t a b l e m a r c h i n g schemes f r e e o f t h e e x c e s s i v e d i s s i p a t i v e e f f e c t s i n t r o d u c e d by s t a n d a r d u p w i n d i n g t e c h n i q u e s . The m a i n e m p h a s i s o f ' t h e s e a l g o r i t h m s i s i n t h e c o m p u t a t i o n o f t h e c o n v e c t i v e t e r m s o f t h e e q u a t i o n s w h i c h , f o r f l o w s a t h i g h R e y n o l d s numbers , seem t o be t h e dom inan t mechan i sm o f t h e d y n a m i c s . The c o m p u t a t i o n a l c o n s e q u e n c e s p r o d u c e d by t h e c o n v e c t i v e t e r m s a r e v e r y u n p l e a s a n t . F o r i n s t a n c e , n o n s y m m e t r i c l i n e a r a p p r o x i m a t e o p e r a t o r s r e s u l t w i t h a p o o r l y r e s o l v e d s p e c t r u m , a s w e l l a s an e x c e s s i v e l y s m a l l t i m e s t e p f o r t h e a p p r o x i m a t e i n t e g r a t i o n . One o f t h e new a l g o r i t h m s d e s i g n e d f o r d e a l i n g w i t h s u c h p r o b l e m s i s t h e so c a l l e d G a l e r k i n - C h a r a c t e r i s t i c Me thod , w h i c h i s b a s e d on c o m b i n i n g t h e method o f c h a r a c t e r i s t i c s w i t h a s t a n d a r d f i n i t e e l e m e n t p r o c e d u r e ( c f . [ 3 ] , [ 1 3 ] , [ 2 7 ] , [ 3 1 ] ) . I n t h e s e p a p e r s , t h e s t a b i l i t y o f t h e method , a s w e l l a s t h e c o n v e r g e n c e and c o n s e r v a t i o n p r o p e r t i e s have been 3 p r o v e d u n d e r t h e a s s u m p t i o n t h a t a l l t h e i n n e r p r o d u c t s a r e e v a l u a t e d e x a c t l y . I n t h i s t h e s i s , we p r o p o s e and a n a l y z e an a l g o r i t h m t o i n t e g r a t e t h e t r a n s p o r t - d i f f u s i o n e q u a t i o n . S p e c i f i c a l l y , we u s e a two s t a g e i n d u c t i v e a l g o r i t h m . T h i s f o r m o f o p e r a t i n g i s t y p i c a l o f t h e methods w h i c h a p p r o x i m a t e t h e t r a n s p o r t - d i f f u s i o n e q u a t i o n a l o n g t h e c h a r a c t e r i s t i c c u r v e s o f t h e h y p e r b o l i c o p e r a t o r . The i d e a i s n o t new by any means, f o r i t ha s been u s e d f o r o v e r 20 y e a r s b y many r e s e a r c h e r s . The n o v e l t y o f o u r a l g o r i t h m i s i n t h e way we a p p r o x i m a t e t h e p u r e l y h y p e r b o l i c f i r s t s t a g e by a c o m b i n a t i o n o f G a l e r k i n - C h a r a c t e r i s t i c and P a r t i c l e methods . E x p l i c i t l y , a s s u m i n g t h a t t h e c o m p u t a t i o n a l doma in ha s b e e n c o v e r e d w i t h a r e c t a n g u l a r g r i d , t h e a l g o r i t h m a s s i g n s one p a r t i c l e t o e a c h node o f t h e g r i d a t t i m e l e v e l t n , and s e e k s i t s l o c a t i o n a l o n g t h e c o r r e s p o n d i n g c h a r a c t e r i s t i c c u r v e a t t i m e l e v e l t n - A t . Then a p a r t i c l e a p p r o x i m a t i o n i s c o n s t r u c t e d and t h i s a p p r o x i m a t i o n i s p r o j e c t e d by t h e G a l e r k i n p r o c e d u r e i n t o a f i n i t e d i m e n s i o n a l s pace . I n t h i s r e s p e c t , t h e p u r e l y h y p e r b o l i c f i r s t s t a g e g e n e r a t e d by t h e a l g o r i t h m can be v i e w e d a s a p a r t i c l e i n c e l l f i n i t e e l e m e n t method. C o n s e q u e n t l y , g i v e n a datum v : -> R m , t h e p u r e l y h y p e r b o l i c f i r s t s t a g e y i e l d s an o u t p u t w = v o X , X b e i n g a mapp ing o f Q g e n e r a t e d by t h e c h a r a c t e r i s t i c c u r v e s o v e r t h e t i m e s t e p A t . On t h e o t h e r hand, t h e s e c o n d s t a g e i n v o l v e s a t i m e p r o g r e s s i o n o f t h e a l g o r i t h m v i a t h e d i f f u s i o n mechan i sm, s t a r t i n g w i t h t h e o u t p u t w f r o m t h e p r e v i o u s s t a g e . The o u t p u t 4 v o f t h i s s e c o n d s t a g e i s t h e i n p u t f o r t h e f i r s t s t a g e a t t h e n e x t t i m e l e v e l . The s e c o n d s t a g e may be t h o u g h t o f a s t h e t i m e p r o j e c t i o n o f t h e L a g r a n g i a n r e p r e s e n t a t i o n o f t h e f l o w i n t o t h e C a r t e s i a n s p a c e - t i m e E u l e r i a n r e p r e s e n t a t i o n , c o o r d i n a t e d w i t h any s t a n d a r d t i m e d i s c r e t i z a t i o n scheme f o r p a r a b o l i c e q u a t i o n s . W h i l e most o f t h e a u t h o r s emp loy t h e b a c k w a r d E u l e r scheme, we, i n c o n t r a s t , a r e i n t e r e s t e d i n u s i n g t h e C r a n k - N i c h o l s o n scheme f o r two r e a s o n s . F i r s t , i n t h e c o n t e x t o f o u r a l g o r i t h m f o r t h e h y p e r b o l i c s t a g e , t h e i m p l e m e n t a t i o n o f t h e C r a n k - N i c h o l s o n scheme i s n e i t h e r d i f f i c u l t n o r e x p e n s i v e . Second , we w i s h t o g e t , u n d e r c e r t a i n c o n d i t i o n s t o be d i s c u s s e d i n C h a p t e r IV, a h i g h e r a c c u r a c y i n t i m e . The s p a t i a l d i s c r e t i z a t i o n o f t h e s e t o f e q u a t i o n s t o be s o l v e d i s made by C ° - f i n i t e e l e m e n t s w i t h b i l i n e a r b a s i s f u n c t i o n s on a g r i d composed o f r e c t a n g l e s . The m a i n a d v a n t a g e o f u s i n g t h i s k i n d o f f i n i t e e l e m e n t s i s a p p a r e n t a t t h e p u r e l y h y p e r b o l i c f i r s t s t a g e as t h e p a r t i c l e s o l u t i o n i s c o n s t r u c t e d . We show t h a t t h e p r o j e c t e d p a r t i c l e s o l u t i o n a t t h e u p s t r e a m l o c a t i o n o f t h e p a r t i c l e s i s e q u i v a l e n t t o c u b i c s p l i n e i n t e r p o l a t i o n . W i t h o t h e r c l a s s e s o f C ° - f i n i t e e l e m e n t s , a s f o r e xamp le , l i n e a r on t r i a n g l e s , q u a d r a t i c p o l y n o m i a l s on r e c t a n g l e s , o r C 1 - f i n i t e e l e m e n t s , i t i s n o t c l e a r how one c a n g e t a s i m p l e r e l a t i o n s h i p b e t w e e n t h e i n t e g r a l s o f t h e i n n e r p r o d u c t s and s i m p l e l i n e a r i n t e r p o l a t i o n a l g o r i t h m s . The e v a l u a t i o n o f s u c h i n t e g r a l s i n t h e c l a s s i c a l G a l e r k i n - C h a r a c t e r i s t i c method p r o p o s e d i n [13] 5 and [31] i s a v e r y c o s t l y p r o c e s s ( c f . [ 1 9 ] ) , w h i c h may e v e n t u a l l y l e a d t o an u n s t a b l e a l g o r i t h m a s M o r t o n e t a l [27] have r e c e n t l y d e m o n s t r a t e d . N e v e r t h e l e s s , we c a n a p p l y o u r p a r t i c l e a p p r o x i m a t i o n t o e v a l u a t e t h e i n n e r p r o d u c t i n t e g r a l s t h o u g h no r e l a t i o n s h i p b e t w e e n them and l i n e a r i n t e r p o l a t i o n a l g o r i t h m s e x i s t s , and s t i l l g e t a s t a b l e a l g o r i t h m , a s we show i n C h a p t e r I I I . I n t h i s c o n n e c t i o n , we m e n t i o n t h a t i n [27] i d e a s r e l a t e d w i t h p a r t i c l e m e t h o d o l o g y a r e u s e d t o compute t h e i n n e r p r o d u c t s o f t h e c l a s s i c a l G a l e r k i n - C h a r a c t e r i s t i c method. We s h o u l d a l s o m e n t i o n t h a t c u b i c s p l i n e i n t e r p o l a t i o n o f t h e v a r i a b l e s a t t h e f o o t o f t h e c h a r a c t e r i s t i c c u r v e s ha s been p r e v i o u s l y u t i l i z e d i n t h e m e t e o r o l o g i c a l l i t e r a t u r e ( c f . [ 3 4 ] ) , b u t b a s e d on h e u r i s t i c a r gumen t s . I n t h i s t h e s i s we s u b s t a n t i a t e t h e o r e t i c a l l y t h e u s e o f c u b i c s p l i n e i n t e r p o l a t i o n made i n [ 3 4 ] . The t h e s i s i s o r g a n i z e d a s f o l l o w s . I n C h a p t e r I I we b r i e f l y i n t r o d u c e t h e c o n t i n u u m QG e q u a t i o n s f o r a t w o - l a y e r b a r o c l i n i c o c e a n and c o n s t r u c t t h e i r n u m e r i c a l f o r m u l a t i o n . C h a p t e r I I I i s d e v o t e d t o t h e f o r m u l a t i o n and a n a l y s i s o f t h e f i r s t h y p e r b o l i c s t a g e o f t h e a l g o r i t h m . Some n u m e r i c a l e x p e r i m e n t s a r e p r e s e n t e d i n o r d e r t o i l l u s t r a t e t h e p e r f o r m a n c e o f t h e a l g o r i t h m w i t h l i n e a r h y p e r b o l i c e q u a t i o n s . I n C h a p t e r IV we g i v e t h e e r r o r e s t i m a t e s w i t h r e s p e c t t o t h e 2 L - no rm o f t h e s e c o n d s t a g e o f t h e a l g o r i t h m i n t h e i n t e g r a t i o n o f t h e p o t e n t i a l v o r t i c i t y e q u a t i o n s . A l s o , 2 L - no rm e r r o r e s t i m a t e s o f t h e a p p r o x i m a t e s t r e a m f u n c t i o n s and v e l o c i t i e s a r e s t u d i e d . 6 E a c h c h a p t e r i s d i v i d e d i n t o s e c t i o n s a n d , when needed , t h e s e c t i o n s a r e f u r t h e r d i v i d e d i n t o s u b s e c t i o n s . The s u b j e c t o f e a c h c h a p t e r i s i n t r o d u c e d by a s h o r t d e s c r i p t i o n , f r e e o f t e c h n i c a l d e t a i l s , o f t h e m a i n a s p e c t s and r e s u l t s o f t h e c h a p t e r . The c r i t e r i o n a d o p t e d t o number t h e e q u a t i o n s i n e a c h c h a p t e r c o n s i s t s o f w r i t i n g f i r s t t h e number o f t h e s e c t i o n f o l l o w e d b y t h e number o f t h e e q u a t i o n i n s u c h s e c t i o n . Thus , ( 3 . 1 2 ) d e n o t e s e q u a t i o n number 12 o f s e c t i o n 3. I f we have t o r e f e r t o a n e q u a t i o n o f a d i f f e r e n t c h a p t e r we do i t by w r i t i n g f i r s t t h e number o f t h e c h a p t e r f o l l o w e d by t h e number o f t h e e q u a t i o n . The t h e s i s i s c l o s e d w i t h t h e C o n c l u s i o n s w h i c h a r e p r e s e n t e d i n C h a p t e r V. Two t e c h n i c a l A p p e n d i c e s have a l s o been i n c l u d e d f o r c o m p l e t e n e s s . The f i r s t one i s a t t a c h e d t o C h a p t e r IV and d e a l s w i t h t h e t e c h n i c a l i t i e s o f t h e p r o o f o f Lemma 2 . 3 o f t h i s c h a p t e r . I n t h e s e c o n d a p p e n d i x , p l a c e d a t t h e end o f t h e t h e s i s , we have c o l l e c t e d t h e i n e q u a l i t i e s w h i c h a r e emp loyed t h r o u g h o u t t h e t h e s i s and t h e L a x - M i l g r a m theo rem. 7 CHAPTER II THE GOVERNING EQUATIONS AND THEIR NUMERICAL DISCRETIZATION T h i s d e s c r i p t i v e c h a p t e r i n t r o d u c e s t h e c o n t i n u u m e q u a t i o n s and t h e p r o p o s e d n u m e r i c a l a l g o r i t h m t o i n t e g r a t e them. I n a d d i t i o n , we a l s o p r e s e n t t h e b a s i c n o t a t i o n u s e d t h r o u g h o u t t h e t h e s i s , and t h e s p a c e s where t h e s o l u t i o n s a r e s o u g h t . 1. G o v e r n i n g E q u a t i o n s We w i s h t o a p p l y t h e p r o p o s e d n u m e r i c a l a l g o r i t h m t o s t u d y t h e c i r c u l a t i o n o f a m i d - l a t i t u d e b a r o c l i n i c o c e a n . I n o r d e r t o f a c i l i t a t e t h e m a t h e m a t i c a l a n a l y s i s we c h o o s e t h e s i m p l e s t e x a m p l e o f s u c h an o cean . Thus, o u r o c e a n mode l c o n s i s t s o f two l a y e r s o f d e n s i t i e s and p 2 > r e s p e c t i v e l y c o n f i n e d t o a m i d - l a t i t u d e , 6 - p l a n e , r e c t a n g u l a r , f l a t b o t t o m doma in D A r o t a t i n g w i t h a n g u l a r v e l o c i t y Cj and w i t h s o l i d b o u n d a r i e s . F i g u r e 1 i l l u s t r a t e s t h e a r r a n g e m e n t o f o u r C a r t e s i a n c o o r d i n a t e s y s t e m , as w e l l a s t h e g e o m e t r i c a l p a r a m e t e r s o f t h e m o d e l . The s t r e a m f u n c t i o n s and r e l a t i v e v o r t i c i t i e s o f e a c h l a y e r a r e d e n o t e d by and u , r e s p e c t i v e l y . The s u b i n d i c e s 1 and 2 s t a n d f o r t h e m a g n i t u d e s r e l a t e d t o t h e u p p e r and l o w e r l a y e r s , wherea s 1- s t a n d s f o r t h e m a g n i t u d e s 8 F i g . 1 The Two Layer Model 9 e v a l u a t e d a t t h e i n t e r f a c e . The f l o w i s d r i v e n by a v a r i a b l e w i n d s t r e s s T(t,x) a c t i n g on t h e f r e e s u r f a c e . The c o u p l i n g b e t w e e n l a y e r s i s t h r o u g h t h e d i s p l a c e m e n t o f t h e i n t e r f a c e . We u s e t h e QG e q u a t i o n s i n t h e v e r s i o n p o t e n t i a l v o r t i c i t y - s t r e a m f u n c t i o n w h i c h c a n be f o u n d i n [ 20 ] . The z - d e p e n d e n c e o f t h e v a r i a b l e s i s m o d e l l e d by e m p l o y i n g c e n t r a l f i n i t e d i f f e r e n c e s i n t he v e r t i c a l d i r e c t i o n . Thus, d e f i n i n g t h e p o t e n t i a l v o r t i c i t y a t e a c h l a y e r by we have Dqi 2 -fif- - AHV'q. + S.2cq. - 5.^ = 0, i n <?r , i =1,2, ( 1 . 2 . 1 ) qAx,t) = on t h e b o u n d a r y T, - ( 1 . 2 . 2 ) qXx.O) = f , ( 1 . 2 . 3 ) where QT = Qx[0, T], D 8 Dt dt f = 2\Q\sin9Q = f + (3y i s t h e C o r i o l i s p a r a m e t e r , w h i c h i s a p p r o x i m a t e d by p r o j e c t i n g t h e e a r t h ' s s p h e r i c a l g e o m e t r y o n t o i t s t a n g e n t p l a n e a t t h e r e f e r e n c e l a t i t u d e 6 = 6^, and /3 = df/dy and a r e p a r a m e t e r s whose m a g n i t u d e s depend on | f i | , Qg 10 and t h e r a d i u s o f t h e e a r t h ( c f . [ 3 0 ] ) . g'= ( )g, h e r e g d e n o t e s t h e a c c e l e r a t i o n due t o g r a v i t y , p2 _ . c u r l T t h e f o r c i n g F = H l ' A„ i s t h e l a t e r a l eddy v i s c o s i t y c o e f f i c i e n t w h i c h i s assumed ti c o n s t a n t o v e r t h e w h o l e doma in Q, c i s a c o n s t a n t l i n e a r b o t t o m f r i c t i o n c o e f f i c i e n t , 8. . i s t h e K r o n e c k e r s ymbo l . F o r a two l a y e r o c e a n , l i n e a r c o m b i n a t i o n s o f and \jt l e a d t o a s e t o f e l l i p t i c e q u a t i o n s w h i c h a r e now f o r m u l a t e d . D e f i n e * ( x , t ) and $ ( x , t ) a s f o l l o w s . *(x, t) = $ (x,t) - ifi2(x,t) , ( 1 . 3 . 1 ) H l * l + H2*2 *(x,t) = ' „ , ( 1 . 3 . 2 ) whe re V(x, t) and $(x,t) a r e t h e s o c a l l e d b a r o c l i n i c and b a r o t r o p i c modes, r e s p e c t i v e l y . A c c o r d i n g t o M c W i l l i a m s [ 2 6 ] , we may f u r t h e r decompose t h e b a r o c l i n i c mode * i n t h e f o l l o w i n g f a s h i o n ¥ ( x , t ) = * (x, t) + C(t)V (x). ( 1 . 4 ) a s By c o n s i d e r a t i o n s o f mass c o n s e r v a t i o n [ 2 6 ] , ^(x,t) s a t i s f i e s t h e g l o b a l c o n d i t i o n - f Kx, t)dQ = 0 , V t € [0,T] . ( 1 . 5 ) J f i 11 I n v i e w o f ( 1 . 4 ) and ( 1 . 5 ) t h e t i m e dependen t f u n c t i o n C ( t ) i s d e t e r m i n e d a t any i n s t a n t t by * (x, t)da C(t) = - — , V t e [0,7]. ( 1 . 6 ) f * (x)dn By v i r t u e o f ( 1 . 1 ) , ( 1 . 3 ) and ( 1 . 4 ) we o b t a i n C V 2 - A 2 ; * (x) = 0 i n Q_ , ( 1 . 7 . 1 ) s 1 * |_ = 1 , V t . ( 1 . 7 . 2 ) s r C V 2 - A 2 ; * a C x , t ) = q2 - q2 i n QT , ( 1 . 8 . 1 ) * |_ = 0 , Vt , ( 1 . 8 . 2 ) a r whe re f2r// + ff ; = V * 3 * 2 • ( i - 8 - 3 ) ffq + ff g vVx,0 = V A ii - / i n Q_ , ( 1 . 9 . 1 ) #| = 0 V t . ( 1 . 9 . 2 ) The s y s t e m s ( 1 . 7 ) - ( 1 . 9 ) a r e c l o s e d w i t h t h e i n i t i a l c o n d i t i o n s * (x,0) = $ ( x , 0 ; = 0 i n Q , ( 1 . 1 0 . 1 ) a o r e q u i v a l e n t l y 12 ip^x.O) = \p2(x,0) = 0 i n ( 1 . 1 0 . 2 ) F i n a l l y , t h e v e l o c i t y v e c t o r u^= (u^,v.) a t e a c h l a y e r i s d e t e r m i n e d a t any i n s t a n t t by t h e r e l a t i o n u i = curty , i = 1,2 . ( 1 . 11 ) 1.2. On B o u n d a r y C o n d i t i o n s . The b o u n d a r y c o n d i t i o n s ( 1 . 7 . 2 ) and ( 1 . 8 . 2 ) s t a t e t h a t t h e s t r e a m f u n c t i o n s and ip^ a r e , a t any i n s t a n t t , c o n s t a n t a l o n g t h e b o u n d a r y T; t h a t i s <p.\r = C (t) , i = 1,2. ( 1 . 12) On t h e o t h e r hand , t h e p h y s i c a l mean ing o f t h e b o u n d a r y c o n d i t i o n ( 1 . 9 . 2 ) f o r t h e b a r o t r o p i c mode i s t h a t t h e v e r t i c a l l y i n t e g r a t e d n o r m a l t r a n s p o r t i s z e r o a c r o s s a s o l i d b o u n d a r y . As i s w e l l known i n t h e s t r e a m f u n c t i o n - r e l a t i v e v o r t i c i t y f o r m u l a t i o n o f t h e N-S e q u a t i o n s , t h e v a l u e o f t h e r e l a t i v e v o r t i c i t y i s unknown "on t h e s o l i d b o u n d a r i e s o f t h e d o m a i n ; howeve r , one c an e a s i l y deduce , f r o m t h e v a l u e s o f t h e v e l o c i t y on t h e b o u n d a r i e s , an a d d i t i o n a l b o u n d a r y c o n d i t i o n f o r t h e s t r e a m f u n c t i o n . S p e c i f i c a l l y , a s s u m i n g t h a t u . l r = g.(x,t) , (1 . 13) and w i t h n and t d e n o t i n g t he n o r m a l and t a n g e n t v e c t o r s on t h e b o u n d a r y , r e s p e c t i v e l y t h e f o l l o w i n g r e l a t i o n s h o l d 13 30. — |_ = -g.ot , (1.14.1) Sn T "i dip — |_ = g.on = 0 . (1. 14.2) at T 6 i I f * 0, t h e n t h e r e i s no s h e a r s t r e s s on t h e s o l i d w a l l , o r i n o t h e r t e r m s , t h e r e l a t i v e v o r t i c i t y i s i d e n t i c a l l y z e r o on t h e w a l l . I n c o n t r a s t , i f = 0, t h e n t h e r e e x i s t s s h e a r s t r e s s on t h e w a l l , and so r e l a t i v e v o r t i c i t y i s g e n e r a t e d . The c o n d i t i o n * 0 i s known i n f l u i d d y n a m i c s l i t e r a t u r e as f r e e - s l i p b o u n d a r y c o n d i t i o n w h i l e t h e c a s e g^ = 0 i s d e n o t e d by n o - s l i p b o u n d a r y c o n d i t i o n . Now, a s s u m i n g t h a t t h e v a l u e o f t h e r e l a t i v e v o r t i c i t y on t he bounda r y i s A , i = 1,2 t he b o u n d a r y c o n d i t i o n s f o r t h e p o t e n t i a l v o r t i c i t y a r e f0 q.. = X + f - -j— C(t) , Vt e 10,TL ( 1 . 15 ) ul 1 g n . 2. N o t a t i o n We now i n t r o d u c e some b a s i c n o t a t i o n w h i c h i s u s ed 2 t h r o u g h o u t t h e t h e s i s . L e t fi be an open s u b s e t o f R w i t h 2 L i p s c h i t z b o u n d a r y T, R i s t h e two d i m e n s i o n a l E u c l i d e a n s p a c e . L e t C^Cfi; be t he s e t o f r e a l v a l u e d f u n c t i o n s on fi w h i c h h a v e c o n t i n u o u s p a r t i a l d e r i v a t i v e s o f o r d e r a t l e a s t k, w h i c h a r e bounded on fi. On C^(Q) we i n t r o d u c e t h e norm 14 \v\. = sup \Dav\ . (2.1) k ' M ' ° xeQ \a\<k B o l d f a c e s y m b o l s d e n o t e e i t h e r m a t r i c e s , v e c t o r s o r s e q u e n c e s ; t h e mean ing w i l l be c l e a r f r o m t h e c o n t e x t , a i s a m u l t i - i n d e x n o t a t i o n . I f IN d e n o t e s t h e s e t o f n o n n e g a t i v e i n t e g e r s a := ( a , , a „ , . . . . a ) , a . e IN, 1 = 1,2 . . ,n 1 2 n l We have t h e f o l l o w i n g d e f i n i t i o n s |a| = a . + a + . . . + a , 1 2 n a £ (3 i f f a i - Pi > Vi = 1,2, . . ,n , (a.+ | 3 J = a i + 0 , Vi = 1,2, . . ,n , ( a i ~ = a i ~ 1 8 i ' V i = 3 ' 2 ' ' a/ = (a'.)(a>) ...(a !) , 1 2 n Ca - &)i = max(a. - p. ; 0) , Vi = 1,2, . . . , n , a , a i , , a.2.. , a n . x = ( x , ; ( x _ ; (x ) , 1 2 n D « d a i a an 3 x , 3 x 2 n We d e n o t e by LP(Q) , 1 s p < co , t h e s p a c e o f a l l e q u i v a l e n c e c l a s s e s o f r e a l - v a l u e d Lebe sgue m e a s u r a b l e f u n c t i o n s . The norm o n lP(Q), p < co i s d e f i n e d by iiuii p n = ( { l"l p<*n J P , (2.2) 15 w i t h Hull = ess sup\u\, Vx € Q . When p = 2 we d r o p t h e s u b i n d i c e s p and Q i f t h e r e i s no c o n f u s i o n . F o r e a c h i n t e g e r m 2: 0 and r e a l p, 1 < p < 00 , we d e f i n e t h e S o b o l e v s p a c e s (Q), a s Wm'P(Q) = j v e L P I TQ ; : D a v € LP(Q), l a l s m . l s p s . o J . ( 2 . 3 ) The s p a c e Mm,P'(Q) i s a Banach space w i t h t h e norm M l n = ( £ IID°Vll _ ) 1 / p , 1 s p < 00 , ( 2 . 4 . 1 ) m ' P ' Q \a\sm P ' Q and t h e semino rm | V L » O = < I l ^ l n O ' ( 2 . 4 . 2 ) When p =00 l lvll _ = max (ess sup\D%\), Vx e Q . ( 2 . 4 . 3 ) I a I sin A l s o , t h e s p a c e W^^Cfl. ) i s s e p a r a b l e f o r 1 < p < 00 and r e p r e s e n t s t h e c o m p l e t i o n o f t h e s pace (fen) i n t h e norm II II m,p,Q L e t X be a B a n a c h s pace w i t h norm II II. I f v(x,t) r e p r e s e n t s a f u n c t i o n d e f i n e d on Q ,^ we s e t 16 LP(0,T;X) = | v: | \\vl\Pdt < oo | , ( 2 . 5 . 1 ) Iv l l = (T \\v\\Pdt LP(0,T;X) VO 1/p ( 2 . 5 . 2 ) T 1^(0, T;X) = | v: J I ID*vl l ' d i < oo, a ^ m ( 2 . 5 . 3 ) l l v l l {/"(O.TjX) m _ -\ 2/2 a=0 ( 2 . 5 . 4 ) a a a h e r e D.denotes ' 3 t a ess supWD v\\ < oo f 0 , 17 , a ^ m j - , ( 2 . 6 . 1 ) Hull = max ess supWD v\\ , m 0 ^'"[O.TjX] Ottm [0,1] ( 2 . 6 . 2 ) I n p a r t i c u l a r LP(0,T;X) = H°(0,T;X), Lm(0,T;X) = W°' °°(0, T;X). Somet imes we u s e t h e s h o r t h a n d n o t a t i o n , i f c o n f u s i o n doe s n o t a r i s e , LP(0,T;X) s LP(X), iT(0,T;X) = ^(X). F o r p = 2, I T ^ ' ^ n ; = [/"(n), where lF(Q) i s t h e H i l b e r t s p a c e o f o r d e r m. The s u b s p a c e H^(Q) of lf(a) i s d e f i n e d by 17 = 0, 0 £ I<x| ^ m-2 ( 2 . 7 ) 3. The Weak S o l u t i o n F o r m u l a t i o n We f o r m u l a t e i n t h i s s e c t i o n t h e weak s o l u t i o n o f t h e p r o b l e m s ( 1 . 2 ) , ( 1 . 7 ) - ( 1 . 1 0 ) and t h e v e l o c i t i e s u^. To b e g i n w i t h , we d e f i n e t h e c l a s s e s o f f u n c t i o n s w h i c h a r e u s e d i n t h e f o r m u l a t i o n o f t h e weak f o r m o f t h e p r o b l e m s . Our b a s i c s p a c e i s H^(£2); howeve r , i t i s c o n v e n i e n t t o i n t r o d u c e some c l o s e d s u b s e t s o f H*(n) w h i c h a r e s p e c i f i c a l l y r e l a t e d t o t he s o l u t i o n s we a r e s e e k i n g . Such s u b s e t s a r e : Thus, t h e s t r e a m f u n c t i o n s i/». € S , whe rea s t h e p o t e n t i a l and l c . r e l a t i v e v o r t i c i t i e s b e l o n g t o S^. The s t e a d y component * (x) o f t h e b a r o c l i n i c mode V(x,t) i s i n S w i t h C = 1. On t h e c o t h e r hand , t h e t i m e dependen t b a r o c l i n i c component * (x, t) 3. and t h e b a r o t r o p i c mode $(x,t) a r e i n HQ(Q) f o r any t € [0,T]. I n o r d e r t o m o t i v a t e t h e a l g o r i t h m t o s o l v e e q u a t i o n ( 1 . 2 ) , we i n t r o d u c e h e r e some i d e a s w h i c h a r e f u r t h e r d e v e l o p e d i n Chap. I I I . R e c a l l t he c h a r a c t e r i s t i c c u r v e s o f t h e l i n e a r - a d v e c t i o n e q u a t i o n (3. 1.1) (3. 1.2) 18 at q(x,0) = qQ(x), ( 3 . 2 . 1 ) s a t i s f y t h e s y s t e m dX(x,S;T) dx Xix, s;s) = x , = U(X(X,S;T),X) , ( 3 . 2 . 2 ) where u i s t h e v e l o c i t y v e c t o r w i t h d i v u = 0. The t r a n s f o r m a t i o n x -» X(x,s;t) d e f i n e s , u n d e r c e r t a i n c o n d i t i o n s t o be s p e c i f i e d be l ow, a q u a s i - i s o m e t r i c t r a n s f o r m a t i o n o f fi i n t o i t s e l f w i t h J a c o b i a n d e t e r m i n a n t e q u a l t o 1 a l m o s t e v e r y w h e r e . To s o l v e t h e p o t e n t i a l v o r t i c i t y e q u a t i o n ( 1 . 2 ) we s e t ( c f . [ 3 1 ] ) ^ = ^(X(x,t:T),r)\T=t , ( 3 . 3 ) whe re d e n o t e s t h e t o t a l d e r i v a t i v e . The r i g h t hand s i d e o f ( 3 . 3 ) r e p r e s e n t s t h e t i m e e v o l u t i o n o f q a l o n g - t h e c h a r a c t e r i s t i c c u r v e s o f t h e f l o w . T h i s c o n s i d e r a t i o n s u g g e s t s t h e f o l l o w i n g t w o - s t a g e i n d u c t i v e a l g o r i t h m . S u p p o s e t h a t a p a r t i t i o n o f [0,T] 19 ? •• = \ 0 = tQ < tt < < tm = T 1 i s s p e c i f i e d . We s h a l l d e f i n e t h e s e m i - d i s c r e t i z a t i o n i n d u c e d by T upon ( 1 . 2 ) i n t h e f o l l o w i n g two s t e p s : 1) G i v e n q\: Q -» R 2 , i = 1,2, 0 s n £ m, we assume t h a t t h e r e i s a ' r e f l e c t i o n ' map E : HS(Q) n La(Q) > HS(Q' ) r\ La(n' ) , s u c h t h a t C^(Q) f u n c t i o n s a r e mapped o n t o C^(Q') f u n c t i o n s , and £ i s l i n e a r and c o n t i n u o u s , and Q' i s d e f i n e d by 1 n+1 n ' * 2 Thus , any f u n c t i o n q : Q' > R c a n be e x p r e s s e d a s q . =Eq. 3 .4 C o n s e q u e n t l y , t h e o u t p u t f r o m s t e p one i s t h e n g i v e n by 5? = S ^ V * ' W V ' V . ( 3 . 5 ) The method t o a p p r o x i m a t e q1} i s one o f t h e c o n t r i b u t i o n s o f t h i s t h e s i s , and i t w i l l be d e s c r i b e d i n d e t a i l i n t h e n e x t c h a p t e r . F o r now, i t i s s u f f i c i e n t t o i n d i c a t e t h a t t h e f u n c t i o n q^.(X,.(x,t , ; t ), t ), where t h e s u b i n d e x h d e n o t e s ^hi hi n+1 n n a p p r o x i m a t e v a l u e s , does n o t b e l o n g t o a f i n i t e d i m e n s i o n a l s p a c e where t h e s o l u t i o n s a r e s o u g h t ; however , t h e 20 f u n c t i o n q\ . must b e l o n g t o H,. I t i s t h e f a s h i o n i n w h i c h we make t h e f u n c t i o n a1}, be i n t h a t makes o u r method d i f f e r e n t hi h f r o m o t h e r G a l e r k i n - C h a r a c t e r i s t i c methods . A wo rd on n o t a t i o n i s now i n o r d e r : We s e t g1? = g.CX1?,) t o r e m i n d u s t h a t ^ 1 ^ 1 i g1? i s computed f r o m t h e v a l u e s g1? t a k e s a t t h e p o i n t s X.(x, t t ). l n+1 n 2) G i v e n t h e o u t p u t f r o m ( 3 . 5 ) , s o l v e t h e f o l l o w i n g : F i n d qn+1, s u c h t h a t qn+1- g " + J e HICQ) and Vr e T l ^i bi 0 n qn+1(x) - g.CX") Au ( - i A t 1 1 , e ) + - 4 - f v<rgf 1 + Ztf)), ve ; di2~^~( q i + i + " V * ^ ' 9 / ) = 5 i / f n + ' Z / 2 ' Q)> V G e H o C n ^ ( 3 . 6 . 1 ) g " t 2 = A n + f - - r ^ - c " i n T , i = 1,2 ( 3 . 6 . 2 ) where (u ,v ) = X_uvdQ , and X7? = X.(x,t ,; t ). Q i i n+1 n To u p d a t e t h e b o u n d a r y v a l u e s A." o f t h e r e l a t i v e v o r t i c i t i e s t/l we a d o p t t h e method o f G l o w i n s k i and c o - w o r k e r s ( c f . [ 1 1 ] , [ 1 6 ] ) . L e t M be a c o m p l e m e n t a r y s ub space o f H^(n); i . e . , ff2rn; = HQ(Q) ® M . ( 3 . 7 . n F i n d ^ " + ^ e s u c h t h a t Vu e W 21 lfjn+1 - a " * 1 , n) = ( V0" + 1, VM ; - J u ds . (3.7.2) 1 1 Next, we calculate the barotropic and baroclinic modes, the stream functions and the velocities. i) Find 9 (x) e S ,c = 1, such that V<£ e HI(Q) s c 0 CV*s, 70 ; + A 2(* . <f>) = 0. (3.8.1) ii) Find <bn+1(x), $ n + 1 ( x ) e n\(Q) such that V0 e fl^ffi; a C/ c /n.n+l „. . . 2,Tn+,l , . , n+1 n+1 , ,_ „ (7* , 70; + X (9 , <(>) = -(q - q_ ; , (3.8.2) 3 3. 1 c. (V9n+1, V<f>) = -(bn+1,4>) , (3.8.3) „ n+1 ., n+1 ,n+l H l q l + H2q2 , where b = - f H 2 + H 2 i i i ) f *n+1(x)dn ^ = - - ^ 1 . (3.9) * (x)dQ From (1.4), (3.8.1), (3.8.2) and (3.9) we are able to determine the baroclinic mode ¥ n +*(x) by V n + 1 ( x ) = <Hn+1(x) + c""1"1* (x) . (3.10) a s Once the baroclinic and barotropic modes have been calculated, at instant t ,, the stream functions and are n+1 1 2 22 o b t a i n e d i n v i e w o f ( 1 . 3 . 1 ) and ( 1 . 3 . 2 ) by s o l v i n g .n+1 ,n+l * 2 " 0 2 = * n+1 n+1 n+1 n+1 ( 3 . 1 1 ) H+H 2 F i n a l l y , u n+1 Cx) i s o b t a i n e d f r o m , n+1 n+1, i n+1 ( 3 . 1 2 ) Remark. E q u a t i o n ( 3 . 8 . 1 ) i s s o l v e d once and f o r a l l a t t h e b e g i n n i n g o f t h e c o m p u t a t i o n s . Our n e x t o b j e c t i v e i s t o show t h a t t h e s y s t e m s ( 3 . 6 ) and ( 3 . 8 ) a d m i t a u n i q e s o l u t i o n i n H*(Q), f o r any t e T. We s h a l l assume t h a t t h e bounda r y v a l u e s and t h e f o r c i n g f u n c t i o n a r e s u f f i c i e n t l y smooth. A l s o , we r e s t r i c t o u r s e l v e s t o p r o v e t h e e x i s t e n c e and u n i q u e n e s s o f t h e s o l u t i o n s o f ( 3 . 6 ) f o r t h e uppe r l a y e r ( i = 1 ) , s i n c e t he p r o o f f o r t h e l o w e r l a y e r ( i = 2) i s e s s e n t i a l l y t h e same. The m a i n t o o l u s e d i n t h e p r o o f s i s t h e L a x - M i l g r a m t h e o r e m ( c f . [2] and A p p e n d i x ) . Towards t h i s end l e t u s f i r s t i n t r o d u c e t h e d u a l s p a c e s V m'P(Q) o f t h e s p a c e s W^'P(Q), and t h e t r a c e s p a c e s (T), where T i s t h e b o u n d a r y o f fi. —m D' F o r 1 £ p < co, t h e d u a l s pace W (Q) i s t h e s p a c e o f c o n t i n u o u s l i n e a r f u n c t i o n a l s d e f i n e d on w"l'P(Q) w i t h t h e norm 0 23 IfII , 0 = s up [ < f ' " > l , u * 0 , ( 3 . 1 3 . 1 : -m, p ,ii „ „ ,m,p Hull _ uelt^j in, p, Q -1 -1 r where p ' s a t i s f i e s p + p ' = i , and <f, u> = f u d f i . I t i s u s u a l t o d e n o t e <f,u> a s d u a l i t y p a i r i n g , where f b e l o n g s t o t h e d u a l s pa ce o f u. The t r a c e s p a c e s a r e d e f i n e d by if1 J 1 / 2 ( D = j <7fa e L2(Q) ; 3 u € ^(Q) y y x = g f o on V J , gj where y . = — i s t he t r a c e o p e r a t o r o f j - t h o r d e r w h i c h c an J dnJ be e x t e n d e d t o c o n t i n u o u s l i n e a r o p e r a t o r s mapp ing Hm( Q) o n t o (D. I n t h e f o l l o w i n g we u se t h e s p a c e H (D w i t h norm d e f i n e d by '•7J 1 ,,o o r = i n f l l u l 1 i n • ( 3 . 1 3 . 2 ) b m-1/2, 2, T uelf1(Q) m>2>& Remark. F o r f u r t h e r p r o p e r t i e s on t h e t r a c e o p e r a t o r s and t h e s p a c e s H"1 ^ ^^2(T) see A p p e n d i x o f C h a p t e r IV. P r o p o s i t i o n 3 . 1 . For q1^.1 € H 1 / 2 ( D and F X I + 1 / Z € H~ltl), problem (3.6) has a unique solution in H*(n). 1/2 1 Proof. S i n c e H (D i s t h e r ange o f i n H (Q), t h e n t h e r e X 1. e x i s t s a q^ e H (Q) s u c h t h a t = c j ^ on T, f o r i = 1, 2. N e x t , c o n s i d e r t h e b i l i n e a r f o r m 24 Ku.v) = (u,v) + v(Vu.Vv), u, v e HQ(&) > LtAH where v = I t i s w e l l known ( c f . [ 2 9 ] ) t h a t I(u,v) i s c o n t i n u o u s and HQ-c o e r c i v e ; t h a t i s , t h e r e e x i s t c o n s t a n t s , > 0 s u c h t h a t f o r u, v i n Hg(Q) \I(u,v)\ < C Hull Hull H0 H0 \I(u,u)\ £ C H u l l 2 . A l s o , I(u,v) i s c o n t i n u o u s and H*- c o e r c i v e i f u,v e H*(Q). Now, t h e p r o b l e m we have t o examine f o r i = 1 (we d r o p t h e s u b i n d e x i ) i s : F i n d qn+1(x) e H1 (SI) s u c h t h a t qnfl(x) - qQ(x) e H^(Q) and I(qn+1- qQ, B) = (qCx"), B) - vCVqCx"), V8) -- Kq0, e) + (Fn+1/2, e) . ve e H^Q) . (3.14) I n o r d e r t o a p p l y , t h e L a x - M i l g r a m theo rem i t r e m a i n s t o show t h a t t h e r i g h t hand s i d e o f (3.14) d e f i n e s a c o n t i n u o u s l i n e a r f u n c t i o n a l ; i . e . , t h e mapp ing 8 > (q(x"), 8) - v((Vq(Xn), VO) - I(qQ, 8) + (FT1+1/2\ Q) b e l o n g s t o H *(&). We f i r s t n o t e t h a t Kqg> 0) i s c o n t i n u o u s , so (Fn+1/2, 8) - I(qQ, B) e H~1(Q). N e x t , . by t h e S c h w a r z i n e q u a l i t y 25 vCVqOf"), VQ) £ C\\Vq(Xn)\\ IIV6II , t where C i s a c o n s t a n t and X i s i n , w h i c h i s t h e image o f Q t h e q u a s i - i s o m e t r i c homeomorphism x —> Xn. By v i r t u e o f Theorem 1 .1 .7 o f [25] and t h e f a c t \\q(Xn)\\ = \\qn\\ , s i n c e t h e J a c o b i a n d e t e r m i n a n t o f t h e q u a s i - i s o m e t r i c homeomorphism i s e q u a l t o 1 a l m o s t e v e r y w h e r e , we have t h a t HVqll i s e q u i v a l e n t t o I IVq n l l . Hence (qCx"), e) - vdqof1), v e ; e H~2(Q) , so t h e r i g h t hand s i d e o f ( 3 .14 ) i s i n H ^ (0.). W i t h a l l t h e a s s u m p t i o n s o f t h e L a x - M i l g r a m t heo rem s a t i s f i e d , t h e e x i s t e n c e and u n i q u e n e s s o f qn+^(x) e H^(Q) f o l l o w s • Propos i t ion 3.2. Problems (3.8.2) and (3.8.3) have a unique solution in HQ(Q), whereas the unique solution of (3.8.1) lies in ^(Q). Proof. We f i r s t p r e s e n t t h e p r o o f f o r t h e P r o b l e m ( 3 . 8 . 1 ) . As b e f o r e , l e t e H*(Q) s u c h t h a t 1 on T. N e x t , we r e c a s t t h e P r o b l e m ( 3 . 8 . 1 ) i n t h e f o l l o w i n g f a s h i o n : F i n d * - * 0 € HI(Q) s u c h t h a t V</> e H\(Q) s 0 0 0 26 w* - vQ), v^ ; + A 2 C * S - * , 0) = -cv*0, Vc5; - A 2 ^ , 0) ( 3 . 15 ) D e f i n e now t h e b i l i n e a r f o r m I'(u,v) a s I'(u,v) = CVu, Vv) + \2(u, v). I' (u, v) i s c o n t i n u o u s and H*- c o e r c i v e , so t h e r i g h t hand s i d e o f (3. 15) b e l o n g s t o H *(£l). T h e r e f o r e , t h e L a x - M i l g r a m theo rem g u a r a n t e e s t h e e x i s t e n c e and u n i q u e n e s s o f t h e s o l u t i o n V (x). N e x t , we p r o c e e d t o p r o v e t h e e x i s t e n c e and u n i q u e n e s s o f t he s o l u t i o n s o f p r o b l e m s ( 3 . 8 . 2 ) and ( 3 . 8 . 3 ) . The p r o o f i s e s s e n t i a l l y t h e same f o r b o t h p r o b l e m s . F i r s t , n o t e t h a t cr?+i a r e i n H*(Q). S e c o n d l y , t h e b i l i n e a r f o r m I"(u,v) d e f i n e d b y I"(u,v) = CVu, Vv) + uCu, v) , u, v € HQ(a), 2 where p. = A f o r P r o b l e m ( 3 . 8 . 2 ) and u = 0 f o r P r o b l e m ( 3 . 8 . 3 ) , i s c o n t i n u o u s and HQ(Q)- c o e r c i v e . S i n c e a l l t h e a s s u m p t i o n s o f t h e L a x - M i l g r a m theo rem a r e s a t i s f i e d , t h e n i t 1. f o l l o w s t h e e x i s t e n c e and u n i q u e n e s s o f t h e s o l u t i o n s ¥ and * i n HQ(Q). • From P r o p o s i t i o n 3 .2 and (3 .11 ) one d e d u c e s t h e e x i s t e n c e and u n i q u e n e s s o f t h e s t r e a m f u n c t i o n s "A "*^ a n d ip^^in t h e s u b s e t s S„ and S „ .of H1(Q), r e s p e c t i v e l y , where C , and C_ Cl °2 1 2 a r e d e f i n e d b y °1 = H1+ H2 ' C2 = ~0-nr2 a n d ^ ^ g i v e n by ( 3 . 9 ) : 27 Remark On Regularity of the Solutions. W i t h r e f e r e n c e t o t he r e g u l a r i t y o f t he s o l u t i o n s t h e L a x - M i l g r a m t h e o r e m s t a t e s t h a t II n H q i 111,2,0 : II* 1 s i " < 1,2,0 c 4 i i g » l l * n | a li,2,n ~ l l * n | I s 1,2,0. C l i b " 6 n n 2-1,2,0 ' -1,2,0 where Cy C^, and a r e c o n s t a n t s w h i c h depend on Q. 2 2 However, one c a n assume t h a t F e L (0,T;L (0)), b e c a u s e t h a l a r g e s c a l e w i n d s t r e s s i s a smooth f u n c t i o n , and t h e i n i t i a l c o n d i t i o n qAx.O) = f e Cm(0). S i n c e Q i s a bounded p o l y g o n w i t h no r e e n t r a n t c o r n e r s , i t i s r e a s o n a b l e t o e x p e c t h i g h e r r e g u l a r i t y o f t h e s o l u t i o n s . I n f a c t , i f one assumes t h a t t h e r e l a t i v e v o r t i c i t y v a n i s h e s a t t h e bounda r y , t h e n a t any i n s t a n t t we m i g h t s uppo se t h a t g . = f + c " € H (T) and by t h e r e g u l a r i t y t h e o r y f o r e l l i p t i c p r o b l e m s ( c f . [ 1 7 ] ) we have " ^ " 2 , 2 ^ ^ 7 - ! " F " + Ai"3/2,2,0 \> (3-16-n where t h e c o n s t a n t depends on 0. When t h e r e l a t i v e v o r t i c i t y doe s n o t v a n i s h a t t h e b o u n d a r y i s 3/2 n o t c l e a r w h e t h e r q^ e H (D; f r o m ou r f o r m u l a t i o n t h e most 2*2 2 / 2 we c a n s a y , i n v i e w o f ( 3 . 7 ) , i s t h a t g^e H (T); t h e r e f o r e , 28 by v i r t u e o f t h e t r a c e theorems ( c f . [ 2 9 ] ) , we have t h a t q n e H1^), V r . (3. 16 .2 ) l n N e x t , we exam ine t h e r e g u l a r i t y o f t h e modes. I f fi i s s u f f i c i e n t l y smooth , t h e n t h e r e i s a r e a l number k £ 0 s u c h t h a t f r o m t h e r e g u l a r i t y t h e o r y o f e l l i p t i c e q u a t i o n s ^ s ( x ) e fTll+2(Q); b u t i n o u r p r o b l e m fi i s a two d i m e n s i o n a l p o l y g o n w i t h no r e e n t r a n t c o r n e r s , so ( c f . [ 1 7 ] ) \(is(x) e H2(Q) (3. 16. 3) The r e g u l a r i t y o f $ n , $ n , w i t h Q s u f f i c i e n t l y smooth , i s g i v e n by " ^ W . n * c 8 < k ' a n * n r * 2 n k . 2 , n ' " • " ' W ^ n 3 c 9 ( k > a ) n b X . 2 , n • S i n c e q1} and q " a r e i n H^(Q) t h e n ¥ n , $ n . e H^(&); h oweve r , t h e c o n s t r a i n t s imposed by t h e geomet r y o f t h e d o m a i n fi i m p l y t h a t $ n g H2(Q). ( 3 . 1 6 . 4 ) a 3 . 2 The F i n i t e E l e m e n t A p p r o x i m a t i o n We now p r o c e e d t o d e s c r i b e t h e f i n i t e e l e m e n t a p p r o x i m a t i o n o f e q u a t i o n s ( 3 . 6 ) - ( 3 . 12 ) . Towards t h i s end , we f i r s t i n t r o d u c e t h e f i n i t e e l e m e n t s p a c e s . L e t h d e n o t e a d i s c r e t i z a t i o n p a r a m e t e r , 0 < h £ < 1, and l e t be a p a r t i t i o n o f fi i n t o s m a l l r e c t a n g u l a r e l e m e n t s fig w i t h 2 v e r t i c e s = C * ^ ' x2j*' ( J > e 1 > 1 - i - 1, 1 - J - J. 29 and s i d e s p a r a l l e l t o t h e c o o r d i n a t e a x e s . F o r e a c h v e r t e x k = (i, j), i and j a r e l o c a l i d e n t i f i e r s i n t h e and x^ d i r e c t i o n s , r e s p e c t i v e l y . The p a r t i t i o n 25^ = U fie i s r e g u l a r i n t h e s e n s e t h a t t h e r e e x i s t s a c o n s t a n t <r > 0 s u c h t h a t f o r he = diaCQ^), p = sup dia(Sph c Sl^) h p . e G i v e n p o s i t i v e i n t e g e r s K and N e > where K i s e q u a l t o t h e numbers o f v e r t i c e s ( o r g r i d p o i n t s ) i n and N g i s e q u a l t o t h e number o f e l e m e n t s Q i n D, , t h e f i n i t e d i m e n s i o n a l s p a c e s e h a s s o c i a t e d w i t h t h e p a r t i t i o n a r e d e f i n e d by Hh = { v h e C°(Q) •• v h \ e p r v n e € Dh) > Hoh = Hh n "l™- sch = sc n Hh> \ h = s c n »h • o. ^ .1: I n ( 3 . 1 7 . 1 ) , i s t h e s pace o f p o l y n o m i a l s i n x ^, x^ o f d e g r e e s 1 ( i = 2 i n ou r c o m p u t a t i o n s ) . The d i m e n s i o n o f i s e q u a l t o K and i t s c a n o n i c a l b a s i s i s t h e s e t \(P^\xy X2^i ' The i n t e r f a c e b e t w e e n e l e m e n t s a r e l i n e s o f = c o n s t a n t and Xg = c o n s t a n t , so t h a t t he r e s t r i c t i o n o f 4>^(x) a l o n g any o f t h e i n t e r f a c e s i s a p i e c e w i s e l i n e a r p o l y n o m i a l i n one v a r i a b l e . T h i s f a c t p e r m i t s w r i t i n g <p,(x) a s a t e n s o r p r o d u c t o f <t>1(x1) w i t h <t>Xx2); i . e . , *k(xV X2} = <t>i(x1><t>/x2)' 1 ~ k ~ K- 1 ~ 1 ~ J ' 1 ~ J ~ J 30 I <t>.(x. ; = 5 . , <f> .(x. ) = 8. , i s r ± I , j * s * J 1 l r l r * j 2 s j s where 4>^(x^ a n d (pXx^) a r e p i e c e w i s e l i n e a r p o l y n o m i a l s i n t h e v a r i a b l e s and x^, r e s p e c t i v e l y . N e x t , a c c o r d i n g t o G l o w i n s k i and P i r b n n e a u [16] we d e f i n e t h e s p a c e W^, t h e f i n i t e d i m e n s i o n a l a n a l o g u e o f t h e s pace H i n t r o d u c e d i n ( 3 . 7 . 1 ) , by M. = •{ u, e H, : u. | _ = 0 Vfi e D, , (2 n T = 0 }• , ( 3 . 1 7 . 2 ) h 1 h h w i fi e h e " ' H. = Hn, ® M, . (3 . 17. 3) h Oh h The d i m e n s i o n o f i s e q u a l t o t h e number o f b o u n d a r y p o i n t s . The c a n o n i c a l b a s i s B, o f M, i s d e f i n e d by h h B = \ w , 1 s i s K., w. . € M , v. .(P.) = i1' l f P i 6 r 1. h I hi b hi h hi l [ _ I v . 0, o t h e r w i s e ' The s u p p o r t o f c o n s i s t s o f t h e e l e m e n t s a d j a c e n t t o t h e b o u n d a r y . The s p a c e s ^Qh^ a r e m e m D e r s t h e f a m i l y of" f i n i t e 1 00 d i m e n s i o n a l s p a c e s Q^c V ' (Q) w h i c h i s c o m p l e t e i n (Q)(HQ(CI)). We now f o r m u l a t e some a p p r o x i m a t i o n and i n v e r s e p r o p e r t i e s o f w h i c h a r e u s e f u l i n t h e e r r o r a n a l y s i s o f t h e a p p r o x i m a t e s o l u t i o n ( c f . [ 2 ] , [ 1 2 ] ) . A l ) I f H.c ^(Q), 1 + I>ntz0, t h e n Vu <= Hr(Q), r*0, 0<s^ min(r.m) h inf llu - all £ Chp\\u\\ " , „ s, 2, fi r, 2, fi * € " h 31 where p ^ min(1+1-s, r-s) . 1 00 A2) I f i n a d d i t i o n , we assume t h a t u e W ' (Q.) t h e n inf llu - Z« „ - Ch l l u l l , * e " h A3) F o r 2 s r * 1+1 inf I llu - *ll + hllu - ^11 + + h f llu - a» n + h l lu _ ; I £ C h r l l u l l _ oo, f2 2, oo, fi J r, 2, L e t k^, be a r b i t r a r y numbers and p, q € [l,a>] s u c h t h a t //^ c t h e n t h e r e e x i s t s a c o n s t a n t C = C(<r, k^.k^, p, q) s u c h t h a t W , e H, , h ,. II ^ „,N/q-N/p+kl-k2.. .. I I ) IIv. II. _ _ ^ Ch IIv. It, , h k2, q, fi h kl,p, fi w i t h W t h e d i m e n s i o n o f t h e doma in fi. 12) IIv II. _ * C h _ 1 l l v , II , n l , co , i ! n 2,2, fi ., „ ^ 2-JV/2. . -1A-1/N,. ,. n oo, fi n 2, 2, fi L e t Qn.(x), * , Cx,), ¥ n . (x.) and <J>fYx,) d e n o t e t h e a p p r o x i m a t i o n s l sh ah h i n H, t o c j . T x ; , * (x), Y (x) and $ Cx^, r e s p e c t i v e l y . To h i s a compute t h e f i n i t e e l e m e n t s o l u t i o n s we expand s u c h a p p r o x i m a t i o n s a s l i n e a r c o m b i n a t i o n s o f t h e b a s i s f u n c t i o n s o f H.. Thus h 32 ^ = p i k * k ( x ) € \n ( 3 - 1 8 1 ) k *sn(x) =l*sk*k(x) *SCn (3'18-2) k *nan(x) =l*ak*k(x) *H0h (3'18"3) k *h(x) = l*k+k(x) *H0n • (3-18-4) k L i k e w i s e , i n v i e w o f (3. 13) t h e s t r e a m f u n c t i o n s c a n a l s o be a p p r o x i m a t e d a s *nih(x) = E ^"hVXJ) ' ( 3 . 1 8 . 5 ) k *2h(x) =l*n2n*k(x) ' (3'18-6) k The f i n i t e e l e m e n t a p p r o x i m a t i o n o f ( 3 . 6 ) - ( 3 . 8 ) i s d e f i n e d as f o l l o w s : G i v e n Q° = f, i =1,2 , = Q° = 0 , l ah h f i n d #, e S „ , , with C = 1, s u c h t h a t hs Ch Then f o r n > 0, a s s um ing (?", , , i//?^  and * , a r e known, s o l v e f o r sh 33 1) < A t > + ^ ( Q T 1 ^ ^ ^ * V + + 8 i 2 - f < J ^ + 3 i ^ ' V = * n ^ U 2 > V ' % e "oh . Qn+1(x)- Q*+1 6 tfn, . (3.20) l b uh . . x -,n+l jn+1 „ , . n ) * , , $> e s u c h t h a t ah n On (Wah , V<0h) + A T* a h , ^ = - ( Q l - Q2 , <t>h), V<ph e H^^ (3.21.1) ' V V = K 1 ' V ' V*h e HOh ' ( 3 " 2 1 - 2 ) , 1^ ah df i CT1 = . (3.21.3) b tf*1 = + c f J * . (3.21.4) h ah h sh H H ,n+l 2 .n+1 .n+1 ,n+l 1 Tn+1 .n+1 * i h B T - * r 2 \ + $ h • ^ h = - n r r r 2 * h + % • ( 3 - 2 2 ) F i n a l l y , we have t o compute t h e v e l o c i t i e s i n e a c h l a y e r f r o m t h e v a l u e s o f t h e s t r e a m f u n c t i o n s and (3.12). I f we t o o k t h e d e r i v a t i v e s o f V^^h W e w o u ^ < ^ o b t a i n v e l o c i t y v e c t o r s w h i c h 2 w o u l d n o t b e l o n g t o H , b u t t h e y w o u l d be i n L (Q); i n o r d e r 34 t o f o r c e them t o be i n H^, we p r o j e c t o r t h o g o n a l l y t he d e r i v a t i v e s o f 0 , „, o n t o H,. F o r i = 1, 2 d e f i n e l,Zh n n+1 n+1 ^ Vn+' • ( 3 . 2 3 . 2 ) l h dx^ lh dXg 2 The o r t h o g o n a l p r o j e c t i o n f r o m L (SI) o n t o i s g i v e n by (Vn+J - V*}*1, = 0, V0. e H , . ( 3 . 2 3 . 3 ) i n i h h h oh I t r e m a i n s t o d e s c r i b e t h e c o m p u t a t i o n o f t h e r e l a t i v e v o r t i c i t y on t h e b o u n d a r y when t he n o - s l i p b o u n d a r y c o n d i t i o n i s u s e d . T h i s i s done by c o n s t r u c t i n g t h e d i s c r e t e a n a l o g u e o f ( 3 . 7 . 2 ) i n t h e s p a c e M^. Thus, by t a k i n g — = 0 i n ( 3 . 7 . 2 ) , we have f o r i = 1,2, t h a t f V ^ ^ V u . d s = - \ J]+J-u,ds, Vu, e N, ( 3 . 2 4 . 1 ) he l h h he l h h h h ' whe re Be i s t h e s u p p o r t o f M^. S i n c e Be i s composed o f t h e r e c t a n g l e s a d j a c e n t t o t h e bounda ry , t h e n we c a n compute e x a c t l y t h e i n t e g r a l s o f ( 3 . 2 3 . 1 ) e l e m e n t by e l e m e n t o f Be. T h i s y i e l d s 35 n+1 ^n+1 _ 3 , .n+1 .n+1 . ih,I „„ ^ A., = - ^ - ( 0 . , .,- i/»., T) - — = - , ( 3 . 2 4 . 2 ) ih . 2 , rih,W ^ih,I 2 ( | n | ; where | n | i s t h e d i s t a n c e a l o n g t h e n o r m a l d i r e c t i o n f r o m t h e w a l l p o i n t V t o t h e f i r s t i n t e r i o r p o i n t I. F o r m u l a ( 3 . 2 4 . 2 ) i s l o c a l l y a s e c o n d o r d e r f o r m u l a known i n f i n i t e d i f f e r e n c e c o n t e x t a s Wood ' s f o r m u l a . I n a more g e n e r a l s e t up; s u c h a s t r i a n g l e s o r c u r v e d e l e m e n t s a s s u p p o r t o f Be, ( 3 . 2 4 . 1 ) y i e l d s an a l g e b r a i c l i n e a r s y s t e m o f e q u a t i o n s w h i c h c a n be e a s i l y s o l v e d by t h e C h o l e s k y method s i n c e t h e number o f b o u n d a r y p o i n t s i s , i n g e n e r a l , n o t v e r y l a r g e ( s ee [16] f o r d e t a i l s ) . By c o m p u t i n g t h e i n t e g r a l s t h a t a p p e a r i n ( 3 . 1 9 ) - ( 3 . 2 1 ) one o b t a i n s t h e f o l l o w i n g a l g e b r a i c l i n e a r s y s t e m s o f e q u a t i o n s : S//* 7 = [RJ , ( 3 . 2 5 . 1) s 1 W i [ Q T 1 } = [ R i 2 ] ' ( 3 . 2 5 . 2 ) S[*n+1] = [R ] , ( 3 . 2 5 . 3 ) a 3 T[*n+1] = [R4J , ( 3 . 2 5 . 4 ) h e r e [ ] d e n o t e s a co lumn m a t r i x . The m a t r i c e s S, V . and T a r e l s p a r s e , banded , s y m m e t r i c , p o s i t i v e d e f i n i t e m a t r i c e s w i t h e n t r i e s g i v e n by t . . = f V<t>u.V<t>u .dQ , ( 3 .26 . 1) 36 s. . = f CV0 .V©>, . + \2<p, .<t>, .)dQ , V l i J = \Q(*hi*hj + ^hi^hj)da ' V 2 i j = J / ^ h A j + v"*hiV+hJ)da ' where v = (—^-AtAu) and u = (1 + Ate) . The c a l c u l a t i o n o f t h e i n t e g r a l s i s p e r f o r m e d b y u s i n g t h e Gau s s ( 2 , 2 ) q u a d r a t u r e r u l e . S i n c e a l l t h e m a t r i c e s a r e p o s i t i v e d e f i n i t e , t h e n t h e f i n i t e e l e m e n t s o l u t i o n e x i s t s and i s u n i q u e . The s y s t e m s ( 3 .25 ) a r e s o l v e d by t h e J a c o b i C o n j u g a t e G r a d i e n t (JCG) method. ( 3 . 2 6 . 2 ) ( 3 . 2 6 . 3 ) (3 .26.4) 37 CHAPTER III A GALERKIN-CHARACTERISTIC ALGORITHM FOR THE HYPERBOLIC STAGE T h i s C h a p t e r i s d e v o t e d t o t h e f o r m u l a t i o n and a n a l y s i s o f t h e f i r s t s t a g e o f t h e a l g o r i t h m . F o r t h i s p u r p o s e , we t a k e as p r o t o t y p e e q u a t i o n t h e l i n e a r a d v e c t i o n ( t r a n s p o r t ) e q u a t i o n , t h e s o l u t i o n o f w h i c h i s c o n s t a n t a l o n g t h e c h a r a c t e r i s t i c c u r v e s o f t h e f l o w . An i n t e r e s t i n g p r o p e r t y o f t h i s e q u a t i o n i s t h a t u n d e r v e r y weak r e g u l a r i t y a s s u m p t i o n s t h e weak s o l u t i o n c o i n c i d e s a l m o s t e v e r y w h e r e w i t h t h e c l a s s i c a l s o l u t i o n . Our ma i n c o n c e r n h e r e i s t h e a p p r o x i m a t i o n o f t h e weak s o l u t i o n by a c o m b i n a t i o n o f G a l e r k i n and C h a r a c t e r i s t i c methods . Towards t h i s end we c o v e r t h e c o m p u t a t i o n a l doma in Q, 2 n £ fij £ R [Q^ i s s upposed t o be a r e c t a n g u l a r d o m a i n ) , w i t h a r e c t a n g u l a r g r i d and a s s i g n one p a r t i c l e t o e a c h node o f t h e g r i d . Then we t r a c e back t h e p o s i t i o n s o f t h e p a r t i c l e s a l o n g t h e c h a r a c t e r i s t i c c u r v e s d u r i n g a t i m e i n t e r v a l x and assume t h a t t h e m a g n i t u d e o f t h e dependen t v a r i a b l e i s t r a n s p o r t e d by t h e p a r t i c l e s , so t h e v a l u e s o f t h e d e p e n d e n t v a r i a b l e a t t h e g r i d p o i n t s x ^ a t i n s t a n t t +T a r e t h e v a l u e s p o s s e s s e d by t h e p a r t i c l e s a t t h e p o i n t s X ( x^ , t+ r ; t ) . He re and i n t h e s e q u e l , X ( x ^ , t + x; t ) d e n o t e s t h e p o s i t i o n a t i n s t a n t t o f t h e p a r t i c l e t h a t a t i n s t a n t t + T i s a t x. . k The m a t h e m a t i c a l " r e p r e s e n t a t i o n o f t h i s p h y s i c a l i d e a l i z a t i o n w o u l d be a l i n e a r c o m b i n a t i o n o f D i r a c d e l t a 38 f u n c t i o n s c e n t e r e d a t X ( x ^ , t + T; t ) . Bu t t h i s r e p r e s e n t a t i o n i s n o t v e r y c o n v e n i e n t f o r n u m e r i c a l c o m p u t a t i o n s , so we must r e g u l a r i z e i t m a t h e m a t i c a l l y . T h i s i s a c h i e v e d i f one s u b s t i t u t e s smoo the r p i e c e w i s e f u n c t i o n s f o r t h e D i r a c d e l t a f u n c t i o n s . From a p h y s i c a l p o i n t o f v i e w , t h i s amounts t o r e p l a c i n g p o i n t p a r t i c l e s by p a r c e l s o f l i m i t e d e x t e n s i o n w h i c h a r e a d v e c t e d w i t h t h e i r c e n t r o i d v e l o c i t i e s . We c h o o s e a s smoo the r p i e c e w i s e f u n c t i o n s t h e b a s i s f u n c t i o n s o f t h e f i n i t e e l e m e n t s p a c e c e n t e r e d a t t h e d e p a r t u r e p o i n t s X ( x ^ , t+x; t ) . T h e r e f o r e , t h e p a r t i c l e s o l u t i o n a t i n s t a n t s = t + x i s g i v e n by u(x, s) = Yp<-k(t)<t>k(x - X(*k> s; s)), k where a^t) d e n o t e t h e v a l u e s o f w(x, t) a t t h e p o i n t s XYx^, s;t). S i n c e ^ . C x ~ ^^xk' S ' S ^ = ^k^X^' ^ n e n w f x , s) b e l o n g s t o t h e f i n i t e e l e m e n t s p a c e H^. To d e t e r m i n e t he w e i g h t s u^t) we p r o j e c t t h e p a r t i c l e s o l u t i o n o n t o and show t h a t t h i s i s e q u i v a l e n t t o c o m p u t i n g t h e v a l u e s o f a A t ) by c u b i c s p l i n e i n t e r p o l a t i o n o f t h e g r i d p o i n t v a l u e s o f a g i v e n f u n c t i o n a l o f u(x, t). We p r o v e t h a t t h i s a l g o r i t h m i s c o n s e r v a t i v e , 2 u n c o n d i t i o n a l l y s t a b l e i n t h e L -no rm and c o n v e r g e n t . M o r e o v e r , f o r s u f f i c i e n t l y smooth f u n c t i o n s t h e s o l u t i o n i s s u p e r c o n v e r g e n t a t t h e f o o t o f t h e c h a r a c t e r i s t i c c u r v e s . 1. P r e l i m i n a r i e s 2 L e t t h e c o m p u t a t i o n a l doma in Q be an open s u b s e t o f R 2 w i t h L i p s c h i t z b o u n d a r y f , R i s t h e t w o - d i m e n s i o n a l E u c l i d e a n 39 s p a c e . We now i n t r o d u c e t he d e f i n i t i o n s o f s p l i n e s and B - s p l i n e s a s w e l l a s some o f t h e i r p r o p e r t i e s . L e t I - [a,b], I = [c,d] and assume t h a t t h e doma in Q = I x I . L e t I, J be 2 1 2 p o s i t i v e i n t e g e r s s u c h t h a t A = ix A = {x .}J a r e ^ & 1 11 1 2 2J 1 p a r t i t i o n s o f I and / r e s p e c t i v e l y , w h i c h s a t i s f y a = x < x < < x = b 1 1 1 2 I I c = x < x < < x = d . 21 22 2J F o r r,s p o s i t i v e i n t e g e r s , we d e f i n e t h e l i n e a r s p a c e o f s p l i n e f u n c t i o n s o f o r d e r r o v e r I a s l S . (I ) = J S(x ) e Hr(I ):DrS(x ) = 0 for x in [x .,x . ]\. r,Ai l 1 l p l l l i i n + i ' for i = 1, , I. Here H^d^) = -{ f e (f 2(^) '• Dr 2f is absolutely continuous, D r _ 1 f e LPaj \ , and (I ) = LV(I ) P i i S p e c i f i c a l l y , a s p l i n e S(x^) € 1 S a P ° l y n o m i a l o f d e g r e e r - 2 w h i c h i s c o n t i n u o u s and w i t h c o n t i n u o u s d e r i v a t i v e s up t o o r d e r r - 2. The l i n e a r s p a c e o f s p l i n e f u n c t i o n s o f o r d e r s o v e r 1^ i s d e f i n e d s i m i l a r l y . We a r e i n t e r e s t e d i n t h e c l a s s o f t e n s o r p r o d u c t s p l i n e s S . © S . d e f i n e d a s f o l l o w s . r , A i s , A 2 40 S . ® S . = •{ S(x ,x ) = S (x )S (x ) : S e S . , S e S . V r . A i s , A 2 ' 1 2 1 1 2 2 1 r , A i 2 s , A 2 ' We n o t e t h a t s . 9 s . c i / 6 ' pcn; , r , A i s, A2 where k = min(r, s). I n o r d e r t o s i m p l i f y t h e n o t a t i o n , we d e n o t e by S. h(&) t h e r e s t r i c t i o n t o Q o f t h e t e n s o r p r o d u c t s p l i n e o f ^ w i t h S . . We now f o r m u l a t e a lemma w h i c h i s a s p e c i a l c a s e s, A2 r o f a r e s u l t due t o W i d l u n d [36] f o r t h e s p l i n e a p p r o x i m a t i o n p r o c e d u r e S. To t h i s end l e t S(Q) be a l i n e a r s p a c e o f f u n c t i o n s i n w h i c h S i s d e f i n e d . T y p i c a l l y LP(£i) c S(Q) f o r 1 i p s n, m = 0, l , . . . k . F u r t h e r m o r e , assume t h a t S s a t i s f i e s t h e f o l l o w i n g p r o p e r t i e s . i ) Uniqueness. F o r any f e S(Q) t h e r e e x i s t s one and o n l y one f . e S. , (Q) s u c h t h a t Sf = f.. h k,h h ii) Stability. T h e r e i s a c o n s t a n t C, s u c h t h a t ii f, ii s en f ii , f e L P e n ; n s e n ; . hp p i i i ) Quasi-linearity in LP (Q). II C f +f ) . - (f +f ) II s || f II + II f -f II , f ,f e 1 2 h 1 2 p l h i p 2 h 2 p i 2 sen; i v ) Optimal accuracy. F o r a l l s u f f i c i e n t l y smooth f u n c t i o n s f II f . - f II * C h k | f | t , h p p,k where k d e n o t e s t h e o r d e r o f t h e s p l i n e . 41 Lemma 1 . 1 . [ 3 6 ] . Let the approximation procedure S satisfy i)-iv), then II f.-f II ^ Chr\f\ . ( 1 . 1 ) h p p,r f o r 0 < r < k . An a l t e r n a t e and c o n s t r u c t i v e d e f i n i t i o n o f S . and r, A i ^ 2 i n v o l v e s B - s p l i n e s a s f o l l o w s [ 7 ] . E n l a r g e t h e s e q u e n c e s {x . }* and {x .} J t o n o n - d e c r e a s i n g s e q u e n c e s {x , } I + r and {x 2^.> i , t h e a d d i t i o n a l p o i n t s b e i n g o t h e r w i s e a r b i t r a r y . F o r i = l , . . . , I + r and j = 1, . . . ,J+s , t h e i-th B - s p l i n e ( j - t h B - s p l i n e ) o f o r d e r r (s) f o r t h e s e s equence s i s B. (x) = (x . - x .)[x ., . . . ,x . ](x-x)/\ ( 1 . 2 ) i,r i i +r i i i i i i + r l + f o r a l l x i n I . S i m i l a r l y f o r B . (y), f o r a l l y i n I . i J>s 2 Here [x ^, .. .x^ ^  ] i s t h e r - t h d i v i d e d d i f f e r e n c e on t h e f u n c t i o n r~ l (x^- x)+ = max(0, x^- x) k e e p i n g x f i x e d . From ( 1 . 2 ) . t h e f o l l o w i n g r e c u r r e n c e r e l a t i o n i s o b t a i n e d . B. (x) = 1 ' X i i ~ X " X i i + i 1 , 1 0 , o t h e r w i s e . ( 1 . 3 . 1 ) X — X . - X — X B. C x ; = - B . (x) + — B . (x). i,r x . - x . i , r - i x . - x . 1 2 + 1 , r - i n + r - i i i 12+r n + i ( 1 . 3 . 2 ) We r e c a l l some p r o p e r t i e s o f t h e B - s p l i n e s w h i c h a r e needed i n t h e t h e s i s ( c f . [ 7 ] ) . 42 Bl) B. (x) = 0 f o r x n o t i n [x .,x . ]. i,r 11 11+r BZ) F o r any Cx ,x ) i n Q 1 2 B3) I E ^ / V ^ ' V " 1 • ( 1- 4 ) 1=1 j = i J S. ,(Q) = span iB. (x )B . (x)\.l'.\ k,h * > i , r l j,s 2 ' i , j = i B4) L e t t h e n o n - d e c r e a s i n g s equence {x . } I + r x {x . } J + S f o r m 1 1 1 2 J 1 t h e i n t e r i o r k n o t s o f t h e c o r r e s p o n d i n g s e q u e n c e o f B - s p l i n e s •{B. (x )B . (x )\-\'J._ • F o r t h e s t r i c t l y i n c r e a s i n g s equence 1 1 V 1 JrS 2 1, J—1 X = ix • y1 x ix • ) J o f d a t a p o i n t s , t h e s p l i n e 1 1 1 2 J 1 S ( x , x ) = V c . .B. f x ; B . Cx ) , ( 1 . 5 ) l 2 IJ i,r 1 j , s 2 w i l l a g r e e w i t h t h e g i v e n f u n c t i o n f a t x i f and o n l y i f S c h o e n b e r g - W h i t n e y theo rem ( c f . [ 7 ] ) g u a r a n t i e s t h e u n i q u e s o l u t i o n o f ( 1 . 5 ) i f and o n l y i f Bl.r<*M>BJ.s<*zJ> # ° ' V ' ( 1 - 6 1 ) o r , e q u i v a l e n t l y , i f and o n l y i f x . < y . < x . , V i , 1 1 1 1 1 J + r ( 1 .6 . 2 ) X2J < * 2 J < X 2 J + S ' V J ' ' 43 2. D e s c r i p t i o n o f t h e A l g o r i t h m 2 .1 . The C o n t i n u o u s P r o b l e m C o n s i d e r t h e Cauchy p r o b l e m f o r t h e s c a l a r a d v e c t i o n e q u a t i o n f o r u ( x , i ) i n t h e c y l i n d e r Q^. , d» + u . v w = = o , ( 2 . 1 . 1 ) at Dt w(x,0) = U (x) i n fi , ( 2 . 1 . 2 ) o where ^ d e n o t e s t h e m a t e r i a l d e r i v a t i v e o f w i n t h e f l o w u. assume t h a t t h e v e l o c i t y f i e l d i s i n c o m p r e s s i b l e , i . e . , V - u = 0 i n Q , ( 2 . 2 ) and t a n g e n t t o t h e b o u n d a r y T o f fi, i . e . , u-n | = 0 . ( 2 . 3 ) We a l s o r e q u i r e u ( x , t ) € LaYo,r;l/,co<'fi,>,> . ( 2 . 4 ) To compute t h e s o l u t i o n o f ( 2 . 1 ) we i n t r o d u c e t h e c h a r a c t e r i s t i c c u r v e s X{x,s;t) o f ( 2 . 1 ) w h i c h s a t i s f y t h e s y s t e m o f d i f f e r e n t i a l e q u a t i o n s 44 X(x,s;s) = x ( 2 . 5 . 2 ) We s t a t e w i t h o u t p r o o f t h e e x i s t e n c e and u n i q u e n e s s o f t he s o l u t i o n o f ( 2 . 5 ) and summar ize some r e g u l a r i t y r e s u l t s . P r o p o s i t i o n 2 . 1 . Under condition (2.4) there exists a unique solution t -» X(x, s;t) of the system (2.5) such that X(x,s;t) is in C° (0,T;Wi,C°(Q)). Furthermore, assume that for some integer k £ 1 u is in Lm(0,T;Wk'CD(Q)), then for a J J a e H n , 1 s |a| s Jt, t -> DaX(x,s;t) e C°(0,T; Lm(Q)). The s o l u t i o n o f ( 2 . 5 ) c an be e x p r e s s e d as X(x,s;t) - x = St u(X(x,s;x),x)dx , ( 2 . 6 ) s where X(x,s;t) d e n o t e s t h e p o s i t i o n a t t i m e t o f t h e p a r t i c l e o f f l u i d w h i c h i s a t p o i n t x a t i n s t a n t s. We n e e d f u r t h e r r e s u l t s o f t h e s o l u t i o n o f ( 2 . 5 ) w h i c h we now f o r m u l a t e . Lemma 2 . 1 . Under assumptions (2. 2),(2. 3) and (2.4) and for \s-t\ sufficiently small, x -» X(x,s;t) is a quasi-isometric homeomorphism of Q into itself with Jacobian determinant equal to 1 a . e. Moreover, L~1\x - y | £ \X(x,s;t) - X(y,s;t)\ £ L\x - y\ , ( 2 . 7 ) whe re L = exp(\s-t|•|u| n ) . 1 i , «>, Q 45 Proof. F o r a l l s,t i n ( 0 , 7 ) we have f r o m P r o p o s i t i o n 2.1 ( c f . [ 1 8 ] ) t h a t t h e r e i s a n e i g h b o r h o o d U(x) f o r w h i c h X(x,s;t) e x i s t s and i s u n i q u e . By f i x i n g s, X(x,s;t) c an be c o n s i d e r e d a s a mapp ing ^ C x J f r o m U(x) i n t o a n e i g h b o r h o o d o f X(x,s;t). P r o p o s i t i o n 2. 1 g u a r a n t i e s t h i s mapp ing i s one t o one and c o n t i n u o u s , a s w e l l a s i t s i n v e r s e = X(X(x, s; t), t; s) = x. On t h e o t h e r hand , s i n c e t h e v e l o c i t y s a t i s f i e s ( 2 . 3 ) , t h e n i t f o l l o w s t h a t X(x,s;t) maps fi i n t o i t s e l f . I t i s e a s y t o o b t a i n t h e i n e q u a l i t y ( 2 . 7 ) by a p p l y i n g G r o n w a l l ' s i n e q u a l i t y t o ( 2 . 5 ) and ( 2 . 6 ) . Hence, x -> X(x,s;t) i s a q u a s i - i s o m e t r i c homeomorphism. The J a c o b i a n d e t e r m i n a n t o f t h e above t r a n s f o r m a t i o n i s g i v e n as J(x,s;t) = det(DX(x, s; t)). P r o p o s i t i o n 2.1 a g a i n g u a r a n t i e s t h a t t -> J(x,s;t) i s c o n t i n u o u s . L i o u v i l l e ' s f o r m u l a and c o n d i t i o n ( 2 . 2 ) y i e l d J(x,s; t) ~ 1 a . e . • Remark. I t c a n be p r o v e d ( see [ 2 5 ] , Th. 1 .1 .7 ) t h a t x -> X(x,s;t) b e i n g a q u a s i - i s o m e t r i c mapp ing o f c l a s s Cml'*(Q), m ^ 1, w h i c h maps Q i n t o Q , t h e norms II • II _ and II • II * a r e m.p.n m,p,Q e q u i v a l e n t . I t i s a s i m p l e m a t t e r t o c h e c k t h a t i f t h e J a c o b i a n d e t e r m i n a n t o f t h e t r a n s f o r m a t i o n i s 1 a . e . t h e n II-II = 11-11 « 2,fi 2,Q . The u n i q u e c l a s s i c a l s o l u t i o n o f ( 2 . 1 ) i s g i v e n by w(x, t) = u (X(x, t;0)) . ( 2 . 8 ) o T h i s c o n d i t i o n h o l d s u n d e r v e r y weak r e g u l a r i t y a s s u m p t i o n s . I t i s w e l l known [29] t h a t f o r some i n t e g e r m i 1, u e 46 Z . e o C O J r ; l / n ' p f n ; ; and u € \T'P(^) the weak s o l u t i o n of the o problem (2 .1 ) be longs to L°°CO, T;]/1' P(Q)) and i s g i v e n by ( 2 . 8 ) . Now, l e t <£(•) € 2)(R 2 ) , where D (R 2 ) = \ <p e C°°(R2) : <p has compact suppor t }• , then f u(x,t)<p(x)dx = f u (y)^(X(y.O;t))dy , (2 .9 ) J R J R s i n c e the J a c o b i a n determinant of the t r a n s f o r m a t i o n x -» X(x , t i s equa l to 1. a . e . In (2 .9) y = X(x,t;0) . 2 . 2 . The D i s c r e t e Problem L e t W be a p o s i t i v e i n t e g e r , At = T/M and t = mAt f o r 0 s m s W - l . L e t u^(x, t.) be an approx imat ion to u(x,t). We assume t h a t u,(x,t) s a t i s f i e s c o n d i t i o n s ( 2 . 2 ) - ( 2 . 4 ) . For s = t , h m+i the approx imate t r a j e c t o r i e s of the p o i n t s x i n the time i n t e r v a l [ t , s] a re g i v e n , a c c o r d i n g to ( 2 . 6 ) , by X, (x , s ; s -x , ) = x - T T u, CX, f x , s ; s-c), s-c)dc , (2 .10) n o h h 0 s T s At For x = A t , X,(x,s;t ) denotes the d e p a r t u r e p o i n t o f the n m t r a j e c t o r y which reaches the p o i n t x at i n s t a n t s=t m+i The e r r o r e(x ) = X(x, s;S-T) - X,(XLS;S-T) n 47 c o m m i t t e d i n a p p r o x i m a t i n g t h e t r a j e c t o r i e s by ( 2 . 10 ) s a t i s f i e s t h e f o l l o w i n g lemma. Lemma 2 . 2 . Assume that u,(x,t) and u(x,t) fulfill h conditions (2.2)-(2.4), then for any integer m in the interval [0,M-1] and for any real x s u c h t h a t 0 s x ± At t h e e r r o r e(x) is bounded by | e ( x ) | * Tn i ( e x P _) - 1) . ( 2 . 11 ) 03, Q Proof. S u b t r a c t i n g ( 2 . 6 ) f r o m ( 2 . 10 ) i t f o l l o w s t h a t | e ( x ) | ^ r T |u, (X, (x, s;s-c), s-c) - u(X(x,s;s-c),s-c)\dc ' ' o 1 h h ^ T T Iu, CX, (x, s;s-c), s-e) - u(X,(x,s;s-c),s-c)\dc o 1 h h h + Si |Vu| |e (e ) |de . 0 1 103, Q < ' A p p l y i n g G r o n w a l l ' s i n e q u a l i t y y i e l d s t h e bound ( 2 . 1 1 ) . • A t any i n s t a n t t we a p p r o x i m a t e t h e weak s o l u t i o n u(x, t) by a two s t e p p r o c e s s . F i r s t , a p a r t i c l e a p p r o x i m a t i o n o f ( 2 . 8 ) i s s e t up , t h e n t h i s a p p r o x i m a t i o n i s p r o j e c t e d o n t o t h e f i n i t e e l e m e n t s p a c e H^. Towards t h i s end we i n t r o d u c e t h e s e t H o f c u t - o f f f u n c t i o n s <p (x) d e f i n e d a s f o l l o w s . F o r k = 1,2, . . .K and 1 £ i £ I, 1 s J < J a) d>, (x) i s b i l i n e a r i n x and x k 1 2 b) supp$k(x) = ^ _ n l i _ 1 ' n l i ^ x ^ - n 2 j _ 1 ' h 2 J - ' ' V i ' J ' w i t h - h . s x £ h,. , - h . z x £ h . . 11-1 l l i 2 j - i 2 2J c) <f>^(x - x^) = <t>k(x .), Q^x) e " i ^ k ^ X ^ k - i ' c a n o n i c a l b a s i s o f 48 Thus ~H = <*k(x) t i N o t i c e t h a t suppfy^Cx) i s t h e s h i f t e d s u p p o r t o f 4>k(x) t o t h e i n t e r v a l s [-h . ,h .]x[-h . ,h .] , V i, j . I t i s c l e a r t h a t 11-1 11 2J-1 2j J — 2 Qj^x) have compac t s u p p o r t i n R and s a t i s f y i) <f>k(x) e l / 1 , 2 C R 2 ; , (2 . 12. 1) i i ) C 4 / u k ; j ^ C x j d x = 1 , ( 2 . 1 2 . 2 . ) where u = meas(supp<p (x)). is. IS. L e t X, Cx, , t ; t ) be t h e d e p a r t u r e p o i n t o f t h e t r a j e c t o r y h k m+i m * w h i c h r e a c h e s t h e g r i d k n o t x, a t i n s t a n t t . L e t us k m+i c o n s i d e r 0. Cx - X , C x , , t ^ ;t )) = 5(x - X, C x , , t ^ ;t )) * k h k m+i m h k m+i m <f>k(x), where S( •) i s t h e D i r a c measure. A n o t h e r p r o p e r t y o f d>, (x - X, C x , , t ; t ) ) w h i c h i s r e q u i r e d f o r o u r a l g o r i t h m t o k h k m+i m ^ make s e n s e i s t h e f o l l o w i n g : i i i ) I f f o r some k and j supp<p.(x) r\ supp<j> .(x) * 0 , K J t h e n s u p p ^ C x - Xh(*k.tm+I;tm)) n s u p p ^ O r - V X j ' W ' V ; * 0 ( 2 . 1 3 ) L e t us c h e c k t h a t i i i ) i s s a t i s f i e d . Assume At = OCh; , by v i r t u e o f ( 2 . 7 ) , ( 2 . 1 1 ) and t h e r e g u l a r i t y o f t h e f i n i t e e l e m e n t p a r t i t i o n one ha s 49 IX. Or ,.t it ) - X , ( x , , t : t ) \ £ \X(x , t - , t ) - X (x t ;t )\ h j m+i m h k zn+i m j m+i m h j m+i m + \ X ( x . , t : t J ~ X,(x t ;t )\ k m+i m h k m+\ m + \X(x.,t ;t ) - X(x ,t ;t )\ j m+\ m k m+i m £ \x . - x , | + OChAt + |u - u. | . - A t ; . (2. 14) J k h oo, Q S i n c e supp<t>,(x) r\ supp<t> .(x) * es i s o f o r d e r 0(h), t h e n i t i s c l e a r f r o m ( 2 . 1 4 ) t h a t iii) i s s a t i s f i e d . F o r r e a s o n a b l y s t r u c t u r e d g r i d s , n o t n e c e s s a r i l y u n i f o r m , t h i s c o n d i t i o n a l s o e n s u r e s t h a t f o r any k i n (I, . . . ,K) t h e r e e x i s t s a t l e a s t an i n d e x j i n (1,...,K) s u c h t h a t x, e suppd> .(x - X,(x.,t ; t ) ) . Nex t , k J h j m+i m assume t h a t y e supp<p (x), t h e n i f one t a k e s y f o r x . i n k J ( 3 . 1 4 ) i t f o l l o w s t h a t suppd>, (x-X, (x, , t , ; t ) ) a p p r o x i m a t e s k h k m+1 m up t o 0(hAt + |u-u, | - A t H h e e l e m e n t whose p o i n t s a r e h oo, Q X,(y,t ^ ;t ) f o r any y i n suppcp. (x). h m+i m k We a r e now i n c o n d i t i o n s t o s t a t e t h e m a i n r e s u l t o f t h i s s e c t i o n and p r o v i d e a d e s c r i p t i o n o f t h e p r o p o s e d a l g o r i t h m t o compute t h e weak s o l u t i o n a t t h e p o i n t s X ,Cx , , t ,; t ). h k m+1 m B e f o r e d o i n g so l e t us i n t r o d u c e some n o t a t i o n a l s i m p l i f i c a t i o n . H e r e a f t e r , we s e t X, (x, , t w i t h r h k m+1 m hk . C a r t e s i a n component s (xf, xf, _,), and d e n o t e t h e m a t r i c e s hkl hkz B. Ax, ) and B. .(x„ ), g e n e r a t e d by t h e B - s p l i n e s B. .(x.) 1,4 lp j,4 2q J 1,4 1 and Bj ^(x^) a t t h e g r i d p o i n t s \x^pWx2qt' 1 ~ P ~ *' 1 ~ g £ J, b y A^and A^  , r e s p e c t i v e l y . Such m a t r i c e s a r e i n v e r t i b l e . L e t " { i ^ - j be t h e v a l u e s o f t h e a p p r o x i m a t e s o l u t i o n a t t h e g r i d p o i n t s a t i n s t a n t t^^, and A a s y m m e t r i c p o s i t i v e 50 d e f i n i t e m a t r i x w i t h e n t r i e s a = (<t> , d> ). The p r o d u c t ( s ee rs r s b e l o w t h e p r o o f o f t h e Theorem 2 .1 ) A[u>m+1] = [B] ( 2 . 1 5 ) s a t i s f i e s t h e t h e o r e m t h a t f o l l o w s . I n ( 2 . 15 ) t h e e n t r i e s b, k o f [B] r e p r e s e n t t h e v a l u e s a t i n s t a n t t o f t h e p r o j e c t i o n o f t h e a p p r o x i m a t e s o l u t i o n w, (x, t ) a t t h e d e p a r t u r e p o i n t s xf\ . h m nk Theorem 2.1 . Consider the set -JB. Jx,)B. Jx„)\I.'J. ,of ' i , 4 1 j,4 2 { i , j = l B-spline basis for the space S „ ,(Q). Then at any instant t , 4, n m 0 s m £ W-2, the entries of the matrix [B] are given by \ j ^ J / ^ A ^ ^ X ^ ; . Vk i n 2,2 K. ( 2 . 1 6 . 1 ) M o r e o v e r , t h e coefficients c"! . satisfy the relation A^A^c" 3 ; = A/"wm7 . ( 2 . 1 6 . 2 ) F o r t h e d e f i n i t i o n o f t h e m a t r i x A see b e l o w t h e p r o o f o f t h e t heo rem . In g e n e r a l , f o r w e l l s t r u c t u r e d g r i d s i t i s a d v a n t a g e o u s t o t a k e A= A f o r r e a s o n s t o be e x p l a i n e d b e l o w . Theorem 2. 1 s i m p l y s t a t e s t h a t a t any i n s t a n t t t h e m a p p r o x i m a t e s o l u t i o n w ^ C x , t^+^) i s u p d a t e d a t t h e g r i d p o i n t s by p e r f o r m i n g c u b i c s p l i n e i n t e r p o l a t i o n o f wCx, t ) a t t h e d e p a r t u r e p o i n t s Thus, t h e a l g o r i t h m t o compute t h e f i r s t h y p e r b o l i c s t a g e c o n s i s t s o f t h e f o l l o w i n g s t e p s : 51 G i v e n [u J : i ) Vm e [0, M-l], compute [cm] by s o l v i n g ( 2 . 1 6 . 2 ) . The most e f f i c i e n t way o f p e r f o r m i n g t h i s s t e p i s u s i n g t h e p r o p e r t i e s ~ m * o f t h e t e n s o r p r o d u c t o f m a t r i c e s . L e t A[w ] = R . The co l umn * v e c t o r R c a n be a r r a n g e d as an I x J m a t r i x R = (r ^ J, 1 s i < I and I s j < j. L i k e w i s e , t h e co lumn v e c t o r [cm] i s - a r r a n g e d a s an I x J m a t r i x C s u c h t h a t CA* = Y . Then, f o r I s J s J , s o l v e F i n a l l y , f o r 1 s i < j , s o l v e A l [ c i j ] = I y i j ] ' 1 ~ J ~ J • * * S i n c e t h e m a t r i c e s A^ and A^ a r e d i a g o n a l l y d o m i n a n t , t h e y c an be e f f i c i e n t l y i n v e r t e d by Gauss e l i m i n a t i o n w i t h o u t p i v o t i n g ( c f . [ 7 ] ) . The number o f l o n g o p e r a t i o n s a t t h i s s t e p i s O(IxJ). i i ) A t any i n s t a n t t^, d e t e r m i n e by s e q u e n t i a l o r b i n a r y s e a r c h t h e i n t e r v a l s where t h e p o i n t s (X™, ) l i e . hk x. i i i ) F o r e a c h p a i r (x f „ x ? x , < x ? , s x , , _ , h p ! hg2 lp-1 hpl lp 2q-l X, _ s x „ , e v a l u a t e t h e B - s p l i n e b a s i s ng2 2q v. = B4 i C X m p l ), f o r i = p, p+2, p+2, p+3 u s i n g t h e r e l a t i o n s ( 1 . 3 . 1 ) and ( 1 . 3 . 2 ) . iv) F o r j = q, q+1, q+2, q+3 f o r m p+3 , ~ m d . = y v.c . . . J • i i j i=p vO E v a l u a t e 52 J Y, d.B . JX? .) . j t 2 J J'4 hcJ2 vi) F i n a l l y , s o l v e t h e s y s t e m ( 2 . 15 ) i n O(IxJ) o p e r a t i o n s . , . . m+1 t o o b t a i n u, k The number o f l o n g o p e r a t i o n s ( m u l t i p l i c a t i o n s and 2 d i v i s i o n s ) t a k e n by s t e p s i i i ) — v) i s 4 r + o(r) p e r p o i n t . H e r e , r d e n o t e s t h e o r d e r o f t h e s p l i n e . Hence, t h e t o t a l number o f o p e r a t i o n s t o c a r r y o u t t h e c o m p u t a t i o n s a t K d e p a r t u r e p o i n t s i s 64K + 0(K). T h i s number i s s l i g h t l y l a r g e r t h a n t h a t o f t h e s t a n d a r d b i c u b i c s p l i n e p r o c e d u r e w h i c h i s 49K + 0(K). Thus , f r o m a c o m p u t a t i o n a l p o i n t o f v i e w t h e u se o f B - s p l i n e s i s more e x p e n s i v e t h a n t h e s t a n d a r d b i c u b i c s p l i n e i n t e r p o l a t i o n p r o c e d u r e ; however , t h e l o c a l n a t u r e o f t h e B - s p l i n e s may o f f e r some a d v a n t a g e s when t h e doma in i s n o t r e c t a n g u l a r o r t h e s o l u t i o n i s n o t g l o b a l l y smooth. Proof of Theorem 2. 1 A c c o r d i n g t o R a v i a r t [32] and M a s - G a l l i c and R a v i a r t [24] we may d e f i n e a p a r t i c l e a p p r o x i m a t i o n o f t h e weak s o l u t i o n o(x, t) a s f o l l o w s . We a p p r o x i m a t e t h e i n i t i a l c o n d i t i o n w 0 ^ y ^ by a l i n e a r c o m b i n a t i o n o f D i r a c measures k=i f o r some s e t iy^P^^ o f . p o i n t s y ^ e Q and P k « n k e R . By s u b s t i t u t i n g ( 2 . 1 7 ) i n t o ( 2 . 9 ) a d i r e c t c a l c u l a t i o n y i e l d s 53 K u (x,t) = I P k u ° k 6 C x - Xh(x ,t;t)) . ( 2 . 18 ) P k=\ Here ^ C x ^ , t ; 0 ; = y . I n [32, T h . 4 . 1 . ] i s p r o v e d t h e c o n v e r g e n c e o f ( 2 . 18 ) t o t h e 0 2 weak s o l u t i o n ( 2 . 9 ) f o r any <f> e C CR .). From a p r a c t i c a l p o i n t o f v i e w i t i s c o n v e n i e n t t o r e g u l a r i z e ( 2 . 18 ) by a c o n v o l u t i o n p r o d u c t w i t h a c u t - o f f f u n c t i o n . S e v e r a l t y p e s o f c u t - o f f f u n c t i o n s have been p r o p o s e d i n t h e l i t e r a t u r e ( c f . [ 3 2 ] ) ; however , we a r e i n t e r e s t e d i n u s i n g c u t - o f f f u n c t i o n s o f low d e g r e e o f smoo thne s s and w h i c h a r e s u i t a b l e t o work w i t h C°-c o n f o r m i n g f i n i t e e l e m e n t s . Nex t , we d e f i n e p a r t i c l e a p p r o x i m a t i o n t o u(x,t) a t i n s t a n t t a s m+i K u (x,t ^ ) = Tp.u..(t )8(x - Xu(x,,t : t ^ ) ) , ( 2 . 1 9 ) p m+i . ^ k h k r n h k m+i m+i k=i where w , , C t ) i s t o be u n d e r s t o o d a s an a p p r o x i m a t e v a l u e t o hk m w(X, Cx, , t ;t ) , t ), p, d e n o t e s t h e v a l u e o f a w e i g h t n k m+i m m k . • « f u n c t i o n a t and x i s a g r i d p o i n t . The r e g u l a r i z a t i o n p r o c e s s o f ( 2 . 1 9 ) amounts t o r e p l a c i n g p o i n t p a r t i c l e s , w h i c h move a l o n g t h e c h a r a c t e r i s t i c c u r v e s p a s s i n g t h r o u g h t h e v e r t i c e s {x^}, b y p a r c e l s o f l i m i t e d e x t e n s i o n mov i ng w i t h t h e i r c e n t r o i d s . We d e n o t e b y w ,(x, t ) t h e r e g u l a r i z e d f o r m ph m+i o f w (x, t ) w h i c h i s c o n s t r u c t e d a s f o l l o w s , p m+i K u> .(x,t ^ ) = Y.p,u,.(t )8{x - Xjx.,t A ;t )) *(pl1 4>.(X)) ph m+i .^^khkrn h k m+i m+1 k k r k=i 54 K hk m Yk h k m+i m+i k=i 2 2 We n o t e r i g h t away t h a t w ,(x, t ) i s i n L (R ) s i n c e by ° ph m+i Y o u n g ' s t h e o r e m e a c h c o n v o l u t i o n o p e r a t o r P^^^ * ' > ' ; ' 2 2 — i s i n £ ( L (R ) ) . By t h e d e f i n i t i o n o f <p and g i v e n t h a t t h e p o i n t s X^(x,s;t) a r e i n fi a s one deduce s f r o m Lemma 2 . 1 , i t i s 1 2 c l e a r t h a t w ,(x,t ) e U ' (Q) f o r any t . F i g . 2 d e p i c t s a ph m+i J m+i & ^ g r a p h i c a l r e p r e s e n t a t i o n o f t h e r e g u l a r i z a t i o n p r o c e s s a t p o i n t xf, w h i c h a r r i v e s a t v e r t e x x, a t i n s t a n t t hie k m+i Now, we l o o k f o r a p p r o x i m a t i n g t h e s o l u t i o n ( 2 . 2 0 ) i n t he s p a c e H.. S i n c e d>. (x-X,(x, , t ;t )) c o i n c i d e s w i t h <b.(x), h rk h k m+i m+i k t h e n we c a n s e t u ,(x, t ) = w, (x, t ) where ph m+i h m+i K %(x'tm+i) =ZQ)k \ ( x ) ' X 6 Q ' ( 2 ' 2 1 ) k=i and s u c h t h a t (w,(x,tm+) , <p.) = Cw, (y,tm ), 4>.(y - X, f x , , t ;t)) , V k, m, h m+i k h m k h k m+i m ( 2 . 22 ) whe re (u, v) = uvdQ . From ( 2 . 2 2 ) one o b t a i n s t h a t t h e ' w e i g h t s s a t i s f y t he a l g e b r a i c l i n e a r s y s t e m A [ u m + 1 ] = [B] . ( 2 . 15 ) Our n e x t s t e p i n t h e p r o o f i s t o show t h a t f o r any k, 1 k £ K, t h e e n t r i e s b o f [B] a r e t h e i n t e r p o l a n t v a l u e s o f a 55 b i c u b i c s p l i n e a t t h e p o i n t s X ^ . By v i r t u e o f t h e p r o p e r t i e s o f t h e t e n s o r p r o d u c t o f f u n c t i o n s on r e c t a n g u l a r g r i d s we c an p r o v e t h i s a s s e r t i o n f o r t h e one d i m e n s i o n a l c a s e w i t h o u t l o s s o f g e n e r a l i t y . L e t a , b be t h e end p o i n t s o f t h e i n t e r v a l I a = x < 1 < x = b and c o n s i d e r i K(Xk) = I w " f <(>Ax - Xk)<f>r(x)dx , V k,r', r J a ( 2 . 2 3 ; where v -( x + h k ; - xk • x € [xk hk'xk] ' 0, otherwise. • x e [ W hk+i] ' D r o p p i n g t h e s ymbo l Z, K(X ) c a n now be w r i t t e n as K(X, - x b X - x )4> (X)dx + L (— )l/> (X)dx 'X h. • T k k+ i J <t> (x)dx a r C l e a r l y , ^ ( ^ ^ ) i s a p i e c e w i s e c u b i c p o l y n o m i a l i n . Now, i f t h e g r i d i s e i t h e r u n i f o r m i n e a c h c o o r d i n a t e d i r e c t i o n o r p r o g r e s s i v e l y s m o o t h l y v a r i a b l e , i . e . a w e l l s t r u c t u r e d g r i d , t h e n by v i r t u e o f ( 2 . 14 ) one ha s t h a t f o r any k t h e r e i s an r s u c h t h a t KCX^J p o s s e s s e s f i r s t and s e c o n d o r d e r c o n t i n u o u s d e r i v a t i v e s i n t h e i n t e r v a l [x ,x ], and w i t h b r e a k i n g r r + i 6 p o i n t s -{x^}-. Such c o l l e c t i o n o f p i e c e w i s e c u b i c p o l y n o m i a l s g e n e r a t e t h e l i n e a r s p a c e > ^ 4 x 3 > ° f p i e c e w i s e c u b i c p o l y n o m i a l s w i t h c o n t i n u o u s d e r i v a t i v e s up t o s e c o n d o r d e r a t 56 t h e b r e a k i n g p o i n t s - { x^ . A c c o r d i n g t o C u r r y - S c h o e n b e r g t heo rem ( c f . [7] ) P , c o i n c i d e s i n t h e two d i m e n s i o n a l i,xr> 3 c a s e w i t h t h e s p a c e S .Cn). Hence, t h e r e e x i s t s a c u b i c 4, n s p l i n e S(x) i n S JQ) s u c h t h a t S(X?, ) = K()CT, ). I n o r d e r t o 4, h hk hk c h a r a c t e r i z e S(x) we have t o a s c e r t a i n t h e v a l u e s S(x ). I t r i s o b v i o u s t h a t i f one t a k e s X f , = x i n t h e two d i m e n s i o n a l hk r a n a l o g o f ( 2 . 2 3 ) , t h e n i t f o l l o w s t h a t SCx ; = AlJ"] , ( 2 . 2 4 ) r where t h e e n t r i e s o f A a r e g i v e n by t h e i n n e r p r o d u c t s (<Pk(x -X, ), d> (x)). I n t h e c a s e o f g r i d s w h i c h a r e u n i f o r m i n e a c h k r d i r e c t i o n A = A. S i n c e any S(x) e S , (Q) c a n be e x p r e s s e d a s 4, n a l i n e a r c o m b i n a t i o n o f t e n s o r p r o d u c t s o f c u b i c B - s p l i n e s , t h e n f o r any k, k hk , L . L i i i,4 lhk 4 2hk I=IJ=I J T a k i n g ( ^ ^ ^ ^"zhk^ = ^Xik' X2k^ a D O V e a n d u s i n g ( 2 . 2 3 ) y i e l d s A*® A* [cm] = A[o>m] 1 2 Remark. I f a n o n u n i f o r m g r i d i n e ach d i r e c t i o n i s u s e d , t h e n A * A. T h i s i m p l i e s t h a t t h e m a t r i x A has t o be r e c o m p u t e d e v e r y t i m e s t e p ; howeve r , f r o m a p r a c t i c a l p o i n t o f v i e w , and s i n c e t h e a l g o r i t h m r e q u i r e s w e l l s t r u c t u r e d g r i d s ( t h i s may be a l i m i t a t i o n ) one m i g h t be tempted t o t a k e A = A. T h e r e a r e s e v e r a l r e a s o n s f o r d o i n g so : 57 i) S a v i n g s i n c o m p u t a t i o n a l t i m e and c o r e memory. The m a t r i x A i s c a l c u l a t e d once and f o r a l l a t t h e b e g i n n i n g o f t h e c o m p u t a t i o n s . ii) The c o n s e r v a t i o n o f a>(x,t), a s we s h a l l s ee i n t h e n e x t s e c t i o n , i s b e t t e r a c h i e v e d . I n f a c t , i f A we re u s e d , u>(x, t) w o u l d be c o n s e r v e d up t o O(At). - O t h e r p r o p e r t i e s a s t h e s t a b i l i t y and t h e c o n v e r g e n c e do no t s u f f e r . i i i ) I f t h e CFL c o n d i t i o n i s l e s s t h a n 1, t h e n A = A w h e t h e r t h e g r i d i s u n i f o r m i n e a c h d i r e c t i o n o r n o t . 3. P r o p e r t i e s I n t h i s s e c t i o n we s t u d y some i m p o r t a n t f e a t u r e s o f t h e a l g o r i t h m . S p e c i f i c a l l y , we show t h a t i t i s c o n s e r v a t i v e and 2 u n c o n d i t i o n a l l y s t a b l e i n t h e L -norm. C o n s e r v a t i o n . Theorem 3 . 1 . Let = meas(suppQ^). For 1 s k £ K and m € [0,M-l] algorithm (2. 15)-(2. 16) conserves £ ^-J^t • k Proof. L e t k -m+i 1 = A/w m+i Then 58 Now, r e c a l l i n g ( 1 . 4 ) , ( 2 .15 ) and ( 2 . 1 6 ) and w i t h A = A i t f o l l o w s i m m e d i a t e l y r- m+i m r o i l k k S t a b i l i t y . Theorem 3.2. Algorithm (2.15)-(2. 16) is unconditionally stable in the L -norm. Proof. By v i r t u e o f ( 2 .22 ) we c a n w r i t e M u l t i p l y i n g by and summing o v e r k i n d i c e s g i v e s II u.(x, t ^ ; i l 2 s II u.(y,t )\\ II Z w f + 1 0 , Cy - x f , )" h m+i h m v. k k hk N o w , i t r e m a i n s t o e s t i m a t e t h e s e c o n d t e r m on t h e r i g h t hand s i d e o f t h e i n e q u a l i t y . To b e g i n w i t h , we r e c a l l t h a t suppip^Cy -) i s t h e s h i f t e d s u p p o r t o f <f> (x). L e t £ and SE. d e n o t e t h e s u p p o r t s o f <p (x) and 0 Cy - x f ,) r e s p e c t i v e l y . Then we may w r i t e JSE ) From ( 2 . 1 4 ) we have t h a t i f supp<j> . r\ supp<p i s n o t empty, t h e n J k supp<pj n supp<f>k = supp<j>Xy - X^j) n supp<pk(y - X™^) + O(hbt), where we have assumed t h a t \u - u, | _ s 0(h). W i t h t h i s 59 i n f o r m a t i o n and t a k i n g i n t o a c c o u n t t h a t t h e amount o f t h e o v e r l a p p i n g i s f i n i t e one e a s i l y g e t s Hence ujx.t )\l s a + CM) II u.(x,t )\\ , ( 3 . 2 ) h m+i h m and so s t a b i l i t y . 4. E r r o r A n a l y s i s L e t um+1(x) = u(x,t ), w h i c h a c c o r d i n g t o ( 2 . 8 ) m+i s a t i s f i e s o>m+1(x) = um(Xh(x t ;t ) ) = C^OO • C4. 1) n m+i m n where x f i s a s h o r t h a n d n o t a t i o n f o r X.(x,t ;t ). For any x h h m+i m i n E^, where E^ i s t h e s u p p o r t o f <p^, t h a t i s , t h e m a c r o e l e m e n t composed o f t h o s e e l e m e n t s w h i c h meet a t t h e v e r t e x x, , c o n s i d e r t h e t r a n s f o r m a t i o n k x -» x - xk + X*k = x + « m k . ( 4 . 2 ] By t h i s t r a n s f o r m a t i o n t h e e l e m e n t E, i s s h i f t e d by a" 3 , . As J k hk i n t h e p r o o f o f s t a b i l i t y , we d e n o t e t h e s h i f t e d e l e m e n t by SE, . L e t us i n t r o d u c e t h e f u n c t i o n o)m+1(x) w h i c h i s d e f i n e d by k 60 w ( x ; = w ( x + a , , ; . ( 4 . 3 ) hk We a l s o c o n s i d e r t h e a p p r o x i m a t i o n o>m*1(x) t o t h e s o l u t i o n wml(x) i n t h e f i n i t e e l emen t s pace . J^^Cx) i s d e f i n e d by ( c f . [ 2 8 ] ) ( u m + 1<xJ>, <p.) = ( J"(X? ), <fi. ) , V k . ( 4 . 4 ) h k h h k 2 L e t IT^ be t h e o r t h o g o n a l p r o j e c t i o n f r o m L (Q) o n t o w i t h r e s p e c t t o t h e i n n e r p r o d u c t (u, v), i . e . , Cn h u, X) = Cu, , V * € # h . ( 4 . 5 ) Then (^^(x) may be e x p r e s s e d a s «f+1Cx; = cxf; . ( 4 . 6 ) h h h h N e x t , f o r any x i n £ , we c o n s i d e r t h e p a r t i c l e a p p r o x i m a t i o n t o (J^*^(x) i n tf, , w h i c h we d e f i n e a s ZTX(x) = l uhAt HAx - Xll1) . ( 4 . 7 ) h u hk m k hk where t h e v a l u e s w. , (t ) a r e o b t a i n e d by t h e r e l a t i o n hk m The i n n e r p r o d u c t s on t h e r i g h t hand s i d e o f ( 4 . 8 ) a r e g i v e n by i n t e g r a l s o f t h e f o r m 61 hE~<(y)~^(y - *^dy • (4-9) k By v i r t u e o f ( 4 . 2 ) we may w r i t e m y = X + %k and c o n s e q u e n t l y t h e i n t e g r a l s o f ( 4 . 9 ) become L wJVx + a*)*.(x)dx , V k . ( 4 . 10 ) 'E, h hk'^k k The f o r m u l a t i o n ( 4 . 10 ) c a n be v i e w e d a s an a r e a - w e i g h t i n g f o r m u l a t i o n a c c o r d i n g t o [ 2 7 ] . By v i r t u e o f ( 4 . 10 ) we may w r i t e ( 4 . 8 ) a s (% ,<pk) = (%(x + «hk),4>k) , V k . ( 4 . 11 ) F i n a l l y , l e t u s d e f i n e m+i, . m+i, . -m+i, . , . .„« v f x ) = w f x > - w, f x ) . (4. 12) n The f o r m u l a t i o n ( 4 . 4 ) i s a n a l y s e d i n [31] and a method t o p e r f o r m t h e c o m p u t a t i o n s o f t h e i n n e r p r o d u c t s on t h e r i g h t hand s i d e i s d e s c r i b e d i n [ 1 9 ] . M o r t o n e t a l [27] have r e c e n t l y shown t h a t t h e a p p r o x i m a t i o n o f t h e r i g h t hand s i d e i n n e r p r o d u c t s o f ( 4 . 4 ) by s e v e r a l q u a d r a t u r e r u l e s m i g h t r e n d e r t h e f o r m u l a t i o n c o n d i t i o n a l l y u n s t a b l e . We w i s h t o 2 a s c e r t a i n , i n t h e L - no rm, t h e e r r o r i n c u r r e d by r e p l a c i n g w>m+1(x) by u^*X(x) a t any i n s t a n t t . Towards t h i s end we J h J m+i 62 s e t m+1, . m+i, . m+1, . -m+i, . -m+i, , m+i, . w ( x ) - w, (x) = w f x ) - w, f x ) + w. f x ) - w, ( x ; . (4. 13) h h h h By t h e t r i a n g l e i n e q u a l i t y „ m+i, , m+i, ... .. m+i, , -m+i, II w ( x ; - u, ( x ; l i s II w ( x ; - w, ( x ; i n h + II ZTX(x) - uT\x)\\ . ( 4 . 1 4 ) The e s t i m a t e o f t h e f i r s t t e rm on t h e r i g h t s i d e o f ( 4 . 1 4 ) i s g i v e n b y t h e f o l l o w i n g lemma. Lemma 4 . 1 . Fo r u(x,t) in L°°CO, T;!^*1'2(Q)), u(x,t) in Lm(0,T;W1'co(n)) and using C° finite elements of degree k, k £ 1 m+i, , -m+i, .„ „ m, . - m , .„ 0) (x) - u, (x)\\ < II u (x) - ujx)\\ h h + |wl . |u - u | _At + C h k + 1 l l uV _ . ( 4 . 1 5 : l.oo,C? h oo,Q k+i,p,Q Proof. See [31] . • To e s t i m a t e t h e s e c o n d t e r m on t he r i g h t hand s i d e o f (4. 14) we c a n a d o p t by v i r t u e o f ( 4 .10 ) t h e method u s e d i n [ 2 7 ] . We o b s e r v e t h a t by t h e t r i a n g l e i n e q u a l i t y -m+i m+i„ _ ,, -m+i ~m+i„ „ ~m+i m+i„ , „ w, - u L II s II w L - w L II + II u u - II . (4. 16) h h h h h h L e t u s s t u d y t h e f i r s t t e r m on t h e r i g h t s i d e o f t h e above i n e q u a l i t y . From t h e r e l a t i o n s h i p 63 i t f o l l o w s t h a t ii sTUx) - Sf*Vx;n * II iM"; - »l(x * , ( 4 . i 7 ) n n h n h hk so t h a t we have t o e s t i m a t e t h e r i g h t hand s i d e o f ( 4 . 1 7 ) . ' U s i n g ( 4 . 1 2 ) and t h e t r i a n g l e i n e q u a l i t y we o b t a i n ii uJVxf; - J?rx+«™ ;» s II umcx?; - < A x + « ? ; » h h h hk h hk + II Ax?; - v "Yx + a™ ; i l . ( 4 . 18 ) h hk The f i r s t t e r m on t h e r i g h t s i d e o f t h e above i n e q u a l i t y i s bounded a s II um(xf) - c/Yx + «J ;|| - C|xf - f x + a " ; | _|vw| _ . ( 4 . 19 ) h hk h hk oo, Q co, Q From ( 2 . 1 4 ) and ( 4 . 2 ) u. I A t ) . ( 4 . 20 ) h t h e n ( 4 . 1 9 ) becomes II (Ax?; - (Ax + a m , ; i l £ C|vu| _ h A t . ( 4 . 2 1 ) h hk oo, Q The s e c o n d t e r m on t h e r i g h t hand s i d e o f ( 4 . 1 8 ) y i e l d s = J (x? - (x + a™ ; ; - D i A F Q f x ; - ; d e , (4 .22) o n hk 0 IX? - fx + a" )l = OihLt + |u h hk H e r e a f t e r we assume t h a t |u - u , | £ 0 ( h ) , n 64 where Fa(x) = x + af, + Gfxf - (x + a*)) . (4.23) 8 hk h hk Thus ii A x J ; - v m ( x + a m k ; n 2 s E f ixjj- x - a m k i 2 [ j iVV e Cx;; i 2 dedfie k E. o k S i n c e FQ(X) i s a q u a s i - i s o m e t r y o f £ ^ o n t o £ k , £ k b e i n g t h e image o f £, by t h e t r a n s f o r m a t i o n x -> X, Cx, t ; t ,), t h e n we 6 k ' h m+i m may change t h e v a r i a b l e s i n t h e s e c o n d i n t e g r a l t o o b t a i n , e m p l o y i n g ( 4 . 2 0 ) , II vmOC?) - vm(x + «™ ; i l £ C h A t l l W m l l , h hk By v i r t u e o f t h e a p p r o x i m a t i o n p r o p e r t i e s o f ( c f . C h a p . I I , S e c t . 3 .2 ) II vm(X?) - vm(x + a" )\\ s CAth2\\ u>m\\ n , ( 4 . 2 4 ) h hk 2 , 2 , Q F i n a l l y , we come t o e s t i m a t e t h e s e c o n d t e r m on t h e r i g h t hand s i d e o f t h e i n e q u a l i t y ( 4 . 1 6 ) . (~u>l+1(x) - co^Cx), <pk) '= (Z>mh(y) - a*(y). \(y - X ^ , V k U s i n g t h e same a rgument f r o m t h e s t a b i l i t y p r o o f y i e l d s ~m+i m+i n _ -m mM , . w, - w, II £ (1 + C A t ) II w, - w, II . ( 4 . 2 5 ) n n h h C o l l e c t i n g t h e e s t i m a t e s ( 4 . 2 1 ) , ( 4 . 24 ) and ( 4 . 25 ) we o b t a i n 65 -m+i m+i„ ^ f, , „ . . ,„ -m m „ u, - w, II s (1 + C At)II w , - w , II h h i n n + C h A t l V w l . + C A t h 2 H wmH _ . ( 4 . 2 6 ) 2 oo, Q 3 2,2,Q Now, f rom ( 4 . 1 4 ) , ( 4 . 15 ) and ( 4 . 26 ) we f i n d t h a t H w m + 1 _ u ™ + 1 || < K f A t ; h 2 | II u II J n + K hAt|Vcj| . h 1 2.2.Q oo,Q 2 oo, Q + (1 + CAt)-{l l u " - um II + II w° - u™ II}- . ( 4 . 2 7 ) A s s um ing t h a t w°= w°= <j°, t h e n s u c c e s s i v e s u b s t i t u t i o n s i n t o h h ( 4 . 27 ) and G r o n w a l l ' s i n e q u a l i t y g i v e II u - w II s K -7-r— -{h II w II _ h A t 1 2 ,2 ,Q m , Q + hAt|Vw| . ]• , ( 4 . 2 8 ) oo,C? 1 where K ( h , A t ) = exp(CT). T h i s r e s u l t b r i n g s us t o t h e f o l l o w i n g t heo rem Theorem 4 . 1 . F o r w(x, t ) i n L°°(0, T ; l / 2 ' 2 ( Q ) n y 1 , c ° ( Q ) ) , u i n L°°(0 ,2 \V , C O ( f2 ) ) and A t = 0(h), algorithm (2. 15)-(2. 16) with C° finite elements of degree k = 1 c o n v e r g e s i n t h e L°°(0, T; LZ(Q)) norm with error 0 ( h ) . C l e a r l y , t h e e s t i m a t e ( 4 . 28 ) i s s u b o p t i m a l . N u m e r i c a l c o m p u t a t i o n s m e n t i o n e d i n [27] show t h a t f o r s u f f i c i e n t l y 3 smooth f u n c t i o n s t h e e r r o r c o m m i t t e d may be 0 ( h ). On t h e o t h e r hand , i n t h e p r e v i o u s a n a l y s i s no c o n s i d e r a t i o n ha s been p a i d t o t h e f a c t t h a t we a r e u s i n g c u b i c s p l i n e t o i n t e r p o l a t e t h e v a l u e s a t t h e d e p a r t u r e p o i n t s , and i t i s w e l l known t h a t 66 4 t h i s m i g h t y i e l d a p o i n t w i s e e r r o r a s h i g h a s 0 ( h ) a t t h o s e p o i n t s . I n o r d e r t o a s c e r t a i n t h e r o l e p l a y e d by t h e c u b i c s p l i n e i n t e r p o l a t i o n i n ou r a l g o r i t h m we now e s t i m a t e t h e e r r o r i n t h e maximum norm. Towards t h i s end , we assume t h a t u(x,t) € Lr(0,T;UZ'm(Q)) n ^'"(Q)) and u(x,t) € Lr(0, T; W1'a>(£})), r and s i n t e g e r s s a t i s f y i n g , i s r s oo, 1 £ s £ 4. A t any i n s t a n t t t h e a p p r o x i m a t e s o l u t i o n u>m+X(x) i s m+i h s i m p l y o>*+1(x) = I (S^CX^J^Xx) = PSU»(X?.) , ( 4 . 29 ) h , h hj j h hj where S i s t h e c u b i c s p l i n e i n t e r p o l a n t o f t h e a p p r o x i m a t e s o l u t i o n D, (x) a t t h e p o i n t s and p i s i d e n t i f i e d a s t h e h hj p r o l o n g a t i o n o p e r a t o r f r o m t he space l/ 1= { 1 - j £ K } i n t o t h e f i n i t e d i m e n s i o n a l s p a ce /Y^.Then m+i, . m+i, . m+i, , r m+i, . u (x) - w, (x) = w (x) - Io) (x) n + Iu>m+1(x) - pS(Ax? J , ( 4 . 30 ) h hj where Iu d e n o t e s t h e i n t e r p o l a n t o f w. The f i r s t t e r m on t h e r i g h t hand s i d e o f ( 4 . 30 ) r e p r e s e n t s t h e a p p r o x i m a t i o n e r r o r w h e r e a s t h e s e c o n d t e rm i s t he e v o l u t i o n a r y e r r o r . I t s e s t i m a t i o n w i l l c o n c e r n us becau se i t a c c u m u l a t e s a t e a c h t i m e s t e p and c o n t r o l s i m p o r t a n t a s p e c t s o f t h e method a s , f o r i n s t a n c e , t h e t r u n c a t i o n e r r o r and t h e n u m e r i c a l d i s s i p a t i o n . A f u r t h e r d e c o m p o s i t i o n y i e l d s i V 7 2 * 1 - P S U W . ) = p o A x " ! ; - c A x J . ; ; + P(u>m(x".) - s c A x " \ ; ; 67 + p(Swm(xl.) - ScAx?.» hj h hj Hence ( d r o p p i n g t h e s u b i n d e x Q) . m+i, . m+i, ^ . m+i, . r m+i, >. u) (x) - w, (x)\ £ u (x) - Iu (x)\ n oo oo + |cAx°!; - a>m(X? .) I J hj co + i (Ax? J - s<Axf.) i h j h j oo + IStAxf J - SfcAx? J | . ( B 1 . . B 4 ) , ( 4 . 31 ) h j h h j co The t e r m B l g i v e s w i t h b i l i n e a r f i n i t e e l e m e n t s \a>m+1(x) - IwmX(x)\ £ Ch2 , ( 4 . 32 ) 00 by i n t e r p o l a t i o n t h e o r y . I n r e g a r d t o B2, f r o m Lemma 2 . 2 we f i n d ( s ee a l s o [ 2 ] , [ 2 8 ] ) \dn(Xm.) - <Ax? J | £ C l V u l M u . - u| n A t ) . ( 4 . 33 ) J hj ca oo, Q h oo, Q A bound f o r B3 i s o b t a i n e d by u s i n g Lemma 1.1, i . e . , IcAxf J - StAxf .) | < C h S | w | . ( 4 . 3 4 ) h j h j oo 's, oo F i n a l l y , i t r e m a i n s t o bound B4. From t h e s t a b i l i t y p r o p e r t y o f t h e s p l i n e s one g e t s iscAxf j - S C A X J J I £ ci<Ax; - <Ax;i . (4 .35) hj h hj co h co From ( 4 . 3 0 ) - ( 4 . 3 5 ) and a s s um ing t h a t w°= co°, i t f o l l o w s on a p p l i c a t i o n o f G r o n w a l l ' s i n e q u a l i t y 6 8 l w m + i _ m+ij s K - ^ f h 0 " + |vw| nhAt) , ( 4 . 36 ) n oo,Q A t oo,Q where o- = min(2,s) and K a p o s i t i v e c o n s t a n t . The r e s u l t ( 4 . 36 ) i s f o r m u l a t e d a s a theo rem. Theorem 4 . 2 . For u(x,t) e La(0, T; W2' °°(Q) n WS,m(n)), a(x,t) e LC°(0,T;V1'a'(Q)), s an integer such that 1 £ s £ 4 and cr = min(2,s); algorithm (2.15)-(2. 16) with C° bilinear finite elements on a rectangular grid and A t = 0(h) converges in the maximum norm as given by (4.36). N o t e t h a t i f u i s s u f f i c i e n t l y smooth, s a y s=4, ( 4 . 34 ) g i v e s a s u p e r c o n v e r g e n t r e s u l t , i n t h e f i n i t e e l e m e n t f r amework , a t t h e d e p a r t u r e p o i n t s . M o r e o v e r , s i n c e t h e e s t i m a t e o f Lemma 1.1 i s l o c a l , we e x p e c t t h a t i n t h o s e r e g i o n s where u(x, t) i s smooth enough t h e i n t e r p o l a t i o n a t X?^. w i l l be o f h i g h o r d e r . Remark. Theorem 4 . 2 may e x p l a i n some o f t h e n u m e r i c a l r e s u l t s m e n t i o n e d b y M o r t o n e t a l . [27] u s i n g L a g r a n g e - G a l e r k i n methods w i t h C° f i n i t e e l e m e n t s t o i n t e g r a t e t h e one d i m e n s i o n a l l i n e a r a d v e c t i o n p r o b l e m . I n t h i s s i m p l e c a s e t h e y were a b l e t o compute e x a c t l y t h e i n t e g r a l s o f t h e i n n e r p r o d u c t s ; b u t a s Theorem .2.1 shows t h i s i s e q u i v a l e n t t o i n t e r p o l a t i o n by c u b i c s p l i n e s a t t h e f o o t o f t h e c h a r a c t e r i s t i c c u r v e s . I f one t a k e s u^= u, t h e n ( 4 . 3 4 ) y i e l d s and e r r o r e s t i m a t e i n t h e L^CO,T: £ 2 ; - n o r m o f 0(h*/At). 69 5. N u m e r i c a l E x p e r i m e n t s To i l l u s t r a t e t h e c o n s e r v a t i o n and s t a b i l i t y p r o p e r t i e s o f o u r scheme a s w e l l a s t o v e r i f y t h e v a l i d i t y o f t h e e r r o r a n a l y s i s we c o n s i d e r two model p r o b l e m s . The f i r s t one i s t h e a d v e c t i o n o f a cone i n a f i x e d r o t a t i n g v e l o c i t y f i e l d Q. The p a r a m e t e r s o f t h i s example a r e o(x.O) = -I * / r ) ' f * " r ) ( 5 .1 ) r.o; { " ( 1 L 0 , otherwise , where r i s a p o s i t i v e p a r a m e t e r , H = 100, and X = (X - X ) + X 1 0 2 u = (-Six , fix ) . ( 5 . 2 ) 2 1 We u s e an u n i f o r m s q u a r e g r i d on t h e d o m a i n [ - 1 , 1 ] x [ - 1 , 1 ] , w i t h h = (1/63), XQ = -15h, r = 8h and t h e CFL c o n d i t i o n e q u a l t o 2 . 9 a t t h e p o i n t o f maximum v e l o c i t y . The number o f t i m e s t e p s t o c o m p l e t e a r e v o l u t i o n i s e q u a l t o 96. F i g . 3a shows t h e i n i t i a l c o n d i t i o n whereas F i g . 3 b d i s p l a y s t h e cone a f t e r 6 r e v o l u t i o n s . The h e i g h t a t t h e v e r t e x i s now 87 and some w i g g l e a c t i v i t y i s o b s e r v e d a t t h e b a s e o f t h e cone . T a b l e 1 shows t h e t i m e e v o l u t i o n o f t h e cone e v e r y 48 t i m e s t e p s (1/2 2 o f a r e v o l u t i o n ) i n t e rms o f mass Tw , s q u a r e o f mass , and maximum and minimum v a l u e s . The mass and s q u a r e o f mass v a l u e s have b e e n d i v i d e d by t h e i r i n i t i a l v a l u e s , w h i l e t h e maximum and minimum v a l u e s a t d i f f e r e n t i n s t a n t s have been d i v i d e d by H. We o b s e r v e t h a t t h e r e i s mass c o n s e r v a t i o n . As f o r t h e d i s s i p a t i v e e f f e c t s , t h e y a r e s t r o n g a t t h e v e r y b e g i n n i n g t o d e c r e a s e as t i m e p a s s e s . F o r e x a m p l e , a f t e r 1/4 70 o f a r e v o l u t i o n t h e r a t e o f d i s s i p a t i o n o f s q u a r e o f mass p e r t i m e s t e p i s 6 *10 5 d e c r e a s i n g t o 3 *10 5 a t t h e end o f t h e 6 - t h r e v o l u t i o n . We a l s o o b s e r v e t h a t t h e e r r o r c o m m i t t e d a t t h e v e r t e x o f t h e cone i s O f l O ) p e r t i m e s t e p . T h i s c o n f o r m s t o o u r e r r o r a n a l y s i s w h i c h i s 0(h/At) i n t h i s c a s e s i n c e u(x, t) € V 1 ' 0 0 . Our s e c o n d t e s t i s a more s e v e r e one. We c o n s i d e r t h e ' s l o t t e d * c y l i n d e r p r o b l e m w h i c h was p r o p o s e d by Z a l e s a k i n 1979 t o t e s t h i g h o r d e r f l u x c o r r e c t e d t r a n s p o r t a l g o r i t h m s . The i d e a b e h i n d t h i s e x p e r i m e n t i s t o compare t he p e r f o r m a n c e o f o u r a l g o r i t h m w i t h t h e r e s u l t s r e p o r t e d by Z a l e s a k [37] and Munz [28] u s i n g h i g h o r d e r f i n i t e d i f f e r e n c e schemes. The p a r a m e t e r s o f t h i s e x p e r i m e n t a r e t h o s e u sed i n [ 2 8 ] . F i g . 4 a shows t h e i n i t i a l c o n d i t i o n , where h = 0 . 0 1 , CFL = 4 . 2 and t h e h e i g h t o f t h e c y l i n d e r H = 4. The number o f t i m e s t e p s t o c o m p l e t e a r e v o l u t i o n i s now e q u a l t o 96. We e m p h a s i z e t h e f a c t t h a t t h i s i n i t i a l c o n d i t i o n i s n o t i n l / 1 , p , b u t doe s b e l o n g t o Lip1', where Lipm,P- i s t h e (m,p) L i p s c h i t z s p a c e . However , we s t i l l c a n a p p l y Lemma 1.1 ( S e e [ 3 3 ] ) . F i g s . 4 b , 4 c , 4 d show t h e c y l i n d e r a f t e r 1/8, 1 and 6 r e v o l u t i o n s r e s p e c t i v e l y . We o b s e r v e i n t h e s e f i g u r e s t h e o u t s t a n d i n g a b i l i t y o f t h e scheme t o m a i n t a i n t h e shape o f t h e i n i t i a l c o n d i t i o n and k e e p u n d e r a s t r i c t c o n t r o l t h e w i g g l e s g e n e r a t e d a t t h e d i s c o n t i n u i t i e s . T h i s i s a r e m a r k a b l e a c h i e v e m e n t o f a scheme w h i c h i s n o t s p e c i f i c a l l y d e s i g n e d t o h a n d l e s t r o n g d i s c o n t i n u i t i e s . The a b i l i t y o f t h e scheme t o keep t h e w i g g l e s 2 u n d e r c o n t r o l i s due t o i t s _L - s t a b i l i t y . T a b l e 2 shows t h e 71 t i m e e v o l u t i o n o f t h e c y l i n d e r e v e r y 24 t i m e s t e p s o f a 6 r e v o l u t i o n e x p e r i m e n t . The c o n c l u s i o n s d rawn f r o m t h i s r e s u l t s a r e , i n some r e s p e c t s , s i m i l a r t o t h o s e o f t h e cone e x p e r i m e n t . T h e r e e x i s t s mass c o n s e r v a t i o n . The d i s s i p a t i o n o f s q u a r e o f mass i s s t r o n g a t t h e i n i t i a l s t a g e s t o d e c r e a s e p r o g r e s s i v e l y a s t i m e p a s s e s . F o r i n s t a n c e , t h e r a t e o f - 4 d i s s i p a t i o n i s 10 a f t e r t h e f i r s t r e v o l u t i o n , t o d e c r e a s e t o 10 5 a f t e r t h e s e c o n d r e v o l u t i o n . The t r e n d i s t o w a r d s a f u r t h e r d e c r e a s e . L a r g e w i g g l e s appea r c o n c e n t r a t e d a r o u n d t h e e x t e r n a l , u p p e r and l o w e r , r i n g s o f t h e c y l i n d e r and t h e s l o t ; b u t a f t e r f ew t i m e s t e p s t h e y a r e damped o u t and t h e r e s u l t s i n d i c a t e t h a t t h e i r maximum and minimum v a l u e s t e n d a s y m p t o t i c a l l y t o + 10% . 72 N°o f t i m e s t e p s C o n s e r v a t i o n mass C o n s e r v a t i o n s q u a r e mass Maximum Minimum 0 .66972175E+04 .33569349E+06 .10000000E+03 0 48 .99999999E+00 . 99706674E+00 .92601255E+00 956667E - 2 96 .10000000E+01 .99512210E+00 .91213264E+00 - 101286E - 1 144 99999999E+OO .99338941E+00 .90342497E+00 - 109947E - 1 192 .99999999E+00 .99178720E+00 .89622887E+00 - 113813E - 1 240 .99999961E+00 .99027655E+00 .88989364E+00 -. 115438E - 1 288 . 10000007E+01 .98883330E+00 . 88427672E+00 -. 118749E - 1 336 . 10000021E+01 .98744122E+00 .87930044E+00 -. 120397E - 1 384 . 10000032E+01 .98608897E+00 .87488915E+00 -. 120812E - 1 432 . 10000037E+01 .98476850E+00 .87096729E+00 -. 120317E - 1 480 .10000035E+01 .98347391E+00 .86746379E+00 -. 119158E - 1 528 .10000026E+01 .98220088E+00 .86431485E+00 -. 117518E - 1 576 .10000010E+01 .98094615E+00 .86144648E+00 -. 118635E - 1 TABLE 1. T ime e v o l u t i o n o f t h e cone N°of t i m e C o n s e r v a t i o n C o n s e r v a t i o n Maximum Minimum s t e p s mass s q u a r e mass 0 .24120000E+04 .96480000E+06 .40000000E+01 0 24 .99999843E+00 .995827685+00 .11428758E+01 - 150388E+0 48 .10000013E+01 .94983986E+00 .11180027E+01 - 131663E+0 96 .10000034E+01 .94457594E+00 .11307321E+01 - 171569E+0 120 .10000019E+01 .94061257E+00 .11386835E+01 - 189159E+0 144 .99999732E+00 .93735256E+00 .11387771E+01 -. 195762E+0 168 .99999179E+00 .93453740E+00 .11348827E+01 -. 197626E+0 192 .99999869E+00 .93203288E+00 .11289023E+01 -. 196171E+0 216 .99999837E+01 .92976022E+00 .11239973E+01 -. 193010E+0 240 .99999827E+01 .92766902E+00 . .11269107E+01 -. 188828E+0 264 .99999836E+01 .92572481E+00 .11285815E+01 -. 184017E+0 288 .99999872E+01 .92390286E+00 .11293093E+01 -. 178815E+0 312 .99999224E+01 .92218474E+00 .11293036E+01 -. 173373E+0 336 .99999870E+01 .92055625E+00 .11287153E+01 -. 167793E+0 360 .10000062E+01 .91900618E+00 .11276561E+01 -. 163329E+0 384 .10000145E+01 .91752550E+00 .11262104E+01 -. 160669E+0 408 .10000233E+01 .91610680E+00 .11244438E+01 -. 157757E+) 432 .10000322E+01 .91474390E+00 .11224079E+01 -. 154651E+0 456 .10000410E+01 .91343160E+00 .11201438E+01 -. 151396E+0 480 .10000497E+01 .91216546E+00 .11176853E+01 -. 148027E+0 504 . 10000579E+01 .91094165E+00 .11150600E+01 -. 144573E+0 528 .10000655E+01 .90975685E+00 .11167015E+01 -. 141044E+0 552 .10000725E+01 .90860816E+00 .11185765E+01 -. 137493E+0 576 .10000788E+01- .90749300E+00 .11291797E+01 -. 133902E+0 TABLE 2. T ime e v o l u t i o n o f t h e s l o t t e d c y l i n d e r . 73 F i g . 2 R e g u l a r i z a t i o n o f the P a r t i c l e Po i n t s 74 3a In c°nd on fo, the COne 7S F i g , 3b The Cone a f t e r 6 Revo lu t i on s 7 6 F i g . 4a I n i t i a l Cond i t i o n f o r the C y l i n d e r 77 F i g . 4b The C y l i n d e r a f t e r 1/8 o f a Revo l u t i on 7 8 F i g . 4c The C y l i n d e r a f t e r 1 Revo lu t i on 79 F i g . 4d The C y l i n d e r a f t e r 6 Revo lu t ions 80 CHAPTER IV ERROR ANALYSIS OF THE POTENTIAL VORTICITY AND BAROCLINIC-BAROTROPIC MODE EQUATIONS The o b j e c t o f t h i s c h a p t e r i s t o d e v e l o p a n e r r o r a n a l y s i s o f t h e s e c o n d s t a g e o f t h e a l g o r i t h m u s e d f o r c o m p u t i n g t h e p o t e n t i a l v o r t i c i t i e s . We a l s o e s t i m a t e t h e e r r o r i n t h e a p p r o x i m a t i o n s o f t h e s t r e a m f u n c t i o n s and t h e v e l o c i t i e s . I n 2 b o t h c a s e s t h e e r r o r a n a l y s i s i s i n t h e L - no rm. I f one t h i n k s o f t h e e r r o r c o m m i t t e d i n t h e a p p r o x i m a t i o n as composed o f two p a r t s , n a m e l y , t h e a p p r o x i m a t i o n e r r o r and t h e e v o l u t i o n a r y e r r o r , t h e n we p r o v e t h a t t h e a p p r o x i m a t i o n e r r o r o f t h e p o t e n t i a l v o r t i c i t i e s , w h i c h i s c o n t r i b u t e d by t h e method e m p l o y e d t o d i s c r e t i z e t h e v a r i a b l e s i n s p a c e ( l i n e a r f i n i t e 2 e l e m e n t s i n o u r c a s e ) , i s o p t i m a l , 0(h ), when f r e e - s l i p b o u n d a r y c o n d i t i o n s a r e u sed ; w i t h n o - s l i p b o u n d a r y c o n d i t i o n s t h e a p p r o x i m a t i o n e r r o r i s o f t he o r d e r 0(h). The se r e s u l t s i m p r o v e t h e e s t i m a t e s g i v e n i n [6] and [14] f o r l i n e a r f i n i t e e l e m e n t methods o f t h e s t r e a m f u n c t i o n - r e l a t i v e v o r t i c i t y f o r m u l a t i o n . I n [6] i s p r o v e d t h a t t h e a p p r o x i m a t i o n e r r o r f o r t h e r e l a t i v e v o r t i c i t y i s o f t h e o r d e r 0(h) w i t h t h e f r e e - s l i p b o u n d a r y c o n d i t i o n . [14] g i v e s an e s t i m a t e f o r t h e r e l a t i v e 1/2 v o r t i c i t y o f o r d e r . 0(h ) when n o - s l i p b o u n d a r y c o n d i t i o n s a r e u s e d . The improvement i n t h e e s t i m a t e s o f t he 81 a p p r o x i m a t i o n e r r o r i s due t o t h e f a c t t h a t t h e G a l e r k i n -C h a r a c t e r i s t i c method, by l u m p i n g t h e n o n - l i n e a r t e rms o f t h e e q u a t i o n s i n t h e m a t e r i a l d e r i v a t i v e o f t h e v a r i a b l e s , c i r c u m v e n t s t h e d i f f i c u l t i e s c a u s e d by t h o s e t e rms . W i t h r e s p e c t t o t h e e v o l u t i o n a r y e r r o r , o u r method l e a d s t o a component w h i c h i s o p t i m a l and more a c c u r a t e (by r e a s o n s t o be s e e n b e l o w ) t h a n t h a t o f c o n v e n t i o n a l t i m e f i n i t e - d i f f e r e n c e methods . However , t h e r e i s a s e c o n d component w h i c h i s o f o r d e r 0(h) and i s c o n t r i b u t e d by t h e p a r t i c l e a p p r o x i m a t i o n t o t h e e v o l u t i o n o f t h e v a r i a b l e i n t h e f i r s t s t a g e . I t i s s u s p e c t e d t h a t t h i s 0(h) t e rm i s i n h e r i t e d by many G a l e r k i n -C h a r a c t e r i s t i c methods w h i c h do n o t compute e x a c t l y t h e i n n e r p r o d u c t s . Our e r r o r a n a l y s i s o f t h e p o t e n t i a l v o r t i c i t y e q u a t i o n s i s b a s e d on t h e m e t h o d o l o g y g i v e n i n [10] and [35, C h a p . 8 ] . The e r r o r a n a l y s i s o f t h e e l l i p t i c e q u a t i o n s f o l l o w s t h e s t a n d a r d a n a l y s i s o f t h e f i n i t e e l e m e n t l i t e r a t u r e . 1. P r e l i m i n a r i e s We i n t r o d u c e some p r e l i m i n a r y d e f i n i t i o n s and r e s u l t s w h i c h a r e needed i n t h e e r r o r a n a l y s i s o f t h e p o t e n t i a l v o r t i c i t y e q u a t i o n s . In o r d e r t o s i m p l i f y o u r e x p o s i t i o n l e t u s c o n s i d e r t h e p a r a b o l i c e q u a t i o n 2 ut - V u = F i n QKIO.T] , u l r = 0 V t , ( 1 . 1 ) u(x, 0) = un(x) . 82 We d e f i n e t h e R i t z p r o j e c t i o n P^ o n t o HQh as t h e o r t h o g o n a l p r o j e c t i o n w i t h r e s p e c t t o t h e i n n e r p r o d u c t (Vu, Vv), so t h a t CVPjU, Vx) = (Vu, VX) , ^X e HQh . ( 1 . 2 ) In f a c t , PjU i s t he f i n i t e e l emen t a p p r o x i m a t i o n o f t h e s o l u t i o n o f - t h e c o r r e s p o n d i n g e l l i p t i c p r o b l e m whose e x a c t 2 s o l u t i o n i s u. The L - p r o j e c t i o n IT^ o n t o i s t h e o r t h o g o n a l p r o j e c t i o n w i t h r e s p e c t t o t h e i n n e r p r o d u c t (u, w), so t h a t ClT h u, x) = (u, X) , Vx e Hh . ( 1 . 3 ) Some r e s u l t s c o n c e r n i n g t h e a p p r o x i m a t i o n p r o p e r t i e s o f t h e p r o j e c t i o n P , p e r t i n e n t t o ou r e r r o r a n a l y s i s , a r e e s t a b l i s h e d i n t h e a p p e n d i x o f t h i s c h a p t e r . N e x t , we i n t r o d u c e t h e " d i s c r e t e L a p l a c i a n " o p e r a t o r A^, w h i c h we t h i n k o f a s an o p e r a t o r f r o m H, o n t o i t s e l f , b y n CA h 0 , x) = ~(V<JJ, Vx), tf>, X e HQH . ( 1 . 4 . 1 ) Note - t h a t A , P , = I L V 2 . ( 1 . 4 . 2 ) h 1 h F o r , w i t h x € G r e e n ' s f o r m u l a y i e l d s , f A ^ u , x) = -(VPf, Vx) = - ( V u , Vx) = (V2u,X) = C T T ^ u , x) • The s e m i d i s c r e t e e q u a t i o n o f ( 1 . 1 ) w h i c h c o n s i s t s o f f i n d i n g u, : [o,T] > H , , t a k e s t h e f o r m h oh 83 ( u h f x ) - r W x ) = ( U h F ' x ) = (Fh' x)> v* e H0h ' o r , s i n c e t h e f a c t o r s on t h e l e f t a r e a l l i n Hn, , Oh V " V h = F h - ( 1 - 5 - n uh(x,0) = (uQ(x), X) = " h 0 • ( 1 . 5 . 2 ) The c o m p l e t e d i s c r e t e s o l u t i o n if1 o f ( 1 . 1 ) i s d e f i n e d b y if1 : T —> Hnu s u c h t h a t On + 7 m (f1 = rCkhJlf1 + k V s.(kL.)F.(t +x .k) , n= 0,1,2,.., ( 1 . 6 ) h l h h n I i=l where w i t h k t h e t i m e s t e p and t = nk, r(X) and - l s . C X H m a r e ^ n 1 i 1 1 r a t i o n a l f u n c t i o n s w h i c h a r e bounded on t h e e i g e n v a l u e s o f kA^, u n i f o r m l y i n k and h, and where -{T y^m a r e d i s t i n c t r e a l numbers i n [0, 1]. D e f i n i t i o n 1 . 1 . The time discretization (1.6) is accurate of order p if and only if i) r(X) = e A + 0(XP+1) as A —> 0 , ii) and for 0 s 1 < p m . t 1 j E T m s a ; = -4iyCeA - E Aj- ) * 0(XP l) as X -> 0 . i=l X j=0 J-To i l l u s t a t e t h e mean ing o f r(X), sAX) and we s h a l l g i v e an e x a m p l e w i t h t h e C r a n k - N i c h o l s o n scheme. F o r t h i s scheme m = 1, x = 1/2 , r(X) = \ + Y2 , s(X) = 1 1 ' v ' 1 - A/2 ' 1 - A/2 ' and t h e r e l a t i o n s i) and ii) y i e l d f o r A —> 0 84 1 + A/2 A n,*3. = e + 0(X ) 1 - A/2 1 = A 1(eX - 1) + O C A 2 ) 1 - A/2 1 = A 2(ex- i - A ; + OCA; 2(2 - A/2; 1 = 2A 3 C e A - 2 - A - A 2 / 2 ; + 0(1) 4(1 - A / 2 ; A l s o , o b s e r v e t h a t | r C A ; i < 1, | sCA ; i < 1, f o r any A. The l a t t e r i n e q u a l i t i e s i m p l y t h e u n c o n d i t i o n a l s t a b i l i t y o f t h e scheme. L e t us now i n t r o d u c e t h e f u n c t i o n s v ^ , 0 £ 1 £ p, d e f i n e d by 1! , „ 4 L XJ\ " 1 v cA; = ^ A J 1 T r ( r a ) -Z-Ar > - I. r.s (X) , 1= 0 p-1 i > o J- 1 = 1 p> p XJ I t f o l l o w s f r o m D e f i n i t i o n 1 . 1 t h a t t h e d i s c r e t i z a t i o n (1.6) i s a c c u r a t e o f o r d e r p i f and o n l y i f vAX) = 0(XP 1) as X —> 0, 1= 0,1,.., p D e f i n i t i o n 1.2. The time discretization (1.6) is strictly accurate of order p^, pn £ p if v^X) = 0 . 1=0,.. -,P0-1 Remark. C r ank - N i c h o l s o n and b a c k w a r d E u l e r schemes a r e 85 a c c u r a t e and s t r i c t l y a c c u r a t e o f o r d e r p = 2 and p = 1, r e s p e c t v e l y . 2. E r r o r A n a l y s i s o f t h e P o t e n t i a l V o r t i c i t y E q u a t i o n s I n t h e e r r o r a n a l y s i s o f t h e a p p r o x i m a t e s o l u t i o n s o f t h e p o t e n t i a l v o r t i c i t i e s ( C h a p . I I . 3 .20) we r e s t r i c t o u r s e l v e s t o t h e u p p e r l a y e r , s i n c e t h e a n a l y s i s o f t h e l o w e r l a y e r s o l u t i o n s i s e s s e n t i a l l y t h e same. H e r e a f t e r we d r o p t h e s u b i n d e x i f r o m t h e e q u a t i o n , u n d e r s t a n d i n g t h a t t h e v a r i a b l e s a r e r e f e r r e d t o t h e u p p e r l a y e r . 2 1/2 L e t u s d e f i n e £ = (1 + |u| ) , where u i s t h e v e l o c i t y v e c t o r and |o| i s t h e E u c l i d e a n norm. L e t oc^, <x^ , be r e a l numbers f r o m t h e i n t e r v a l [0, 2n] s u c h t h a t c o s a ^ = 1/^ , c o s a . = u ./£ , j = 1, 2. W i t h t h i s n o t a t i o n t h e t o t a l J J d e r i v a t i v e o f q c a n be w r i t t e n as t h e d e r i v a t i v e i n t h e d i r e c t i o n T = (cosa^, cosa.^, cosa^). Thus Dq c, dq ^ „ dq . ,. dq(X(x,t;t), t) ,„ .. R e c a l l f r o m C h a p t e r I I an e q u i v a l e n t d e f i n i t i o n o f t h e t o t a l d e r i v a t i v e , Dq = _ d q _ ( x ( X t t ; s ) t s ; | ( 2 . 2 ) Dt ds v v ' ' 's=t I n t e r m s o f ( 2 . 1 ) t h e e q u a t i o n o f t h e p o t e n t i a l v o r t i c i t y i n t h e u p p e r l a y e r i s : 86 g dq(X(g*' t : s ) ' s ) = AHV2q(X(x,t;s),s) + F(X(x, t; s) \ g=t, ( 2 . 3 . 1 ) q l r qb , Vt , ( 2 . 3 . 2 ) q(x, 0) = f . ( 2 . 3 . 3 ) The c o r r e s p o n d i n g s e m i d i s c r e t e v e r s i o n o f ( 2 . 3 ) i s ( s u p p r e s s i n g f r o m t h e n o t a t i o n t h e dependence on X^(x, t;s) f o r b r e v i t y ) : VST- W + F*n > (2-4-15 Q * l r = 0 , ( 2 . 4 . 2 ) Q*(x, 0) = f , ( 2 . 4 . 3 ) * 1 h e r e Q d e n o t e s t h e f u n c t i o n Q - Q', Q' e H (Q) s u c h t h a t V t Q ' 'r " %n a n d F h = Vh + ? h ^ f - W ' V ° + 2 1/2 |u^| ) w i t h u^ b e i n g t h e s e m i d i s c r e t e a p p r o x i m a t i o n o f u. * n F rom C h a p t e r I I ( ( 3 . 1 8 . 1 ) ) t h e e x p r e s i o n o f Q i n t e rms o f t he c a n o n i c a l b a s i s o f Hn, i s uh Q*n(x) = I Q*k\(*> • Vt n e T . k A l s o , f r o m C h a p t e r I I ( 3 .20 ) t h e c o m p l e t e d i s c r e t e e q u a t i o n f o r Q i s ( d r o p p i n g t h e s u p e r i n d e x * f o r b r e v i t y ) : 87 Qn+1(x) - Q(X") ( R T , 0 H ; + AHCJ(vlQ n Ux) + ,2Q(^h) ), V V = (FZK> V ' ^ h&HOh ' ( 2 . 5 ) where y^, y ^ a r e p a r a m e t e r s f r o m /'O, 1] s u c h t h a t £ y^, and We f u r t h e r r e c a l l f r o m C h a p t e r I I I , S e c t i o n 2, t h a t Q(X^) = Q(x, tn) = I Qk4>k(x> i s s u c h t h a t (QCx, t n ; , <(>k) = (<?"(>;, 0 k C y - t n + 2 ; r ^ J ) , Vk Hence A[Q] = [B] , and f r o m Theorem 2.1 o f C h a p t e r I I I , t h e e n t r i e s o f [B] a r e g i v e n by c u b i c s p l i n e i n t e r p o l a t i o n o f A[Qn]. Lemma 2 .1 . Assuming that bounded at any instant t, the 2 scheme (2.5) is unconditionally stable in the L -norm. Proof. T a k i n g i n ( 2 . 5 ) jn+1 <Ph = y 2 Q n + (x) + V2Q(X^) y i e l d s + AH)\V(?1Qn+1+ 72Q(X^ )\\2 £ \\Fn+K\\2\\<t>h 88 Now, s i n c e (Qn+1, Q(X?)) < l/2(\\Qn+1 II 2 + IIQII2; and t h e s e cond h t e r m on t h e r i g h t hand s i d e i s p o s i t i v e , we f i n d t h a t I I Q n + I H 2 - HQH 2 < 2 A t C I I F n + K l l C l l Q n + I H + IIQIi; . Hence I IQ n + 1 H s ||Q|| + 2 A t l l F n + ' C | l s (l + C A t ; i l Q n H + 2 A t l l f n + ' C l l , ( 2 . 6 ) where t h e s t a b i l i t y theo rem o f C h a p t e r I I I ha s been u s e d t o o b t a i n t h e l a s t i n e q u a l i t y . • We n e x t w i s h t o p r o v e t h e a c c u r a c y o f t h e scheme ( 2 . 5 ) . F o r s i m p l i c i t y , we r e s t r i c t o u r s e l v e s t o t h e c a s e y^ = n^ = 1/2, w h i c h c o r r e s p o n d s t o t he c l a s s i c a l C r a n k - N i c h o l s o n scheme. To t h e a u t h o r ' s knowledge t h i s i s t h e f i r s t t i m e t h a t C r a n k - N i c h o l s o n scheme i s a n a l y s e d i n t h e c o n t e x t o f G a l e r k i n - C h a r a c t e r i s t i c methods. We p r o v e t h a t f o r A t = 0(h), o u r 2 method g i v e s an e v o l u t i o n a r y e r r o r o f o r d e r OCAt + h) i n s t e a d 2 o f OCAt ) a s one w o u l d e x p e c t f r o m t h e C r a n k - N i c h o l s o n scheme. On t h e o t h e r hand , f o l l o w i n g t he s t e p s o f o u r p r o o f i t i s e a s y t o e s t a b l i s h t h a t t h e t i m e a c c u r a c y o f t h e b a c k w a r d E u l e r scheme i s o f o r d e r 0(M+h). I n t h i s s e n s e , t h e l a t t e r scheme i s o p t i m a l when a p p l i e d i n c o n j u n c t i o n w i t h o u r method, w h i l e t h e C r a n k - N i c h o l s o n scheme i s s u b o p t i m a l . However , a s o u r m a i n t h e o r e m shows, t h e C r a n k - N i c h o l s o n scheme i s more a c c u r a t e t h a n t h e backwa rd E u l e r scheme f o r l a r g e r t i m e s t e p s . 89 I n o u r a n a l y s i s we need a p r o p e r t y o f t h e d i s c r e t e L a p l a c i a n o p e r a t o r w h i c h i s f o r m u l a t e d i n t h e n e x t lemma. Lemma 2.2 The discrete Laplacian operator - A ^ is a self-adjoint, positive definite operator in considered as 2 inner product space with respect to the L -inner product. Proof. R e c a l l t h e d e f i n i t i o n o f A, , i . e . , h (AhiP, X) = -(Vij>, Vx) , <P, X e HQh L e t x £ ^Oh ^ e s u c n t h a t t h e n (Ahip, x) = - C V 0 , Vx) = (*p, Lhx) = (ip, yp) = H 0 I I 2 , so i f A^i/( = 0 , t h e n ip = 0 . m By v i r t u e o f Lemma 2.2 t h e r e e x i s t s a s e t o f f u n c t i o n s KO "{XJ^J w n i c n a r e t h e o r t h o n o r m a l e i g e n f u c t i o n s o f -A^ , w i t h t h e a s s o c i a t e d s e t i^j} of e i g e n v a l u e s . The f u n c t i o n s •{Xj} c o n s t i t u t e an o r t h o n o r m a l b a s i s o f /7_, . Thus , any V, e Hn, i s Oh h Oh e x p r e s s e d a s Vh = I (Vh' V * k * k Assume t h a t G(X) i s 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 t h e 2 s p e c t r u m o f -A^ . F o r v e L (Q) we s e t G(Lh) = I G(-X )(v, Xj)x, , j J J J 90 t h e n IGCA. ; v l l s IIGCA, ; i l llvll , n h and by t h e P a r s e v a l ' s r e l a t i o n \G(h,)\\ = max\G(-\ .) I h j j F o r e xamp le , I I I I = max\-X .\ . The scheme ( 2 . 5 ) , w i t h y = r 2 = 1 / 2 c a n b e w r i t t e n a s Qn+1 = r(kL.)Q(X?) + ksCkA, ;F. (X?+1/2), ( 2 . 7 ) h h h h h where t h e b o u n d a r y c o n d i t i o n s have a l r e a d y been impo sed , so n+1 Q i s i n HQ^* A t d e n o t e d by k, and I + - ^ - A 2 + . r(kA.) = — - = £—+ , 2 s J s K_ , ( 2 . 8 ) r * A J - 4"* • 1 ~ — A h 2 J S'*V = r " -J-k- > l s J * K o . ( 2 ' 9 ) Lemma 2 . 3 . l / i t h P^ defined by (1.2) and q € HS(Q), 1 s s s 2+2 s 2, the following inequality holds. IP q - q\\ + h\\V(P q - q)\\ £ ChS\\q\\ , ( 2 . 1 0 ) X X .Sf & f \L where, for our problem, the constant C = KRe , with K another constant which is independent of h, and Re is the Reynolds number. 91 Proof. See A p p e n d i x C h a p t e r IV • We now p r o c e e d t o s t a t e t h e ma i n r e s u l t o f t h i s c h a p t e r w i t h r e s p e c t t o t h e r a t e o f c o n v e r g e n c e o f t h e a p p r o x i m a t e s o l u t i o n Q(x,t). I f t h e p o t e n t i a l v o r t i c i t y q(x,t) i s a r e a s o n a b l y smooth f u n c t i o n i n t i m e and s p a c e ( see c o n d i t i o n s o f Theorem 2.1 t h a t f o l l o w s ) t h e n " t h e r a t e o f c o n v e r g e n c e o f Q(x,t) t o q(x,t) i s q u a d r a t i c i n t i m e and l i n e a r i n s p a c e . A f u r t h e r a n a l y s i s o f t h e g l o b a l e r r o r shows t h a t t h e a p p r o x i m a t i o n e r r o r , w h i c h i s t h e e r r o r p e r p e t r a t e d by t h e method u s e d t o d i s c r e t i z e t h e s o l u t i o n i n s p a c e , d e p e n d s on t h e t y p e o f bounda r y c o n d i t i o n s f o r t h e p o t e n t i a l v o r t i c i t y . Thus , we p r o v e t h a t w i t h f r e e - s l i p b o u n d a r y c o n d i t i o n s t h e a p p r o x i m a t i o n e r r o r p r o d u c e d by l i n e a r f i n i t e 2 2 e l e m e n t s i s 0 ( h ) w i t h r e s p e c t t o t h e L —norm, w h i l e w i t h n o - s l i p b o u n d a r y c o n d i t i o n s and l i n e a r f i n i t e e l e m e n t s the a p p r o x i m a t i o n e r r o r i s 0 ( h ) . T h i s l o s s i n t h e r a t e o f c o n v e r g e n c e w i t h n o — s l i p bounda r y c o n d i t i o n s i s due t o t h e f a c t t h a t i n t h i s c a s e q(x,t) b e l o n g s t o (Q.) a c c o r d i n g t o t h e r e g u l a r i t y c o n d i t i o n s o f C h a p t e r I I . The p a r t i c l e a p p r o x i m a t i o n i n t r o d u c e s an e r r o r t e r m o f o r d e r 0 ( h ) r e g a r d l e s s o f t h e t y p e o f b o u n d a r y c o n d i t i o n s emp l o yed t o s o l v e t h e e q u a t i o n o f q(x, t). 92 Theorem 2 . 1 . Suppose i) q(x,t) € ffS(T2; p| W1'a>(n), 1 £ s £ 2, Vt e 10, T] i i ) qt e L2(HS) 2 3 .... dq r2,„s. „2 d q .2,.2. d q .2...1. m ) —r— e L (H ), V ^ - e L (L ), ^ - e L (H ) , " T . 2 ,3 dx dx where denotes the derivative along the characteristic curves. Then II? - 0 1 1 ^ , C^h*{ \%nSi2tQ + ^ t h 2 ( H S ) + llgllL» (HS) + "^Th2(HS) } + C 0 2 k 2 + C 0 3 H ^ L , Q T • ( 2 - U ) where and C^^ are directly proportional to Re, and s = 2 for free—slip boundary conditions and s = 1 for no—slip bounday conditions according to Chp.II (3.16.1) and (3.16.2), respectively. Proof. S e t Qn - q(x, t ) = (Qn - P2q(x, t )) + ( P gC"x, t ) - q(x, t^) = e " + P " . ( 2 . 1 2 ) By Lemma 2 . 3 l lp n H £ ChS\\q(x, t n n s 2 Q = C h S { l l g 0 l l + l l g t " L 2 r / / s ; } . ( 2 . 1 3 ) I t r e m a i n s t o e s t i m a t e 9 n e H„, . We f i r s t n o t e t h a t t h e R i t z Oh p r o j e c t i o n o p e r a t o r P^g (x, t> = w^(x, t) € , f o r g e , s a t i s f i e s t h e f o l l o w i n g Lemma. 93 Lemma 2 . 4 . Let P^ be the Ritz projection operator defined in (1.2). Then an aM* p eJS- = e L _ . 2 q dr q dx Proof. L e t Sx be a d i f f e r e n t i a l o f a r c a l o n g t h e c h a r a c t e r i s t i c s . |5x | = LU1 + \^\2)1/2. Then « » * / ,. e q (X) - q (X - ST) 1 L I M T C T 1  l i a x i ^ o 1 5 x 1 P q (X) - P q (X - 8r) 1 im ' i-|Sx|-»(A A t At->0 P2q(X) - P2q(X - 5x) dP q* lim £ rjr-, = ?—-w-— . • By v i r t u e o f t h e above Lemma, w, (x, t) s a t i s f i e s i n e a c h h i n t e r v a l It , t ,/ t h e s e m i d i s c r e t e e q u a t i o n n n+1 3w * ^ ^ ' W h = p^-af " W i * = prds * (\(-.t;S),s)\s=t ' + W / = (\Fh + \ ( P 1 ~ 1)}4s <I<Xh<*.t:s).s)\sst = F. (X, (x, t;s),s)\ - t * s,t £ t , , (2.14) h h s=t n n+1 where q* = q - q', q' e L2(HS) s u ch t h a t q' \ = qb V t € 10, T] Remark. N o t e t h a t A, w, makes s en se b e c a u s e w, (X, ) i s d e f i n e d by h h h h -94 wAX?) = P,q*(X?) , where by P,q*(X?) we d e n o t e t h e s o l u t i o n o f h h 1 h I n (VP^, VX) = (Vq*(fy, VX) , X e HQh , x £ € Q . The s o l u t i o n l / 1 o f t h e c o r r e s p o n d i n g t o t a l l y d i s c r e t e scheme i s g i v e n , a c c o r d i n g t o ( 2 . 7 ) , by 1 / 1 + 2 = + - ( 2 - 1 5 ) where £ , , = r ( k A , ,) and kh h S . . F . C x f ; = sfkA, CX. (x, t t +k/2),t+k/2) kh h h h h h n+1 n n We f u r t h e r decompose 9n a s 9 n = (Qn - l/1) + (W" - wjx, t )) = z" + Rn . (2. 16) h n He re x z " + 3 = ^u2<x?) + u s . ( i - P 7;|^ , k h h kh h h 2 3 T h e n c e i+l„ ^ ,r- h,;/«ni. . n~ I „ / T _ o »fl<7* IIZ" + III * i£khnzcxJJ;ii + kis^ i iehma - ry|f II - (2 .17) S i n c e t h e scheme i s s t a b l e , t h e n | £ . . l < 1 and |S | < 2; kh kh a l s o , we assume t h a t |£ | i s bounded. From t h e s t a b i l i t y t h e o r e m o f C h a p t e r I I I we have IZCx"; i l * (1 + CJOIIZ71!! . n A l l p i e c e s t o g e t h e r g i v e 95 \\7n+1\\ - l i z " l l £ CkWT^W + kC'WCI - Pjp- II , 1 O T where C and C a r e c o n s t a n t s . By v i r t u e o f G r o n w a l l ' s i n e q u a l i t y and Lemmae 2 . 3 and 2 . 4 IIZ""*"1!! £ C h S | l l q 0 l l + 1 | - 4 T _ L L L 2 C f f S ; } ' ( 2 . 18 ) where C = KReexp(K'T), K and K* a r e c o n s t a n t s i n d e p e n d e n t o f h and A t . The most d i f f i c u l t p a r t o f t he p r o o f i s c o n c e r n e d w i t h t he e s t i m a t e o f Rn. N o t e t h a t Rn+^ s a t i s f i e s Rn + 1 = V n + 1 - w,(X, t .) = E , ,CWCX " ; - w, e x " ; ; + h n+1 kh h h h + (E.,w,(X?) - w,(x, t .) + kS.jP.Z, p- - AuA,P.q*)(X,n) . kh h h h n+1 kh 1 h S T H h 1 h ( A l , A2, A3) (2 .19) To e s t i m a t e A l we n o t e t h a t \E,,\ < 1 , so kh \\Ai\\ £ \\w(x?) - wjx?)\\ £ i i i/cx"; - W \/X " ; I I + h h h h h h l l w . f x " ; - w . f x " ; i l . (2 .20) h h h h The f i r s t t e r m on t h e s e c o n d i n e q u a l i t y i n (2 .20) i s bounded by i i i / rx " ; - w.(x?)\i £ (i + CJOM/ 1 - wjx, t ;n , (2 .21) h h h h n a c c o r d i n g t o t h e s t a b i l i t y t heo rem o f C h a p t e r I I I . To bound t h e l a s t t e r m on t h e r i g h t hand s i d e o f (2 .20) we m i m i c t h e p r o o f g i v e n i n C h a p t e r I I I t o e v a l u a t e ( 4 . 1 6 ) . Thus , 96 i d e n t i f y i n g f u n c t i o n s C h a p t e r I I I ( 4 . 16 ) P r e s e n t A n a l y s i s u, (x) > w, J^Ax) => w h ^ ^ h ^ ' w, (x) > v h h h we have f r o m ( 4 . 1 6 ) and (4 .17 ) o f C h a p t e r I I I t h a t »V U C X " ; - wjX?)\\ £ UwjX?) - w*(X?)\\ + \\w*(X?) - v.(X?)\\ . h h h h h h h h h h h h (2 .22) From ( 4 . 7 ) - ( 4 . 1 0 ) o f C h a p t e r I I I i t f o l l o w s V = (w> + V > v* • N e x t , we n o t e t h a t t h e f u n c t i o n v(x) d e f i n e d by ( 4 . 1 1 ) i n C h a p t e r I I I t a k e s t h e f o l l o w i n g e x p r e s s i o n i n t h e p r e s e n t s i t u a t i o n : v n + 1 ( x ) = q(x, t n + 1 ) - Pla(x, t^) U s i n g t h e p r o c e d u r e d e s c r i b e d i n C h a p t e r I I I and Lemma 2.3 y i e l d s llw. (X?) - v.(X?)\\ £ CJik\Vq\ _ + C.hSk\\q(x, t ; i l _ „ h h h h 2 oo, QT 3 n s, 2,u Thus , f r o m ( 2 .20 ) and (2 .22) t h e t e rm A l i s bounded by \\W(X?) - wjX?)\\ £ (1 + C.kH]/1 - wjx, t )\\ + h h n 1 n n + C -Wc|v * g| + CjfkWqCx, t )\\ , V[t , t ]. (2 .23) 2 oo, Q j . 3 n s, 2, Q n n+1 97 The t e rm A2 may be w r i t t e n a s T a y l o r e x p a n s i o n s a l o n g t h e c h a r a c t e r i s t i c c u r v e s Y(X(x,k;k),k) = Y(X(x,k;0),0) + k-^Y(X(x,k; s), s) | s = Q + ds ^ 0 ds 2 2 Jc 2 k d Y(x(x,k;s),s)\_=Q +J K 2 / y c x r x , i c ; 0 ; , e ; d e g i v e * 2 = " P l { " " V + k - d T A ' s - 0 + 2 ^ . 2 's=0 ds tn+2 et - T ; .3 >> — r ^ T , ^ q C X J x . t ;T),T) dr\ . ( 2 . 24 ) J tn ' dx J L i k e w i s e , t o e s t i m a t e A3 we expand i n T a y l o r s e r i e s a l o n g t h e C h a r a c t e r i s t i c c u r v e s and o b t a i n t n n 2 1 d x 3 h i Utn+2 2 N . ( t n + 4 " - ^ V / 1 7 2 } • ( 2 - 2 5 ) 98 Then, A2 + A3 = 2 kj , , dJ * - l-jr i/kA^P* q <%) + * J + tfj + , ( 2 . 26 ) j=0 J J ds where TJ^, , s t a n d f o r t h e i n t e g r a l r e m a i n d e r s i n (2.24) and (2.25). L e t wjj d e n o t e Pf^—. <J(^> • The f u n c t i o n s KkA A)wJ, dsJ J a d m i t a s p e c t r a l r e p r e s e n t a t i o n i n t e rms o f t h e e i g e n f u n c t i o n s o f kA^A^. I n d o i n g so , t h e f a c t o r s lj(kA^A^) a r e g i v e n i n t e r m s o f t h e e i g e n v a l u e s o f k A^A^ by 1Q(X) = 1 - r(X) + s(X) = 0 1JX) = 1 - s(X) + (X/2)s(X) = 0 1AX) = 1 - s(X) = - X 2 W ' v ' 1 - X ' so t h a t t h e f i r s t t e r m on t he r i g h t hand s i d e o f (2.26) becomes j=0 J J ds where i s a p o s i t i v e c o n s t a n t l e s s t h a n 1. We now t u r n o u r a t t e n t i o n t o w a r d s t h e t e r m s 3?n „ „. N o t e t h a t tn+2 3 * + IIK^H s eye \\p - U L ||ds . (2.27.2) ^ tn ds A l s o 99 l l f t " l l £  c 6 k \ l l i i H A h P 3 2 " d S ' ( 2 . 2 7 . 3 ) i n d s I n v i e w o f ( 2 . 23 ) and ( 2 , 2 7 . 1 . 2 . 3 ) and t a k i n g i n t o a c c o u n t t h a t |£ , , | < 1 b e c a u s e t h e scheme i s s t a b l e , we have \\Rn+1\\ = l l l / n + 2 - wjx, t JII - III/5 - wjx, t )\\ £ -h n+1 h n C.kW]/1- wjx, t )\\ + C.hk\Vq\ _ ^ _ , s , „ , . ... 2 h n 2 ^ oo,Q + C„h k l l qCx , t ; i l r 3 n s, 2, n - .2 * An+1 ^ ^ + C4k sap II V ^ l - V + C 5 k < l ^ - i L * „ + II V H * , ^ " ' ^ d s ^ t n 1 2 , 3 Hh 1,2 1 t£s£t , ds ds n+1 ( 2 . 28 ) d2a* d3a* I t r e m a i n s t o bound t h e t e rms IIA.P, %-\\ and IIP, 5-11. From h 2 , 2 2 , 3 d s d s ( 1 . 4 . 2 ) we have ,2 * _ ,2 * IIA P -±4_|| £ IIV 2 d-4-l l . h 1 , 2 ,2 ds ds 3 # d3 * d3 * To e s t i m a t e \\PJ q II we n o t e t h a t IIVP^-^-^H £ I I V — - ^ l l . S i n c e d s d s d s P^o i s i n H t h e n by t h e d i s c r e t e F r i e d r i c h ' s i n e q u a l i t y ( c f . [ 3 8 ] ) t h e r e e x i s t s a c o n s t a n t C i n d e p e n d e n t o f h s u c h t h a t 3 * 3 * 3 * IIP3 D q ii * cnypI-d-^n s C I I V - ^ I I . ds ds ds By G r o n w a l l ' s i n e q u a l i t y ( 2 . 28 ) becomes 100 - w.(x, t s C,hSt llgll.co. <>. + C_hSllqnll , n h n+1 6 n L (H ) T 0 s,2,Q. + 1 t IIV + c8mn\vq\ n ds' ds ( 2 . 29 ) Hence , p u t t i n g t o g e t h e r ( 2 . 1 3 ) , ( 2 . 1 6 ) , ( 2 . 18 ) and ( 2 . 29 ) y i e l d s t h e e s t i m a t e ( 2 . 1 1 ) . i) N o t i c e t h a t t h e te rms m u l t i p l y i n g k a r e t h e norms o f t h e d e r i v a t i v e s o f q a l o n g t h e c h a r a c t e r i s t i c c u r v e s o f t h e f l o w . F o r c o n v e c t i o n d o m i n a t e d f l o w s t h e v a r i a t i o n o f q i s l e s s r a p i d a l o n g s u c h c h a r a c t e r i s t i c c u r v e s t h a n i n t h e t d i r e c t i o n , so t h e scheme p e r m i t s t h e u se o f l a r g e r t i m e s t e p s w i t h o u t l o s s o f a c c u r a c y . i i ) The l a s t t e rm i n ( 2 . 29 ) comes f r o m t h e p a r t i c l e a p p r o x i m a t i o n t o t h e e v o l u t i o n o f t h e v a r i a b l e s a l o n g t h e c h a r a c t e r i s t i c c u r v e s o f t h e f l o w ; t h e r e b y , t h i s t e rm b e l o n g s t o t h e e v o l u t i o n a r y e r r o r o f t h e a p p r o x i m a t i o n r a t h e r t h a n t o t h e a p p r o x i m a t i o n e r r o r o f t h e f i n i t e e l e m e n t s , w h i c h i s g i v e n s ui by 0(h ). From t h i s p o i n t o f v i e w , i f one t a k e s A t = 0(h ), 1/2 s m £ 1, t h e n t h e C r a n k - N i c h o l s o n scheme g e t s c l o s e r t o i t s " o p t i m a l i t y " w i t h l a r g e r t i m e s t e p s . i i i ) A s i m i l a r a n a l y s i s o f t h e b a c k w a r d E u l e r scheme shows t h a t t h e t i m e e r r o r i s now Oik), w h i c h i s o p t i m a l . From t h i s and r emark i i ) one deduce s t h a t t h e C r a n k - N i c h o l s o n scheme i s more a c c u r a t e w i t h l a r g e r t i m e s t e p s . Remarks. 2 101 iv) Our a n a l y s i s shows t h a t G a l e r k i n - C h a r a c t e r i s t i c methods f o r t h e c o n v e c t i o n - d i f f u s i o n e q u a t i o n i n w h i c h t h e i n n e r p r o d u c t s a r e n o t e x a c t l y c a l c u l a t e d , a s i n t h e methods p r o p o s e d by M o r t o n e t a l [27] and Benque e t a l [4] , l e a d t o 2 an e r r o r e s t i m a t e i n t h e L - norm w i t h a t e rm o f o r d e r 0(h) a p p e a r i n g i n t h e e v o l u t i o n a r y component o f t h e e r r o r , r e g a r d l e s s o f t h e d e g r e e o f t h e f i n i t e e l e m e n t s u s e d . Thus one m i g h t c l a s s i f y t h e a p p r o x i m a t e G a l e r k i n - C h a r a c t e r i s t i c methods 2 a s 0(h) methods i n t h e L - norm. 3. E r r o r A n a l y s i s o f t h e B a r o t r o p i c and B a r o c l i n i c Modes I n t h i s s e c t i o n we s t u d y t h e a p p r o x i m a t i o n e r r o r o f t h e b a r o t r o p i c , $, and b a r o c l i n i c , * , modes. B o t h modes s a t i s f y t o s e c o n d o r d e r e l l i p t i c e q u a t i o n s ( r e c a l l e q u a t i o n s ( 1 . 7 ) - ( 1 . 9 ) o f C h a p t e r I I ) . Our a n a l y s i s i s ba sed on t h e s t a n d a r d t h e o r y o f t h e a p p r o x i m a t i o n o f e l l i p t i c p r o b l e m s by f i n i t e e l e m e n t s ; a d e t a i l e d a c c o u n t o f i t c a n be f o u n d i n C i a r l e t [ 1 2 ] . F o r c o m p l e t e n e s s we w r i t e h e r e t h e e q u a t i o n s s a t i s f i e d by 4> and t h e i r f i n i t e e l emen t a p p r o x i m a t i o n s r e s p e c t i v e l y . Thus C V 2 - A 2 ; * = 0 , i n Q_ , s T (3. 1) 102 ( V 2 - A 2 j ) * a = Q l - q 2 , i n Q T * |_ = 0, V t , a I V 2 $ = b , i n Q r * l r = 0 , V t , whe re ( 3 . 2 ) ( 3 . 3 ) H l q l + H2q2 B - H\ +H2 ~ F ' ( 3 " 4 ) * f x , t ; = * Cx, t ; + c a ; * f x ; , ( 3 . 5 ) a s and C(t) i s d e t e r m i n e d by Chap. I I ( 1 . 6 ) . The f i n i t e e l e m e n t a p p r o x i m a t i o n s s a t i s f y (™sh> V V +  x2(%n> V = 0 ' V*h € * 0 h ' ( 3 - 6 ) where ¥ , € S w i t h c - 1. sh ch F o r any i n s t a n t £ e !P ( h e r e a f t e r we d r o p t h e s u p e r i n d e x n ) , r v ^ , % ; + A 2 C * a h , <f>h) = - < Q l - Q 2 , 4>h), V 0 h £ / / 0 h , ( 3 . 7 ) C V V = (bh, 4>h), % e H 0 h , ( 3 . 8 ) where * and $ € /7_, . ah Oh N o t e t h a t 0^ - Q £ i n ( 3 . 6 ) and b f t i n ( 3 . 8 ) a r e f i n i t e e l e m e n t a p p r o x i m a t i o n s o f - i n ( 3 . 2 ) and b i n ( 3 . 3 ) , 103 r e s p e c t i v e l y . Due t o t h i s f a c t some p r e l i m i n a r y work i s r e q u i r e d b e f o r e p r o c e e d i n g w i t h t h e s t a n d a r d a n a l y s i s . L e t <p(h) and <p be t h e s o l u t i o n s o f t h e e l l i p t i c p r o b l e m s 2 V <p(h) + c<p(h) = &h, i n fi <p(h)\T = 0 . 2 \7 <p + c<p = 6 , i n fi (3. 9 . i : ( 3 . 9 . 2 ) <p\r = 0 , 1 1 where c i s a p o s i t i v e c o n s t a n t , 9. e H, c H (Q), 9 e H (Q) and h h 2 s u c h t h a t 9^ —» 9 w e a k l y i n L (Q). L e t be t h e f i n i t e e l e m e n t a p p r o x i m a t i o n t o <p(h) i n S i n c e fi i s a c o n v e x 2 p o l y g o n a l doma in , t h e n by r e g u l a r i t y t h e o r y (p(h) , cp € H (Q) f] Hg(Q) and *9n2.2.a £ c 2 l l e " - ( 3 - 1 0 - 2 ) where C a r e c o n s t a n t s i n d e p e n d e n t o f h. Lemma 3.2. <p —> cp strongly in H^(Q). Proof. From ( 3 . 9 . 1 ) and ( 3 . 9 . 2 ) one o b t a i n s 2 -V (<p - <p(h)) + c(<p - <p(h)) = 0 - 9h i n fi , ( 3 . 11 ) <p - <p(h)\ = 0 104 S i n c e 9^ —> 6 w e a k l y i n L (Q), <p(h) —> <p w e a k l y i n H (£2) and by t h e u s u a l c o m p a c t n e s s argument lim \l<p - <p(h)\\ = 0. m Lemma 3.2. Let <p(h) and &p be the solutions of (3.9.1) and (3.9.2), respectively. Let <p^ e be the finite element approximation to <p(h). Then the following inequality holds " ~ ^i,2,n * c , l e - V V"J ^ ( h ) ~ ^i,2,n • <3-12) where C = C(Q) is a positive constant. Proof. L e t u. , </>, € Hn. , t h e n h h Uh (v(9h - <ph), vu h ; + c(<ph - <t>h, uh) = (V(<p(h) - <t>h), Vph) + c(<p(h) - <t>h, ph) . (3 . 13) T a k i n g = <p^ - <f>^ and u s i n g F r i e d r i c h ' s i n e q u a l i t y ( 3 . 1 3 ) y i e l d s ^h-^i,2,Q^Ki^(h) -^i,2,n- ( 3 - 1 4 ) We n e x t w r i t e " Vl,2,fi * I* - *<h)ti,2,Q + **(h) ~ Vl,2,fi From ( 3 . 1 4 ) 105 " - *hh,2.a * ^ ~ * ( h ) l i . 2 , a + K 2 i n f ^ ( h ) - *hl1.2.a *neHOn ( 3 . 15 ) I t r e m a i n s t o bound t h e f i r s t t e rm on t h e r i g h t hand s i d e o f ( 3 . 1 5 ) . N o t e t h a t by v i r t u e o f t h e L a x - M i l g r a m t heo rem a p l i e d on ( 3 . 1 1 ) one o b t a i n s " • ^ l l 1 2 , 0 4 j f 3 , f l - f l h l - U D -From t h e d e f i n i t i o n o f t he d u a l norm ( s ee Chap. I I ) f o l l o w i n g by an a p p l i c a t i o n o f Schwarz and F r i e d r i c h ' s i n e q u a l i t i e s one g e t s '* " * ( h ) i 1 . 2 . a * V 9 " V • ( 3 - 1 6 ) By s u b s t i t u t i n g ( 3 . 16 ) i n t o ( 3 . 15 ) y i e l d s ( 3 . 1 2 ) . • We a r e now i n c o n d i t i o n s t o a p p l y t h e a r gument s o f t h e s t a n d a r d f i n i t e e l e m e n t a p p r o x i m a t i o n t h e o r y f o r e l l i p t i c p r o b l e m s . Theorem 3 .1 . Suppose i) Conditions i) - iii) of Theorem 2.1 hold. ii) Regularity conditions (3.15.3) and (3.15.4) of Chapter II hold. Then for any instant t e T J n " * a " * a h " * C / { *Q1 ~ Q2ti- + W*l ~ V ' - . Q . } + *" °" T ' 106 II* - * II £ Ch2i lib, II + |vfg , - q.)\ _ i + H.O.T. n Z y t\ 1 Z 03, J ™ s - * s h * * C / W 3 / 2 . 2 . T ' ( 3 - 1 ? ) where are positive constants independent of h but directly dependent on Re, g = 1 on the boundary T and H.O.T. stands for higher order terms. 2 Proof. We f i r s t e s t i m a t e t h e e r r o r f o r * and * i n t h e L - n o r m a u s i n g d u a l i t y i d e a s . S p e c i f i c a l l y , we emp loy t h e A u b i n - N i t s c h e method ( s ee C i a r l e t [ 1 2 ] , Theorems 3 . 2 . 4 and 3 . 2 . 5 ) . I n so d o i n g , we have t h a t a t any i n s t a n t t I* - * .11 £ tf,hll* - * J I , . _ , ( 3 . 1 8 . 1 ) a ah 1 a ah 1,2, fi II* - * J I £ W_h l l * - $ j | , ( 3 . 1 8 . 2 ) h 2 h 1,2,0. where a r e c o n s t a n t s i n d e p e n d e n t o f h. From Lemma 3 . 2 , t h e e q u i v a l e n c e be tween I ° I j 2 fi a n 0 " " ° " l 2 fi * n HQ(^)> t h e a p p r o x i m a t i o n p r o p e r t i e s o f H^ (See Chap. I I , S e c t . 3 . 2 ) and Theorem 2.1 we g e t '*a ' *ah" * C l h 2 { "Ql ~ <V + | V ^ 1 -*2)[o3,QT } + °-{ "bh" + | V ^ 1 " + II* - #|| £ CJi2\ lib. II + IvCg, - q0)\ n } + H.O.T h 2 I  1 2 oo,QT ' 2 The e s t i m a t e o f i n t h e L -norm i s g i v e n by a d i r e c t a p p l i c a t i o n o f Theorem 8 .7 i n Oden and Reddy [ 2 9 ] . Thus , 107 3/2, 2, Q ' Now, f r o m Theorem 3.1 and t h e r e l a t i o n s (Chap. I I ( 3 . 11 ) and ( 3 . 2 2 ) ) i t f o l l o w s i n m e d i a t l y t h a t To e s t i m a t e t h e v e l o c i t i e s we u se t h e r e l a t i o n s (Chap. I I ( 3 . 1 2 ) , ( 3 . 2 3 . 2 ) ) and ( 3 .16 ) o f t h i s c h a p t e r t o g e t Theorem 3. 1 s t a t e s t h a t t he a p p r o x i m a t e s o l u t i o n s o f t he b a r o t r o p i c and b a r o c l i n i c modes c o n v e r g e q u a d r a t i c a l l y i n t he 2 L —norm t o t h e e x a c t s o l u t i o n . The same d e g r e e o f c o n v e r g e n c e i s o b t a i n e d f o r t h e a p p r o x i m a t e s t r e a m f u n c t i o n s b e c a u s e t h e y a r e l i n e a r c o m b i n a t i o n s o f t h e modes. However , s i n c e t h e v e l o c i t y component s a r e e x p r e s s e d i n t e r m s o f t h e f i r s t d e r i v a t i v e o f t h e s t r e a m f u n c t i o n s , t h e n i t i s n u m e r i c a l l y r e a s o n a b l e t o e x p e c t t h a t t h e y w i l l c o n v e r g e w i t h a l o w e r r a t e . I n f a c t , ( 3 . 20 ) shows t h a t t h e r a t e o f c o n v e r g e n c e o f t h e v e l o c i t y i s o f f i r s t o r d e r . 4. N u m e r i a c l Exper iments I n o r d e r t o v e r i f y t h e v a l i d i t y o f o u r e r r o r e s t i m a t e s f o r ( 3 . 19 ) l l u . - U. l l = 0(h). ( 3 . 20 ) l l 108 t h e v o r t i c i t y and t h e s t r e a m f u n c t i o n , we p r e s e n t h e r e t h e r e s u l t s o f some n u m e r i c a l e xamp le s . We s h o u l d e m p h a s i z e t h a t t h e p u r p o s e o f s u c h e x p e r i m e n t s i s n o t t o s i m u l a t e a ' r e a l ' s i t u a t i o n o f a b a r o c l i n i c o c e a n , i n s t e a d we o n l y w i s h t o i l l u s t r a t e o u r t h e o r e t i c a l a n a l y s i s o f t he G a l e r k i n - C h a r a c t e r i s t i c method p r o p o s e d i n t h i s t h e s i s . H o p e f u l l y , c o m p u t a t i o n s w i t h t h i s method i n more r e a l i s t i c o c e a n o g r a p h i c a l s i t u a t i o n s w i l l be c a r r i e d o u t i n t h e f u t u r e . We have d e c i d e d t o t e s t t h e method w i t h t h e n o - s l i p b o u n d a r y c o n d i t i o n f o r two r e a s o n s . The f i r s t one i s b e c a u s e t h i s b o u n d a r y c o n d i t i o n i s c o m p u t a t i o n a l l y more d i f f i c u l t . The v a l u e s o f t h e v o r t i c i t y a t t h e b o u n d a r y a r e unknown, t h e y have t o be c a l c u l a t e d as p a r o f t h e n u m e r i c a l p r o c e d u r e . ( see C h a p t e r I I ) . The s e c o n d one i s b e c a u s e ou r e r r o r e s t i m a t e f o r t h i s c a s e i s O(Reh) no m a t t e r i f u b e l o n g s t o a s p a c e o f o r d e r h i g h e r t h a n 1. I n o u r e x p e r i m e n t s , t h e doma in fi i s a s q u a r e (1000 km x l O O O k m ) , w i t h no r o t a t i o n and c o n s t a n t d e n s i t y . No te t h a t f r o m o u r a n a l y s i s , r o t a t i o n and number o f l a y e r s do n o t i n t r o d u c e any a d d i t i o n a l n u m e r i c a l f e a t u r e t o t h e n o n r o t a t i n g and homogeneous c a s e . Under t h e s e c o n d i t i o n s i t i s e a s y t o c hoo se an e x a c t s o l u t i o n (OJ,I/») t o t e s t t h e n u m e r i c a l l y c a l c u l a t e d s o l u t i o n . 2 The e x a c t s o l u t i o n i s , w i t h V 0 = w , 2 nxl 2UX2 ip(x,t) = A(t)sin —^— sin —^— , . *'r = *n'r = 0 • 109 _ 2 27rx. 0 7ix_ 2nx_ _ n x . o)(x,t) = — — A ( t ) [ c o s — ^ — sin —j— + cos—j— sin —^—J , u(x, t ) \ r = ub , where A(t) = A(1 - exp(-t/T)). By d i r e c t s u b s t i t u t i o n o f w and iji i n t o t h e e q u a t i o n D W = A.A + F Dt H one compute t h e f o r c i n g f u n c t i o n F(x, t). We have r u n s e v e r a l e x p e r i m e n t s w i t h d i f f e r e n t v a l u e s o f h, R e y n o l d s number Re, and C o u r a n t — F r i e d r i c h — L e w y c o n d i t i o n At " f ~ f h " CFL = max|u|—r—. T a b l e s 3 , 4 , 5 g i v e t h e v a l u e s o f ln-h . ~, — — ~ ~ j l f | | f o r t h e v o r t i c i t y and s t r e a m f u n c t i o n i n e a c h e x p e r i m e n t . The e x p e r i m e n t s were r u n u n t i l A(t) = 0.64A, w i t h At c o n s t a n t f o r a l l t h e e x p e r i m e n t s , e x c e p t t h o s e w i t h h 2= 50, whe re i t was d e c r e a s e d i n o r d e r t o make t he s o l u t i o n c o n v e r g e . max|u| was a l w a y s c o n s t a n t . C o n s e q u e n t l y , t he CFL c o n d i t i o n i n c r e a s e s a s h 2 t a k e s t h e v a l u e s 20, 25, .. . ,40 and d e c r e a s e s a s h 2 = 50. A l i n e a r r e g r e s s i o n o f t h e v a l u e s o f t h e v o r t i c i t y e r r o r i n T a b l e s 3, 4, 5 shows t h a t II w - o> II L N IT—!T = A. + B .hRe , 1 = 3, 4, 5 , Hull 1 1 where 4 0 = - 2 . 7 5 , B 0 = 0.95, A,, = -3.19, B-. = 0.87, Ac = 3 3 4 4 5 -3.96, Bc = 0.80 . A c c o r d i n g t o Theorem 2. 1 t h e c o e f f i c i e n t s B s h o u l d be e q u a l t o 1; howeve r , t h e r e e x i s t s d i s c r e p a n c y i n t h e v a l u e s o f B^ and B^. T h i s m i g h t be due t o t h e s m a l l number o f p o i n t s 110 used i n f i t t i n g the s t r a i g h t l i n e i n both cases . The v a l u e of By wh ich was a d j u s t e d w i t h a number of p o i n t s tw ice l a r g e r than the one used i n B^ and B^ , i s v e r y c l o s e d to 1. S i m i l a r l y , a l i n e a r r e g r e s s i o n f o r the the v a l u e s of the s t ream f u n c t i o n e r r o r shows tha t these a re p r o p o r t i o n a l to 2 h Re, w i t h s i m i l a r d i s c r e p a n c i e s as i n the v o r t i c i t y case . F i g u r e s 5a and 5b show the exact and the approximate s o l u t i o n s , r e s p e c t i v e l y of the stream f u n c t i o n i n the run Re = 1000, h * = 50 At = 2.30h. F i g u r e 5c i s the p o i n t w i s e e r r o r of the st ream f u n c t i o n . F i g u r e s 6a and 6b r e p r e s e n t the exact and approx imate s o l u t i o n s of the the v o r t i c i t y i n t h e same run . F i g u r e 6c i s the p o i n t w i s e e r r o r of the v o r t i c i t y . We observe i n t h i s f i g u r e tha t the v o r t i c i t y e r r o r s are l a r g e r at and near the b o u n d a r i e s , p a r t i c u l a r l y the e a s t e r n and n o r t h e r n b o u n d a r i e s . T h i s f a c t deserves more t e s t i n g w i t h d i f f e r e n t type of end c o n d i t i o n s f o r the s p l i n e procedure . In the p r e s e n t exper iments we have used the second type of end c o n d i t i o n s recomended i n [4] , we have a l s o observed tha t the n a t u r a l end c o n d i t i o n s gave ve ry poor r e s u l t s . I l l h " 1 CFL V o r t . E r r o r S t r . F u n c . E r r o r Re. Number 20 1.27 - 1 . 2 5 - 4 . 26 100 25 1.60 - 1 . 4 2 - 4 . 71 30 1. 90 - 1 . 55 - 5 . 01 35 2 .22 - 1 . 70 - 5 . 51 40 2 .37 - 1 . 90 - 5 . 7 3 50 2 .30 - 2 . 10 - 6 . 2 0 T a b l e 3. L o g a r i t h m o f r . m . s . e r r o r s h " 1 CFL V o r t . E r r o r S t r . F u n c . E r r o r Re. Number 35 2. 22 - . 6 3 - 3 . 50 1000 40 2. 37 - . 7 8 - 3 . 7 7 II 50 2. 30 - . 90 - 4 . 20 II T a b l e 4. L o g a r i t h m o f r .m. s . e r r o r s h " 1 CFL V o r t . E r r o r S t r . F u n c . E r r o r Re. Number 35 2. 22 - . 3 6 - 2 . 50 3000 40 2. 37 - . 48 - 2 . 81 II 50 2 .30 - . 65 - 3 . 2 5 II T a b l e 5. L o g a r i t h m o f r .m. s . e r r o r s 112 F i g . 5a Exact Stream Funct ion S o l u t i o n 113 F i g . 5b Approximated Stream Funct ion 114 5c Po in tw i se E r r o r f o r Stream Funct ion 115 Fl9. 6a F y , „ + 116 117 118 APPENDIX TO CHAPTER IV The m a i n p u r p o s e o f t h i s a p p e n d i x i s t o p r o v e Lemma 2 . 3 ; i n o r d e r t o do so we n e e d some r e s u l t s on t h e t r a c e o p e r a t o r s y . and t h e i r a p p r o x i m a t i o n p r o p e r t i e s by f i n i t e e l e m e n t s . These r e s u l t s a r e e n u n c i a t e d w i t h o u t p r o o f . 1. The T r a c e Theorem 2 L e t n be a bounded L i p s c h i t z c o n t i n u o u s s u b s e t o f R w i t h b o u n d a r y T. We d e f i n e t h e t r a c e o p e r a t o r s y ^ and y ( j an i n t e g e r £ 1), f o r a f u n c t i o n u d e f i n e d i n Q by y Q u = u | r , 2 _ j 1=1 j i (A. 1) ( A . 2 ) where a r e t h e component s o f t h e u n i t n o r m a l v e c t o r on T. Theorem A l . Let SI be a domain defined as above and s a real number. Then the trace operators y^., 0 s j < s - 1/2, j integer, can be extended to continuous linear operators mapping H (Q) onto HS 3 ^^2(D. Moreover, the operator y : = \t Q> VJ' ' ' '*p^ i s a j. • i- rrS/n\ 4. S-l/2 ..S-j-1/2 . continuous linear mapping of H (Q) onto f | j = n ^ (F). Proof. L i o n s and Magenes [20 p. 42] • As a c o n s e q u e n c e o f t h i s theorem we have t h a t " t h e r e e x i s t p o s i t i v e c o n s t a n t s c and c s u ch t h a t 119 c . Hull _ _ £ lly .ull . _ _ £ c . llu II , n • (A.3) 1 s,2,Q j s-j-l/2,2,Q 2 s,2,Q r-l/2 The norm i n H (D i s d e f i n e d by If II _ _ = inf \lv\l _ _ . (A. 4) r-l/2,2,Y r , 2, fi r0v=f I n we have t h e f o l l o w i n g c h a r a c t e r i z a t i o n Kerj^ Kerj 2 = H2(Q) . 2. B o u n d a r y F a m i l i e s . C o n s i d e r t h e f a m i l y o f f i n i t e d i m e n s i o n a l s p a c e s {Q^}- L e t e be a f i n i t e d i m e n s i o n a l s p a ce s u c h t h a t H. c //Yn; P , C n ; C H, , 1 + 1 2: 02 £ 0 2 n R e c a l l t h e a p p r o x i m a t i o n p r o p e r t i e s o f ff^ ( C h a p . I I ! inf llu - *|| . < Ch0"lluII „ _ (A. 5) „ s, 2, fi r, 2, fi X*Hh a- = min( 1 + 1-s, r-s) ( A . 5 ) h o l d s f o r a r b i t r a r y r, s e ven r, s £ 0 ( c f . [ 1 ] ) . 120 Theorem A2. Let Q. be a domain defined as above. Then the traces of functions U e with property (A.5) generate families H^(D of fund ions on T such that 1 + 1/2 > m - 1/2 > 0 . Proof. F o r a p r o o f o f t h i s t heo rem see B a b u s k a - a n d A z i z [ 1 ] . • The f o l l o w i n g r e s u l t s on t h e a p p r o x i m a t i o n p r o p e r t i e s o f t h e b o u n d a r y s p a c e s H^CD a r e due t o B r amb le and S c h a t z [ 7 ] . L e t \yj}J=Q be a s y s t e m o f n o r m a l t r a c e o p e r a t o r s o f o r d e r q^, and l e t t h e r e be g i v e n a f i n i t e d i m e n s i o n a l s p a c e H^(Q), t h e n t h e f o l l o w i n g i n e q u a l i t i e s h o l d : i) For u ., A . r e a l numbers such that 0 £ 1 - q . - 1/2, 0 £ J J J A . £ J + 1/2 - q . - u . J J J 1-1 q .+u . inf ( £ h J J lly .u - y .(/II ; < h J J 1-1 q .+u .+A . C £ h J J J lly ull , (A. 6) j=o J W 2 ' where C is a constant independent of h and y . ii) Let a. be a real number such that <7j_^ + 2/2 < a < J + I and 0 < u . £ 1 - q . - 1/2. Then f o r u € /Y°Y£2; J J i _ i V1/ oc-1/2 inf V h J J lly .u - y l/ll . _ £ Ch Hull _ _ . (A. 7 ) UeHh j=0 J 3 *f2'r a ' 2 > n 121 4. P r o o f o f Lemma 2 . 3 To prove Lemma 2 . 3 we r e c a l l that the R i t z p r o j e c t i o n P^g = w s a t i s f i e s f o r q € HS(Q) f| H2(Q) , 1 £ s £ 1 + 1 (Vw, VX) = CVg, V*; , V* 6 HQh , ( A . 8 . 1 ) v | r - q b h . ( A .8.2 ) We f i r s t e s t i m a t e the term llvYw - q)\\. Towards t h i s end, we d e f i n e two a u x i l i a r y problems. Le t w, e H,(Q) be the f i n i t e element a p p r o x i m a t i o n of 1 h -AuV2q = f , (A .9 . 1) H q\T = qh • ( A . 9 . 2 ) Then (Vw., VX) = CVg, V*; - (Vw , Vx) , *X e Hn, , ( A .10) where w e H (Q) i s such that w^lp = q f o . L e t w^ be the f i n i t e element a p p r o x i m a t i o n of - V 2 g ' = 0 , ( A . 10 . 1 ) Then g ' l r = q b h - qh . (A .10.2] (Vwy VX) = - f V w 4 , Vx) , Vx e HQh , ( A .11) 122 where w . € H (Q) i s s u c h t h a t w. |_ = q, , - q, . 4 4 Y oh o By t h e t h e o r y o f f i n i t e e l e m e n t s f o r e l l i p t i c p r o b l e m s ( c f . [26], T h e o r . 8.5) we have liq - wJI. _ _ £ Ch0"llqll . _ , cr = mind, s-1), (A. 12) 1 1,2,0. s, 2,Q where C depends on A . By v i r t u e o f t h e L a x - M i l g r a m t h e o r e m and n t h e t r a c e t heo rem , w 3 " ] > 2 i n i C " b " V i / 2 l 2 , r • ( A- 1 3 ) S i n c e q ^  i s t h e f i n i t e e l e m e n t a p p r o x i m a t i o n o f q ^ e H^Cfi), t h e n using A. 6 with q = 0 and X^ . = r llvJI. , _ £ Chr~1/2\\q.\\ , _ i C_h S _ 1 llqll , _ . (A. 14) .3 2,2, fi 2 b^ r+2/2,2, T 2 s, 2, fi Now, IIVCW -q )\\2 = cvcw - q;, vcw - q;; = = CVCw - q;, ~V(V - w ) + V(w1 - q)) £ IIVCw - q)\\ IIVCw - v ;il + HVfw - q)\\ IIVCw^ - qjll . Hence Wv - q;il * II fv - V"2,2,fi + llw^  - qHlf2>Q = "w3Ui,2,n + nwi ~ *h.2.a From (A.12) and (A.14), t h e f o l l o w i n g e s t i m a t e h o l d s IIVCw - q;il £ ChS 2 l lqll s 2 fl • (A. 15) 123 It remains to estimate the term llv - qll. We proceed by defining an auxiliary el l iptic problem with homogeneous boundary 2 2 conditions. Let \p e H (U) and tp e L (Q). Furthemore, -AUV2\I) = <p in f2 , (A. 16. 1) n 0 1 = 0 . (A.16.2) By regularity theory \\if>\l2 2 n s Cllf>ll . (A. 18) Then Cw - q, <p) = -(w - q,V2ip) = CVCw - q), Vip) - <-|jj-p 7Q(^ ~ For any x e HQh(Q) (w - q, <p) = CVCw - q;, V C 0 - - < - | n L ' - q;>p (A.19) By virtue of the following items: i) Approximation properties of H ^  (A. 5) I V C 0 - s K2hll^ll2 2 n = /C2hll<pll i i ) The trace theorem l l l I / 2 , 2 ( r S V " l 2 ) 2 , Q S V " i i i ) Inequality A.6 and the trace theorem 124 where K_ are constants, the inequality (A.19) becomes (w - q, <p) £ K6h\\V(w - q)\\ llipll +K7hS\\q\\s 2 Q \\<p\\ . Taking <p = w - q and using (A. 15) the Lemma is proven. • 125 CHAPTER V CONCLUSIONS We have formulated and analysed an algorithm of the type Galerkin-Characteristic methods to integrate convection-dominated diffusion problems. The proposed algorithm has been combined with a mixed finite element method, similar to the one given in [11], to produce a formulation of the quasi-geostrophic equations (potential vorticity — stream function equations) for a mid—latitude baroclinic ocean. The integration of the transport-diffusion equations by our algorithm is essentially a two stage process. The f i rst stage corresponds to the integration of the advection operator along the characteristic curves of the flow in combination with Galerkin method. For this stage, we show that a reinterpretation of the usual Galerkin-Characteristic method - in terms of particle methodology, with rectangular grids, yields a computationally efficient scheme, which consists of interpolating by cubic splines the grid point values of a functional of the dependent variable at the departure points of the particles. The scheme thus devised is conservative, unconditionally stable and convergent. Our error analysis in the maximum norm for this stage proves that for sufficiently smooth functions the approximate solution is super convergent at the foot of the characteristic curves. To assess the performance of our algorithm for the hyperbolic stage we have 126 carried out two types of advective experiments. The f irst one is a fairly hard problem which consists of advecting a cone in a rotating flow field. Munz [28] has reported the results of this problem obtained by high resolution finite difference schemes of the type total variation diminishing (TVD) such as MUSCL, UNO, etc. , which are considered to be the best finite difference schemes for hyperbolic problems. A visual comparison of our results with those portrayed in figures 6 to 16 of [28] shows that: i) Our scheme is able to keep the shape of the cone better than any high resolution scheme. i i ) The 'numerical diffusion' of our scheme is lower than that of the high resolution finite difference schemes, with the possible exception of the superbee scheme. i i i ) Our scheme exhibits small wiggle activity at the base of the cone, whereas the high resolution finite difference schemes are wiggle free because they possess the TVD property. iv) Our scheme is able to use much larger time steps than any high resolution finite difference scheme because ours is unconditionally stable with respect to the 2 L —norm. This property is responsible for keeping the wiggles under strict control. Our second numerical experiment consists of advecting a 'slotted' cylinder in a rotating flow field. This is a very hard problem because of the discontinuities on the lateral surface and the 'slotted' region, respectively. A visual 127 comparison of our results with those reported in [37], using SHASTA and modified SHASTA schemes, and [28], specifically figures 6 to 16, using the aforementioned high resolution schemes, yields the same conclusion as in the cone experiment. The second stage corresponds to the time progression of the algorithm via the diffusion mechanism starting with the output of the previous stage. The time progression is carried out in our analysis by the Crank-Nicholson scheme. The error 2 analysis with respect to L -norm, based on techniques developed in [10] and [35], reveals for At = 0(h) the presence of a term of order 0(h) in the evolutionary component of the error. This term is due to the particle approximation of the f i rst stage and is suspected to be inherited by those Galerkin-Characteristic methods which approximate the inner products by alternative techniques, such as the ones proposed in [27] and [5]. In this sense, the Crank—Nicholson scheme is suboptimal when used in conjunction with our method; however, for larger time steps it leads to more accurate results than the backward Euler scheme , which appears to be optimal for At = 0(h). Our error analysis technique is more general than the ones used in [13], [31] and [33], which cannot be used for the Crank—Nicholson scheme. The constants C Q 1 and C Q 2 in Chap.IV (2.11), which multiply the terms of the evolutionary error, are the product of the exponential constant of the Gronwall's inequality with the norm of the derivatives of the variable along the characteristic curves. For convection dominated flows the 128 speed of variation along the characteristics is less rapid than in the t direction, so the algorithm permits larger time steps without loss of accuracy. On the other hand, since the algorithm in the second stage is also unconditionally stable, then the exponential constant of the Gronwall's inequality is less than one; so that as time progresses the influence of the evolutionary error on the global accuracy of the method decreases, and eventually for T —> oo the' error of the approximate solution is the approximation error. The latter 2 type of error is 0(h ) with free-slip boundary conditions, and 0(h ) with no—slip boundary conditions. These estimates are one order higher than previous estimates given in the literature. Finally, we prove that the approximation error 2 2 with respect to L -norm for the stream functions is 0(h ), 2 which is optimal, while the L —norm approximation error for the velocities is 0(h). Numerical experiments with no-slip boundary conditions shows the validity of our error estimates. As a final remark, we should mention that our analysis can be extended to approximate the solution of Navier—Stokes equations by the proposed algorithm. 129 REFERENCES 1. Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975) 2. Babuska, I. and Aziz, A.K.: "Survey Lectures on the Mathematical Foundations of the Finite Element Method," in Aziz, A.K., Edi., The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations. Academeic Press, N.Y. (1972) pp 5-359 3. Bardos, C. , Bercovier, M. , and Pironneau, 0. : The vortex method with finite elements. Math. Comp. 36, 119-136 (1981) 4. Behforooz, G. H. and Papamichael, N. : End conditions for cubic splineinterpolation. J. Inst. Maths. Applies., 23, 355-366. (1979) 5. Benque, J. P. , Ibler, B. , Keramsi, A. , and Labadie, G. : A Finite Element Method for Navier-Stokes Equations. -Proceedings of the third international conference on finite elements in flow problems. Banff, Alberta, Canada, 10-13 June(1980) 6. Bermejo, R. : "An analysis of a Quasi-Geostrophic Finite Element Method," in Graham, G.A.C. and Malik, S. L. Eds., Continuum Mechanics and its Applications. Hemisphere Publ. Co. New York (1989) pp 825-835 7. de Boor, C. : A Practical Guide to Splines.Springer-Verlag. New York, Berlin, Heidelberg (1978) 130 8. Boris, J.P., and Book, D.L.: Flux corrected transport. SHASTA. J. Comp. Phys. 11, 36-39 (1973) 9. Bramble, J.H. and Schatz, A.: Least squares method for 2mth order el l ipt ic boundary—value problems. Math. Comp., 25, 1-32. (1972) 10. Brenner, P. , Crouzeix, M. and Thomee, V. : Single step methods for inhomogeneous linear differential equations in Banach spaces. RAIRO, Anal. Numer., 16, 5-26 (1982) 11. Bristeau, M.O., Glowinski, R. and Periaux, J. : Numerical methods for the Navier—Stokes equation. Applications to the simulation of compressible and incompressible viscous flow. Comp. Phys. Rep., 6, 73-183 (1987) 12. Ciarlet, P.: The Finite Element Method for Elliptic Problems. Amsterdam: North-Holland (1978) 13. Douglas, J. , and Russell, T. F. : Numerical methods for convection dominated diffusion problems based on combining the method of the characteristics with finite elements or finite differences. SIAM J. Numer. Anal. 19. (1982) 14. Fix, G. , Gunzburger, M., Nicolaides, R. and Peterson, J. : "Mixed Finite Element Approximation for the Biharmonic Equations. , Proc. 5th International Symposium on finite Elements and Flow Problems, Austin, Texas, (1984) 15. Girault, V. and Raviart, P.-A.: Finite Element Approximation of the Navier—Stokes Equations. Lecture Notes in Mathematics, vol. 749. Springer Verlag. Berlin (1979) 131 16. Glowinski, R. and Pironneau, 0. : Numerical methods for the f irst biharmonic equation and for the two—dimensional Stokes problem. SIAM. Rev., 17, 167-212 (1979) 17. Grisvard, P.: Boundary Value Problems in Non-Smooth Domains. Univ. of Maryland, Dept. of Math. Lecture Notes, 19 (1980) 18. Hale, J. K. : Ordinary Differential Equations. 2 n d Edition. Robert E. Krieger Publish Co. Malabar Florida(1980) 19. Hasbani, Y., Livne, E., and Bercovier, M.: Finite elements and characteristics applied to advection-diffusion equations. Comput. Fluids. 11, 71-83 (1983) 20. Holland, W.R. and Rhines, P.B.: An example of eddy-induced ocean circulation. J. Phys. Oceang. , 10, 1010-1031 (1980) 21. Hughes, T. J. R., Tezduyar, T. E. , and Brooks, A.: "Streamline Upwind Formulation for Advection-Diffusion, Navier-Stokes and First Order Hyperbolic Equations". Fourth Internat. Conf. on Finite Element Methods in Fluids, Tokyo (1982) 22. Johnson, C. , Navaert, U. , and Pitkaranta, J. : Finite element methods for linear hyperbolic problems. Comp. Meth. Appl. Mech. Engrg. 45, 285-312 (1984) 23. Lions, J.L. and Magenes, E. : Problems aux Limites Non-Homogeneous at Applications, vols 1 and 2, Dunod. Paris (1968) J 132 24. Mas-Gallic, S. , and Raviart, P-A.: A particle method for f i rst order symmetric systems. Numer. Math. 51,323-352 (1987)25. Maz'ja, V. G. : Sobolev Spaces. Springer Verlag. New York, Berlin, Heidelberg (1979) 26. McWilliams, J.C.: A note on consistent quasi-geostrophic models in multipled connected domains. Dyn. Atmos. Ocean. 1, 427-441 (1977) 27. Morton, K. W., Priestley, A., and Suli, E.: Stability of the Lagrange-Galerkin method with non-exact integration. RAIRO. M2AN. 4, 225-250 (1988) 28. Munz, C-D.: On the numerical dissipation of high resolution schemes for the hyperbolic conservation laws. J. Comp. Phys, 77. 18-39 (1988) 29. Oden, J.T. and Reddy, J.N.: An Introduction to the Mathematical Theory of Finite Elements. Wiley-Interscience, New York (1976) 30. Pedlosky, J.: Geophysical Fluid Dynamics. Springer Verlag, New York (1979) 31. Pironneau, 0. : On the transport diffusion algorithm and its applications to the Navier-Stokes equations. Numer. Math. 38, 309-332 (1982) 32. Raviart, P-A.: An analysis of particle methods. In Numerical Methods in Fluid Dynamics. F. Brezzi. Edi. Lecture Notes in Mathematics, vol. 1127. Springer Verlag, Berlin (1985) 133 33. Suli, E. : Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations. Numer. Math. 53, 459-483 (1988) 34. Temperton, C. , and Staniforth, A. : An efficient two-time level semi-Lagrangian semi-implicit integration scheme. Q. J. R. Meteorl. Soc. 113, 1025-1039 (1987) 35. Thomee, V.: Galerkin Finite Element Methods for Parabolic Problems. Lecture Notes in Mathematics, vol. 1054. Springer Verlag, Berlin (1984) 36. Widlund, 0. : On best error bounds for approximation by piecewise polynomial functions. Numer. Math. 27, 327-338 (1977) 37. Zalesak, S. T. : Fully multidimensional flux-corrected transport. J. Comp. Phys. 31, 335-362 (1979) 38. Zeniseck, A.: Discrete forms of Friedrichs'inequa;lities in the finite element method. RAIRO. Anal. Numer. 15, 255-286 (1982) 134 APPENDIX ON INEQUALITIES We collect here some inequalities which have been used throughout the thesis,as wellas the Lax-Milgram theorem. 1. Gronwall Inequalities If <f>, a are real valued and continuous for a s t £ b, fi(t) ^ 0 is integrable on [a, b] and then <p(t) £ oc(t) + f !B(s)a(s)(exp(\ p(u)du))ds , a £ t £ b . (2) For a proof of (2) see [18]. The discrete version of the above inequality goes as follows. Let (an)' ^ n^' ^Cr? b e ^ r e e sequences of positive real numbers such that (c ) is monotonically increasing and a (1) n n-1 a + b £ n n n * 1, A > 0 , (3) n then a + b £ c exp(Xn) , n ^  0 . (4) n n n For a proof ao (4) see reference [15] Chap.V. 135 2. Schwarz Inequality 2 2 Let n c R and f.geL (Q), then f fgdfi s f f 2d£> f g 2 d n . (5) J r> J n J n 3. Young Inequality Let e > 0, and a, b e R, then ab £ -:—a + eb . (6) 4c 4. Friedrichs Inequality If Q is connected and bounded at least in one direct ion, then for each integer m £ 0, there exists a constant K = K(m, Q) > 0, such that Ivll _ s K\v\m _ , Vv e fljjfn; . (7) The proof of (7) can be found in [1]. A discrete version of this inequality is given in [38]. 5. Lax-Milgram Theorem Let a(u,v) be a continuous and elliptic form on the space V, i.e., \a(u,v)\ £ tfllull llvll , Vu, v e V \a(u,u)\ £ allull2^ , Vu e V. 136 Then , the problem a(u,v) = <l,v> , Vv e V and 1 e V has a uniqe solution in V. Proof. See [2]. 137 

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