A Generalized Procedure for Constructing an Upwind - Based TVD Scheme

This paper is dzclnred a work of the U.S. Co\ernrnenl and i s not subjecl l o copyright protection in Ihe United Stales. f l u x d i f f e r e n c e s i n ou r case) t o meet such r e q u i r e ments as ( 1 ) y i e l d i n g monotonic and sharp rep re s e n t a t i o n o f jumps. and ( 2 ) s a t l s f y i n g t h e en t ropy c o n d i t l o n so as t o a u t o m a t i c a l l y s e l e c t p h y s l c a l l y r e l e v a n t s o l u t i o n s . a t t a c k e d t h i s p rob lem and a r r i v e d a t h i s secondo r d e r scheme. The c r u x o f t h e m a t t e r I s t o f o r m a n o n l i n e a r comb lna t ion o f t h e u n d e r l y j n g f l r s t o r d e r scheme and t h e rema in ing a n t i d i f f u s i v e terms. Hence we v iew t h a t i n secondand h i g h e r o r d e r TVD schemes t h e a n t i d i f f u s i v e terms b a s i c a l l y p l a y seeming ly uncompromlslng r o l e s o f b o t h i n c r e a s i n g accuracy and d l m i n l s h l n g t h e t o t a l v a r i a t i o n . We a l s o remark t h a t a l l TVD schemes c l t e d above use some f o r m o f an upw ind ing scheme as t h e u n d e r l y l n g f i r s t o r d e r scheme. I n t h i s paper we app ly t h i s p r l n c i p l e t o f o r m u l a t e our TVD scheme w l t h emphas l s on n o n l i n e a r systems o f c o n s e r v a t l o n laws, namely E u l e r equa t ions . T h i s approach appears n a t u r a l and s t r a i g h t f o r w a r d f o r e x t e n s i o n t o m u l t i s p a c e d imens ions . A l though r l g o r o u s mathem a t i c a l p rocedures a r e f o l l o w e d t o ensure H a r t e n ' s TVD c o n d l t l o n s , t h e a c t u a l imp lemen ta t i on f o r s o l v i n g t h e m u l t l d i m e n s i o n a l E u l e r equa t ions I s q u i t e s imp le and g e n e r a l . I n numer i ca l t e s t s we even c a r r i e d over t h e same d l f f e r e n c e schemes, p roved t o be TVD i n t h e h y p e r b o l l c system, d l r e c t l y t o t h e s teady E u l e r equa t ions whose t y p e I s n o t known. The same c h a r a c t e r l s t l c s as t h a t ob ta lned by s o l v i n g uns teady equa t lons were found. t h e n o n l i n e a r s c a l a r equa t ion . The procedure f o l lows c l o s e l y t o t h a t o f Swebys, e s p e c i a l l y I n t h e TVD p r o o f , b u t t h e y d i f f e r i n t h e d e f l n i t l o n o f a b a s i c f l r s t o r d e r scheme. The en t ropy cond i t i o n i s a l s o examined. S e c t i o n 3 shows t h e e x t e n s l o n t o t h e one-d imens iona l system o f cons e r v a t l o n laws and t h e co r respond ing TVD p r o p e r t y f o r a l i n e a r i z e d system. F i n a l l y . a d e s c r l p t i o n o f t h e g e n e r i z a t l o n t o t h e E u l e r equa t ions o f gas dynamlcs i s g i v e n i n S e c t i o n 4. A p p l i c a t l o n t o s teady equa t ions , use o f b o t h e x p l l c l t and i m p l i c i t schemes, c e n t r a l and upwlnd ing d i f f e r ences a r e a l s o i n c l u d e d . Van Leer1 f i r s t s u c c e s s f u l l y S e c t i o n 2 shows t h e p resen t f o r m u l a t l o n f o r Second-order TVD Scheme f o r N o n l i n e a r S c a l a r Equa t lon To f a c l l i a t e unders tand ing o f t h e p resen t f o r m u l a t i o n . f i r s t we cons ide r t h e numer l ca l s o l u t i o n o f t h e s c a l a r n o n l i n e a r c o n s e r v a t l o n law i n one space ____. ____ _____-___ u + f ( u ) = 0 , t > O , = < x < = u(x,O) = u ( x ) t x (2 .1 ) 0 For Eq. (2 .1 ) t o be o f h y p e r b o l i c t y p e , t h e f l u x f u n c t l o n f ( u ) has a r e a l d e r l v a t l v e a = d f / d u w l t h c h a r a c t e r i s t i c cu rve desc r ibed by d x / d t = a . Suppose a secondo r h i g h e r o r d e r ( c e n t e r e d o r one-s lded b iased) d i f f e r e n c i n g scheme I s g l v e n (e .g . , Eq. ( 2 . 5 b ) ) . t h e p resen t f o r m u l a t l o n makes Copwight IC' 1987 American Institute of Aeronautics and Astronautics. Inr. Nocopyright is asurtrd in Ihe Unitrd S t a l e undcr Titlc 17. US. Code. The U.S. Govrrnmcnl has a royally-free license lo cxcrcix all righls under the copyrighl claimcd hcrein for Governmental purposes. All other rights are reserved by Ihc copyriEhl owncr. 1 i t T V D s a t i s f y i n g by the f o l l o w i n g s teps : ( 1 ) Form a f i r s t o r d e r , upwind d i f f e r e n c i n g , (Eq. ( 2 . 5 a ) ) ; ( 2 ) Car ry ou t s u b t r a c t i o n o f t h e d i f f e r e n c i n g s and o b t a i n t h e rema in ing terms, c a l l e d a n t i d i f f u s i v e terms, (Eq. (2 .6 ) ) ; ( 3 ) L i m i t t h e a n t i d i f f u s i v e terms, (Eq. ( 2 . 7 ) ) , u s i n g t h e s o c a l l e d f l u x l i m i t e r f unc t i on3 ; * and ( 4 ) De te r mine t h e bounds of t h e l i m i t e r f rom H a r t e n ' s cond i t i o n s 2 , as w e l l as spec i f y i t s f u n c t i o n fo rms. s p l i t t i n g a ( u ) ( a i s r e a l f u n c t i o n ) i n t o "+" and 'II' p a r t s , l h e upwind ing i s conven ien t l y ach ieved by t a = a + a (2 .2a) where a t 0. a5 o (2.2b) l h u s t h e f l u x f i s s p l i t a c c o r d i n g l y by r e q u i r i n g : ( i ) f = f t t f (2.3a) (11 ) at = (ft),,, a= ( f ) u (2 .3b) We n o t e t h a t how the s p l i t t i n g ( n o t un ique) i s done i s i r r e l e v a n t i n t h e p resen t a n a l y s i s . I t i s needed o n l y i f an ac tua l c a l c u l a t i o n i s made. We a l s o remark t h a t Roe's f l u x d i f f e r e n c e s p l i t t i n 9 3 can be a p p l i e d equa l l y w e l l , as w i l l become c l e a r l a t e r . L e t us use t h e n o t a t i o n (2 .4 ) A ( ) j = ( ) j ( ) j 1 We now approx imate f x : f x = ( A + f t A-f+)/Ax, f i r s t o r d e r upwind ing (2 .5a) = ( A + f t A'f)/2Ax, c e n t r a l d i f f e r e n c i n g (2.5b) S u b t r a c t i n g Eqs. (2.5a) f r o m (2 .5b) . we have t h e a n t i d i f f u s i v e te rm A ( A t f ' A t f ) / 2Ax (2 .6 ) The f l u x d i f f e r e n c e s A t f t a r e l i m i t e d by ass ign i n g t h e l i m i t e r funct ' lons cpt and cpr e s p e c t i v e l y , and we w r i t e t t t f x = ( A t f t Af t ) /AX + A ( c p A f cp-.Atf-)/2AX (2 .7 ) . . _._ . *S ince I t i s t h e f l u x d i f f e r e n c e s , r a t h e r than f l u x e s themselves, t h a t a r e l i m i t e d , i t m i g h t be more a p p r o p r i a t e t o c a l l t h i s t h e f l u x d i f f e r e n c e l i m i t e r . I t i s n o t e d t h a t a l l o w i n g d i f f e r e n t f u n c t i o n s Q+ and cpt o be a s s o c i a t e d w i t h f t and f i s e s s e n t i a l f o r t h e p r o o f o f TVD. Now. t h e i n t e g r a t i o n scheme f o r Eq. (2 .1 ) can be a p p l i e d . To i l l u s t r a t e some p r e l i m i n a r i e s o f TVD and H a r t e n ' s c o n d l t l o n s , we c o n s i d e r t h e f i r s t o r d e r E u l e r e x p l i c i t scheme w i t h t h e use o f upwind d i f f e r e n c i n g , Eq. (2 .5a) . Then we have