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.. NASA Contractor Report 178235ICASE REPORT NO. 87-6

I C A S Ec

ERROR ESTIMATES FOR CELL-VERTEX SOLUTIONSOF THE COMPRESSIBLE EULER EQUATIONS

(SASA -Ca -1782 .35 ) ERBCIi ES'11HAlES FOBL L L - V E R T E X 3 C L t ' l l C Y S C P ' I t E C C B E h E S S I B L EEDLER EkUATPC!BS F i n a l he pc r t ( B A S A ) 43 pC S C L 0 1 A ~ 3 / i l 2

P. L. Roe

C o n t r a c t No. NASl-18107J an u a r y 1987

- _- - - _ _ _- -~~-_ _

INSTITUTE FOR COMPUTER APPLICATIONS I N SCIENCE AND ENGINEERINNASA Langley Research Center, Hampton, V irginiaOperated by the Universities Space Research Association

23665

NASANationalAeronautics andSpaceAdministrationLangley Research CenterHampton, Virginia 23665-5225

N87- 18534Unclas43560

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ERROR ESTENATES FOR CELL-VERTEX SOLUTIONS OF TEECOMPRESSIBLE EULER EQUATIONS

P. L. RoeC r a nf i e ld I n s t i t u t e of Technology, United Kingdom

ABSTRACTThe ce ll -v er te x schemes due t o Ni and Jameson, e t a l . , have been

s u b j e c t e d t o a t h e o r e t i c a l a n a l y s is of t h e i r t r u n c a t i o n e r r o r . The a n a l y s i sconf irms t he au thors claims f o r second-order accuracy on smooth g r ids , bu tshows that the same accuracy cannot be obta ined on a r b i t r a r y g r i d s. I t i sshown th a t th e schemes have a u ni qu e g e n e r a l i z a t i o n t o ax is y mm etr i c f l ow t h a tpres erve s th e second-order accuracy.

T h i s r e s e a r c h w a s suppor ted under t h e Nat i onal Aeronau t ics and SpaceAd mi ni st ra ti on under NASA Co nt ra ct No. NAS1-18107 w h il e t h e a u t h o r was i nr e s i d e n c e a t t h e I n s t i t u t e or Computer App l ica t ion s i n Science a n dEn gi ne er in g (ICASE), NASA Lan gley Res ear ch Ce n t er , Hampton, VA 23665-5225.

i

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1. I n t r o d u c t i o nMost Eule r codes wr i t te n fo r computing in vi sc id aerodynamic f lows have

made use of t he f inite-v olum e form ulat ion. An e a r l y example was d es c r i b ed b yRi zz i and Inouye 111. E s s e n t i a l l y , t h ey a r e g e n e r a l i z a t i o n s of t h e se co nd -order accurate Lax-Wendroff [2 ] a lgor i thm t o non rec tangu la r mesh geomet r ies .Conservat ion (which i s a neces sary ing red ien t toward ensur ing convergence t oco r r e c t weak s o l u t i o n s [ 3 ] ) comes f rom regarding the numerical representat ionof th e flow as compris ing average s t a t e s within each computat ional c e l l . Ther u l e s f o r u p da t in g t h e s e v a l u e s employ i n t e r f a c e f l u x e s which t r a n s f e r t h emass, momentum, and energy between neighboring c e l l s , l e a v i n g t h e t o t a l quan-t i t i e s pr es en t unchanged except f o r boundary efEects. There seems t o bel i t t l e an a l y s i s t o i n d i c a t e how accu r a t e t h e s e s chemes are on nonuniformgr ids . They are c l ea r l y s econ d- or de r a ccu r a t e o n u n if o rm , r e c t a n g u l a r g r i d s ,and i t has been ge ne ra l l y assumed th a t second-order accuracy w i l l . s t i l l beob ta ined on a " s u f f i c i e n t l y smooth" i r r e g u l a r g r i d .

Fini te-volume schemes based on ce nt ra l - d i f f er en ci ng algo r i thm s havebecome ve ry p opular fol low ing th e work of Jameson e t a l . [ 4 1 . Because theseschemes upda te t he so lu t i on v ia a f i v e - c e l l r a t h e r t h an n i n e - c e ll s t e n c i l ,t h e i r a c c u r a c y i s eas i e r t o a n al y ze and h a s bee n t r e a t e d t h e o r e t i c a l l y b yTurke l [51, whose r e su l t s have led to modi f ied schemes o f f e r in g b e t t e raccuracy. Numerical exper iments conf irming t h i s are repor ted by Turkel e t a l .[61

Y e t a n o t h e r p o s s i b i l i t y i s the scheme recent ly in t roduced by N i [7].Here t h e n u m er i ca l v a l u e s r e p r e s en t s t a t e s f o un d a t t h e c o rn e rs ( v e r t i c e s ) o ft h e co m p u t a t i o n a l c e l l s . N i o r i g i n a l l y d e s c ri b e d h i s scheme i n f i n i t e -d i f f e r e n c e terms, with t he value s approximat ing po in t samples of th e cont inuum

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-2- I

so lu t ion . Subsequen t ly , i t has been descr ibed i n f in i te-e lement terms witht h e v e r t ex v a l u e s i m pl y in g a b i l i n e a r i n t e r p o l a t i o n f u n c t i o n w i t h i n e ac h c e l l I(Davis e t a l . [8], Lohner e t a l . [ 9 1 ) . This l a t t e r view enables a p r e c i s e

I

I

i n t e r p r e t a t i o n of c o n se r v at i o n t o be gi ve n. A t a t i m e when var i ous sc ho ol s of 1thought are l e a r n i n g t o u se each o t h e r' s l ang u ag e and i d ea s , we p r e f e r t o

Ic l a s s i f y N i ' s method under t he n eu tr al term of " c e l l ver tex '' scheme. Onea t t r a c t i o n of N i ' s scheme (which has been very s i gn i f ic an t l y r e f in ed by H a l l[ l o ] ) i s t h a t i t seems t o hand le a rb i t r a ry , even comple te ly unsmooth , meshes I

i n a very n a t u r a l way. However, w e w i l l show that claims of uniformly second-order accuracy are based on a n in co r r ec t a rgument. The ob j ec t o f t h i s paperi s t o p ro vi de a co r r ec t an a l y s i s of t h e accu racy o f c e l l - v e r t ex s ch em es .

Th is i s made e a s i e r by the f ac t t h a t t h e s e schemes, i n a s e n s e , f a c t o r i z eth e p roces s o f upda t ing th e so l u t io n . Each c e l l i s examined t o see i f t h ef luxes a round i t are i n b al an ce ; i f not, some changes are made. The up da ti ngp r o ces s a t e ach v e r t ex u s e s n i n e v e r t ex s ta te s s o t h a t , o v e r a l l , t h e scheme i sas complex as Lax-Wendroff. However, t h e t e s t f o r b a lan ce i n v o l v e s o nl y f o u rv e r t i c e s , making t h e f i r s t s t a g e v er y amenable t o a n a l y s i s . T hi s f i r s t s t a g ei s t h e on ly p a r t t o be ana l y zed h e r e , b u t t h i s i s t h e o n l y an a l y s i s n eeded t ode te rmine th e accuracy of th e s teady s t a t e which i s reached when a l l c e l l s a rei n equi l ibr ium. That i s t o s ay , t h e i n t e g r a l

eva lua ted by the t r apez ium ru l e , van ishes a round every c e l l ( F i g u r e l a ) . N ot et h a t t h e r e a r e g e n e r a l l y more v e r t i c e s t h a n c e l l s s o t h a t t h i s c o n di t io nle av es the so lu t i on undetermined. Boundary co nd i t io ns need to be added, but

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-3-

c a r e f u l l y . I f to o many are p r e s c r i b ed , t hen t h e i n t eg r a l c an no t v an i s h ev er y -

e t a l . [13] have proposed a scheme which a l s o makes use of f l o w v a r i a b l e s

ver t i ces ou tnumber c e l l s by roughly 2:l; s o t h e problem would be overcon-s t ra in ed even without boundary cond i t io ns . In s t ea d, Jameson's procedure i ss u ch t h a t t h e s o l u t i o n ceases t o change when th e f l ux in te gr a l van ishes a rounda l l co nt ro l volumes such as th e one shown i n Figu re lb.

The an a l y s i s co n t a i n ed i n t h i s p ape r a p p l i e s e q u a l l y t o N i ' s and t oJameson's schemes beca use i t eva lua tes the accuracy o f th e approx imate f l uxi n t e g r a l i n terms of th e geometry o the polygon around which i t iseva lua ted . The claim, ment ioned ear l i e r , of second-order accuracy a r i s i n gf rom the t r apez ium ru le i s f a l l a c i o u s b ecau s e , as w e r e f i n e t h e g r i d , t h e con-tour around which w e i n t e g r a t e d oes n o t r em ai n f i x ed b u t s h r i n k s w i t h t h e g r i dand always con t a ins j u s t a few in te rv a l s . In f a c t , i t i s easy enough t o makean ana l ys i s based on a l o c a l two-dimensional Ta yl or exp ansi on which demon-s t r a t e s t h a t t h e i n t eg r a l s a r e o nl y ev a l u a t ed t o s eco nd -o rd er a ccu r acy if t h epolygons meet c e r t a i n g eo me tr ic c r i t e r i a . These c r i t e r i a a re n ot m e t by any

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t r i angl e or by any quadr i l at er al except par al l el ogr ams. They ar e sat i sf i ed byal l r egul ar n- gons (n > 3 ) and i n a seem ngl y spor adi c manner by var i ousi r r egul ar n- gons (n > 4 ) . I n pr act i ce, t he er r or s wi l l be accept abl e . i f the

Ipol ygons have suf f i ci ent l y smal l er r or const ant s, t he f or mul ae or whi ch ar e I

I

gi ven.There ar e t wo of f shoot s to our anal ysi s. One i s t hat we der i ve a con- I

si st ent f i ni t e- el ement i nt er pr et at i on of t he cel l - ver t ex schemes. We showt hat i t gi ves r esul t s whi ch di f f er f r om t hose of t he f i ni t e- di f f er ence i nt er - Ipret at i on by t erms of t he same magni t ude as t he t r uncat i on er r or i nherent i n I

Iei t her . Fi nal l y, we consi der t he appl i cat i on of t hese schemes to pr obl emswhi ch are pseudo- t wo- di mensi onal , or exampl e, axi al l y symmet r i c. We showt hat out of a f am l y of pl ausi bl e ext ensi ons, onl y one member i s capabl e ofr et ai ni ng second- order accur acy near t he axi s.

2. Descr i pt i on of Cel l - Vert ex SchemesWe consi der st eady i nvi sci d t wo- di mensi onal compressi bl e f l ow governed by

t he Eul er equat i ons

wher e

wher e

aGF- + + - - 0ax aY

P =- (y-l) p [ h - 1 ( u2 + v211Y

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-5-

and h i s t otal speci f i c ent hal py, assumed const ant . We i nt end t o sat i sf yt hese equat i ons i n an appr oxi mat i on t o thei r i nt egr al f or m

wher e t he i nt egr al i s ar ound any ar bi t r ar y cl osed cont our . To t hi s end, t hepl ane i s di vi ded i nt o cel l s t hat are ei t her r egul ar l y ar r anged quadr i l at er al s( Fi gur e la) or unst r uct ur ed t r i angl es ( Fi gur e lb). I n ei t her case, t hever t i ces ar e number ed, and a f l ui d st at e xi , wher e u = ( p , pu, pv), i sassi gned t o t he i t h ver t ex. N s scheme f or quadr i l at eral s appr oxi mat es (2. 4)by t r apezi um r ul e i nt egr at i on ar ound each cel l . Thi s i s

-

where t he summat i on i s over al l si des of t he cel l s, and al l di f f er ences ar et aken ant i cl ockwi se. Not e t hat D has di mensi ons of t he t i me der i vat i ves oft he act ual conser ved quant i t i es (not t hei r densi t i es), so t hat i ndi cat eshow masses, etc. , associ at ed w t h t he poi nt s A, B, C, D ar e to be changed.Di f f er ent t i me- mar chi ng schemes r esul t f rom speci f yi ng di f f er ent wei ght s wi t hwhi ch the changes are di st r i but ed over A , B, C, D N s or i gi nal reci pe [ 7]r epr oduced a t ype of Lax- Wendr of f al gor i t hm and Hal l [ l o ] has present ed aRunge- Kut t a ver si on. Her e, however , we are i nt er est ed onl y i n t he st eady

-

st at e whi ch we ass ume i s charact er i zed by t he vani shi ng of D f or ever ycel l . I n pr act i ce, t hi s wi l l onl y happen i f t he boundar y condi t i ons ar especi f i ed car ef ul l y. ( For exampl e, on a gr i d of n x n cel l s t her e wi l l ben2 equat i ons and (n+1)2 unknowns; hence 2n+ boundar y condi t i ons must be

-

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p r e s c r i b e d . ) Also, t h e r e s i d u a l s d o n o t i n g e n e r a l v a n i s h n e ar a c a p t u r e dshock i l l ] . This i m p l i e s t h a t o ur i n t e r e s t h e r e i s c h i e f l y i n t h e a c cu ra c yobta ine d i n smooth p a r t s of t he flow.

I n Jamesons t r ia ng ul ar gr id scheme [13 ] , t he average f lu x acr os s eachl i n e j o i n i n g two v e r t i c e s i s e v a l u a t e d as

1 1E =- ( F +EB) (J, - yA> - - ( G cB) xB - XA)*-AB 2 -A 2 -AThen masses, e t c . , p ro po r t i ona l t o E A t a re added a t X

-ABand sub-

t r a c t e d a t Y. When t h i s ha s been done f o r a l l edges i n the mesh, t .ien ev erypoint has been changed by a ne t amount p ropor t ion a l t o the f lu x ac ro ss thepe ri me te r of t h e polygon Formed by a l l t r i a n g l e s m e e t i n g a t t h a t p o i n t . A tt h e s t e a d y s t a t e , w e a g a i n assume t h a t a l l t h e s e f l u x i n t e g r a l s v a n is h - duecare being ta ken a t the boundary.

I t i s t he a b i l i t y t o c h a ra c t er i z e st ea dy s t a t e s o l u t io n s i n t h i s r a t h e rsim ple way t ha t makes th e accuracy of c el l- ve r t ex schemes much e a s i e r t oana ly ze than Lax-Wendroff schemes where th e st ate me nt of lo ca l equ il ib r i umt y p i c a l l y i n v ol v es n i ne f l u i d s t a t e s and thi r ty - tw o mesh co ord ina tes . Thes t a t emen t of equ i l ib r ium fo r a Jameson-type scheme on q u a d r i l a t e r a l s i n v o l v e so nl y f i v e f l u i d s t a t e s , b u t a n a l y s i s of i t s acc ura cy needs twenty-four meshc o o r d i n a t e s .

3 . F in i te-D if ference Analysi sWe wish t o know how ac cu ra te ly th e formula

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ap pr o xi m at e s t h e i n t e g r a l-F = $ F d y ( 3 . 2 )

tak en around th e same polygonal contour. A t t h i s s t a g e , t h e v a l ue s of Fare thought of as poi n t s amples , i n the s p i r i t of f i n i t e d i f f e r e n c e m e t h o d s .T h i s i s most e a s i l y done by assuming th at - i s r ep r e s e n t ab l e by a Taylorexpans ion about some ar bi t r a ry nearby or i g i n . We co n s i d e r e ach s i d e of t h epolygon i n tu rn . The numer ical approx imation t o th e in t eg ra l i s

-

( 3 . 3 )

4+ 0 ( h 1.

The leading terms i n t h e ex a ct i n t e g r a l are found by introducing a l o c a lc o o r d i n a t e 5 v a r y i n g l i n ea r l y f r o m 0 t o 1 over AB , s o t h a t a t y p i c a l p o i n ton AB i s def ined by

and

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F-XY

) F ] + O(h) .2 2+ (YA + YAYB + YB -yyS u b t r a c t i n g (3.5) f rom (3.3) g iv es the er ro r on AB as

I

5 I

(3.5)

Hence the t o t a l er r o r may be ar r iv ed a t simply by summing the contribu-t ion f rom a l l s i d e s . C l e a r l y w e can f i n d t h e e r r o r i n t h e - i n t e g r a ls i m i l a r l y . The f i n a l r e s u l t i s

/ F d y = l F A y - E F + 2 E F - E F2 -xx 3 -xy 4 -yy-/ G ~ X = ~ G A X - E ~ ~ ~ + ~ EG - E G- - 2 -xy 3 -YY

(3.6a)

(3.6b)

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whereE l = iij- 1 (Ax)3

=-1 ( A X ) 2AyE2 12=- 1 AX (Ay) 2E3 12

E 4 = iij- 1 ( A Y l 3 -

( 3 . 7 )

I t i s p l e a s i n g t h a t t h e e r r o r c o n s t a n t s t a k e t h e s e s i m p l e forms and use-f u l t h at t h er e are only four of them. Observe t ha t th e qu an t i t i es w e wish t oapproximate are O (h 2) and t h a t , s u p e r f i c i a l l y , t h e e r r o r c o n s t a n t s areO(h 1, making the method f i rs t - or de r acc ura te . However, some ca nc el la t i on3t a ke s p lace i ns i de t he summations i n ( 3 . 7 ) , which ho lds ou t t he p rospec t ofbe t te r accuracy , depend ing on geomet r ic p roper t i es of the g r id which w e nowi n v e s t i g a t e .

4. Polygons with Vanishing Error Cons tantsI t i s easy t o see t ha t any po lygon wi th c en t r a l sy m m et ry w i l l have E l

t o E4 i d en t i c a l l y z e r o b ecau se co n t r i b u t i o n s fro m o p p o s i t e s i d e s w i l lv an i s h . Also, giv en one er ro r- f r ee polygon, we can form ot he rs by permutingthe s ides and forming mirror images . I t i s s t r a i g h t f o r w a r d t o show t h a t ane r r o r - f r ee p ol yg on r em ai ns e r r o r - f r ee u nd er r i g i d r o t a t i o n s . C e r t a i n o t h e rr e s u l t s are a l s o o b t a i n a b l e .

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-10-

Theorem IA q u a d r i l a t e r a l i s e r r o r - f r e e i f f i t ,s a para l le logram.

Proof: For a q u a d r i l a t e r a l ABCD , t h e e r r o r c o n s t a n t s c an be m an i pu l at ed I

where

and S i s t h e a r e a of t h e q u ad r i l a t e r a l . The o n ly terms which may va ni sh i nX +x >, y -y +y -y >. T h e s e q u a n t i t i e s ,h e s e ex p r e s s i o n s are

d i v i d e d by two, are the components of the vec to r which jo in s t he midpoin t s ofth e two dia gona ls. Thus E l t o E4 v a n i s h i f f t h e d i ag o n a l s b i s e c t e ac h o t h e r ,which only happens i f t h e q u a d r i l a t e r a l i s a para l le logram.

(xA B C D A B C D

Coro l la ry : No t r i a n g l e i s e r r o r - f r e e .

P r o of : S i n ce t r i a n g l e s are d e g en e r at e q u a d r i l a t e r a l s , t he y are excludedby Theorem 1.

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-1 1-

Theorem 2: Any r e g u l a r n-gon ( n > 3) is e r r o r - f r e e .

Proof: For n even th e r e s u l t is t r i v i a l by symmetry. A proof , whichin cl ud es n odd, fo l lows by con sid er i ng th e polygon i n th e complex p lane .Without lo s s of ge ne ra l i ty , one s id e can be made p a r a l l e l t o the x -ax i s, ande a c h s i d e is then r epr ese nte d by one of the n th roo ts of u n i t y , s a yz k ( k = 0 , ... -1) where

2nk 2n kz = CO S - i s i n-n nthen2s k s in2 2ak 3 2.nk3 c o s- - - i s i n --n -nk2nk 2 2nk= 1 cos - 3- cos -k k n n n n n n'k

2nk 2 2nk- 3 cos- 1 - cos -)nks i n-2nk= 1 cos3= 3 1 co s -n n n nk- i s in-nk (1 - cos 2 -)nkn n

= 4(E1 + i E 2 )

= (by a s imi la r argument) - 4(E t iE4) -3But

3k=n-13k=O k=O

k=n- 1 1z =2;" - 1

= 3z -11

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-12-

The numerator vanishes because zn = 1. So t h e e x p r e s s i o n e q u a l s z e r o1un l ess the denomina to r a l s o van i shes ; t h a t i s , u n l e s s n = 3 (when t he ex pre s-s i o n e q u a l s 3 ) . So u n l e s s n = 3, E = E = E = E = 0.1 2 3 4

For n > 4 , we can f ind a v a r i e t y o f a s y m m e t r i c er ro r - f r ee polygons. Thegenera l p rocedure i s t o p r e s c r i b e (n -2 ) v e r t i c e s . Then r e q u l r i n g t h a t E lt o E4 vanish g ives four equat ions for two unknown coordinate pai rs . Unfor-t u n a t e l y , t h e s e are s imul t aneous cub ic equ a t ions fo r wh ich no sys t ema t i cs o l u t i o n i s apparent . To g ive a f l a v o r of t h e p o s s i b i l i t i e s , w e c o n s i d e r a

pentagon wi th p resc r ibed v e r t i ce s a t (O,O), (O, l ) , and (1 ,O) . We m a i n t a i nsymmetry by supposing t h a t an err or- f ree p en ta go n e x i s t s wi t h a d d i t i o n a lv e r t i c e s a t ( l + a , b ) , ( b , l + a ) . Then w e r e q u i r e

3 3 31 AX)^ = 1 ( A Y ) ~ 1 - (l+a-b) + a - b = 0and

21 ( A x ) ~ A ~ 1 (Ay2)Ax = ab2 - (l+a-bI3 - ba = 0 .

Condi tion (4.2 ) s i mp l i f i e s t o

3 ( a - b ) ( a + l ) ( b - l ) = 0 ; (4.4)

i f (a -b ) = 0 , t h e r e are no s o l u t i o n s t o ( 4.3) . I f b = 1 , t hen (4.3 ) y i e l ds

2a 3 + a - a = ~

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-13-

whereas, i f a = -1 t h en2b 3 - b - b = O

12= 0, - 1 i 47).

Alt..ough there appear t o b e s i x s o l u ti o n s , L e y a c t u a l l y ap p ea r i np a i r s . In each p a i r , t h e s i d e s are permutati ons of th e same s e t of vectors .The so lu t io n t o th e p roblem co ns i s t s of th re e independen t se t s of vec to r s ,each of which, by per mut ati on, ge ne ra te s 24 d i f f e r en t p en tago ns , i f w e a l l o wt h e g t v en v e c t o r s t o p er mut e a l s o . However, out of a g i v e n s e t of v e c t o r s ,only two order ings w i l l produce a convex polygon, t h a t i s , those which aremonotonical ly orde red by or ie nt at io n. Those two are mirror images of eachot he r. The independent convex pentagons de ri va bl e from (4.2) and (4.3) a reshown i n Fi gure 2. The square i s a degenera te case f o r which one s i devan ishes .

E r ro r - f ree po lygons can a l so be cons t ruc ted by un i t ing one o r more e r ro r -f r e e polygons th at have edges i n common; b ut , though the c on s tr uc t i on ofe r r o r - f ree polygons has some mathematical interest , i t seems u n l i k e l y t h a t a n yp r a c t i c a l mesh could be cons t ruc ted from them, except f o r completely r eg ul armeshes. We t u r n , t h e r e f o r e , t o t h e more p r a c t i c a l q u e s t i o n of f i n d i n gpolygons f o r which th e e r r o r c ons t an t s are accep t ab l y small.

5. Polygons with Small Error ConstantsThe only complete r e s u l t s that have been obtained r e l a t e t o q u a d r i -

l a t e r a l s , f o r which t h e e r r o r co n s t an t s are giv en by (4.1). The l o c a l tru nca -

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t i on er r or wi l l be Oh2) pr ovi ded t hese expr essi ons are Oh4). We can Icl ear l y i dent i f y f act or s whi ch ar e Oh2), and t he onl y possi bi l i t y f or Isecond- order accur acy i s t hat

15. l a)XB + xc - XD) = 0 (h 1,( xAand

(5. l b) I

Al t hough t hese condi t i ons ar e not met by ar bi t r ar y quadr i l at eral s, t heyar e met by quadr i l at eral s whi ch ar e pr oduced as par t of an anal yt i c gr i d.Suppose we have gr i d gener at i ng f unct i ons I

and t he mesh poi nt s ar e t he i nt ersect i ons of f or i nt egerval ues of m n. Then, i f t he mesh i s syst emat i cal l y r ef i ned, AS, A n t endt o zer o, and

5 = mAS, n = nAn,

pr ovi ded the cross- der i vat ves ar e not l ar ge.Not e t hat si nce t he er r or t er ms i nvol ve onl y t he l ocal i nt r i nsi c geomet r y

of each cel l , nei ghbori ng cel l s may be of mar kedl y di f f er ent shape, si ze, oror i ent at i on. That st at ement cannot be made f or f i ni t e- vol ume schemes

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[ 5 , 1 1 ] . I nd ee d, t h e l o c a l q u a l i t y of ce ll -v er te x schemes makes them second-o r d e r a ccu r a t e o n g r i d s g en e r a t ed more s t r a i g h t f o r w a r d l y t h an t h o s e d e s c r i b edabove. Suppose a c o a r s e g r i d i s d et erm in ed a r b i t r a r i l y , and f i n e g r i d s areco n s t r u c t ed f r o m i t by r ep ea t ed l y b i s ec t i n g t h e c e l l edges. Within each oft h e o r i g l n a l c e l l s , PQRS, t h e r e i s a mapping defined by

(5.3a)

(5.3b)

which meets t h e co n d i t i o n s ( 5 . 2 ) .R e l a t ed r e s u l t s have b een f oun d by P a i s l ey [ l l ] , who f i n d s t h a t i n t h e

ce l l -ver tex method , i t i s a l l o w ab l e to p e r t u r b a re ct an gu la r mesh by O(h2)3whereas with Jameson's [ 4 ] scheme, th e a l lowable pe r t u r ba t i on i s only O(h ).

6 . A Finite-Element ApproachA s mentioned i n the i n t ro du ct io n, th e schemes under s tudy have been

descr ibed e l sewhere as f i n i t e e lement schemes . Whether t h i s i s j u s t i f i a b l e o rn o t , t h e r e is one ob je c t io n th a t can be removed. Th is ob jec t io n i s t h a t t h eformula

i s o b t a i n ed by i n t e g r a t i n g o ve r each t r i a n g u l a r ( q u a d r i l a t e r a l ) e le me ntassuming a l i n e a r ( b i l i n e a r ) d i s t r i b u t i o n or b o t h - and -, which i s incon-

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si st ent because t her e i s a nonl i near f unct i onal r el at i onshi p bet ween - andG A consi st ent f i ni t e- el ement model can be der i ved by assum ng t hat t hevect or w var i es l i near l y ( bi l i near l y) over t he el ement s, wher e-

- = p2 (i) .These var i abl es wer e i nt r oduced by t he aut hor t o f aci l i t at e t he con-

They have t he propert y t hatt r uct i on of an appr oxi mat e Ri emann sol ver [14].ever y component of 2, -, - i s bi l i near i n t he component s of - Thus

F =c

" l W2'+hw2 +-+ 2 y- 1 21 2y w2 - 7g w3w2w3

wher e h i s a const ant f or st eady f l ow, equal t o the speci f i c totalent hal py. I n comput i ng a t ypi cal term of (6.1) such as

I AB= I, B dyt her e w l l be subt er ms of t he f or m

I AB, j J A w w dy .The consi stent eval uat i on of t hi s t er m i s

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+ w w + w , w +2 w w 1= ( yB 'A [ 2W BW B i A j B i B j A i A j A

1- yB - y )(F2 A -Bhe f i r st t er m cont r i but es t he ' nai ve' appr oxi mat i on +EA);t he second t er m i s par t of a cor r ect i on t hat makes t he appr oxi mat i on sel f -consi st ent . The l eadi ng cont r i but i ons t o i t ar e due t o f i r st der i vat i ves of- The cor r ect i on t erm i n ( 6 . 5 ) i s , to l eadi ng order ,

When we sum such t erms over ever y si de of t he pol ygon, expr essi ons ar i sepr oport i onal t o t he const ant s E l , E*, E3, E 4 gi ven by equat i ons (3.7).Wher e t hese err or const ant s are smal l enough, t he er r or due t o i nconsi st encyi s negl i gi bl e, and bot h t he consi st ent and i nconsi st ent f or mul ae appr oxi mat ethe f l ux i nt egr al s t o Oh2).

7. An I nt egr al For mul at i on of Axi symmet r i c Fl owFl ow whi ch i s i dent i cal i n ever y pl ane t hr ough some axi s of symmet r y i s

gover ned by di f f er ent i al equat i ons t hat can be wr i t t en i n ei t her of t he f orms

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u + F + G = ( v / r ) u- -x -r - (7.1)or

( r 2l t + ( r F I x + ( r -I r = - (7.2) Iwher e E, -, - ar e t he usual conser ved quant i t i es i n t wo- di mensi onal f l ow,except t hat r i s now a r adi al coor di nat e, and v a r adi al vel oc i t y. Her e- i s the vect or (0, 0, p , O)T.

A s a basi s f or numeri cal work, (7.1) i s t empt i ng because i t r equi r es onl ysl i ght modi f i cat i on t o a t wo- di mensi onal code. However , ( 7 . 2 ) i s a bet t erbasi s f or shock- capt ur i ng because i t expr esses conser vat i on more exact l y. Tosee t hi s, i magi ne t hat we ar e perf orm ng a t r ul y t hr ee- di mensi onal comput a-t i on, usi ng the same gr i d i n ever y axi al pl ane, s o t hat each 3-D cel l i s a 2-Dcel l r ot at ed t hr ough some angl e 9 w t h r espect t o t he axi s (Fi gur e 3 ) .The exact angl e does not mat t er , so l ong as we avoi d t he ext r emes Q scd 27.

I f we make a f l ux bal ance on such a cont r ol vol ume, t he cont r i but i onsf r om t he f our f aces whi ch do not l i e i n t he radi al pl anes can be eval uat edanal yti cal l y, by i nt egr at i ng w t h r espect t o t he angl e. The r esul t i s

a- /{ r u dA - { (rx dr - r Gdx) - I/ p dA = 0- -an na t n ( 7 . 3 )wher e n i s t he i nt er i or of t he 2-D c el l , and an i t s boundar y.

I n order to check t hat a pr oposed sol ut i on was conser vat i ve ( and hence acor r ect weak sol ut i on) , we woul d check t hat al l such i nt egr al equat i ons wer esat i s f i ed. A numer i cal met hod whi ch i mposes t hi s as a const r ai nt wi l l be acor r ect shock- capt ur i ng met hod. I n t hi s way, we can gi ve an i nt er pr et at i on t oconservat i on of " r adi al moment um" vi a t he t hi r d equat i on i n (7.3).

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Equat ion ( 7 . 3 ) can be t h e b a s i s f o r ex ten di n g t o axi s ym m et r ic f lo w e i t h e ro f th e ce l l -v er t ex methods under d i scuss ion . In N i s scheme, one in te gr a t est h e q u a n t i t i e s r F , -G around qu ad r i la te ra ls ; i n Jamesons scheme, th eq u an t i t y r ( F A y - ~ A X ) i s eva lua te d f o r each s id e and used t o upda te t head j o i n i n g v e r t i c e s . The p ro bl em w e now address i s how to ev a lu a te t he seth ings numer ica l ly .

8. E r r o r A na l vs i s f o r t h e Axisvmmetric MethodsIn e i t h e r method, t h e c e n t r a l i n g r e d i e n t i s an ex p r e s s i o n of the form

B r F d r o r lABr- dx.L -C on si de r t h e f i r s t of t h e s e. Two p l au s i b l e co m p u t a t i o n a l f or mu lae are

1I AB

=- ( r B - r A ) (rA + r B )[ F ~AB 4A s i n the two-d imens iona l case, we f i n d what t h e exact v a l u e of t h e

i n t e g r a l s ho ul d be i n terms of a Taylor expans ion abou t an a r b i t r a r y nearbyp o i n t . The r e s u l t i s

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+ 3 ( r : + r r + r 2 )FA B B -r

124 A A A A B B A B A B+ r x 2+ 2r x x- 3 r x + 2r x x + rA%

+- ( 3 r A x A + 2rArBxA+ rB xA + r g 2XA + r A2 r B1 22 3 ) F ] + ... .3 2+ (rA + rA rB + r r + rB -rr

A B

(8.3)+ 3r x 2 ) ~B B -XX

2 r r x + 3 r B xB) xxrA B B

I f w e make s imi lar expansions f o r the approximations (8.11, (8.21, andsubt rac t f rom them (8 .3) , we f i n d e r r o r e s t i m a t e s as fo l lows .

1 - 1 1 3IAi- IA B= ( rB-rA)L#rB-rA)(x -x )F + :(r - r A ) - F -B A -X 6 R -L- z ( ~ B - ~ A ) ( 3 r A ~ A+ r x - r x - 3rBxB) FA B B A -xx- -(r -1- ) (3rAxA - r x + r n - 3rBxB) F-xr2 B A A B B A+ 1 ( rB- rA) ( rB 2r ) F ] + ...

A -rr

xAB - = ( r B - r A > [ - -(r -r >(x -x F -- ( r B - r A l 2 F-r2 B A B A -X 12+- (xB-xA> (rAxB - r x ) F12 B A -XX

112 B A -xr- rB-rA) (rAXB - r X ) F ] +

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Combning terms f o r a pa r t i c u l a r po lygona l co n t r o l volume, w e o b t a i n t h ef o l l o w i n g eq u a t i o n s r e l a t i n g t h e ex act i n t e g r a l t o t h e ap pr o xi m at i on s ( 8 .1 ) ,( 8 . 2 ) .

wherein we a g a i n i n d t h e e r r o r c o n s t a nt s d e f i n e d i n ( 3. 7). A t f i r s t s i g h t ,t h i s i s a s a t i s f a c t o r y r e s u l t s howing t h a t t h e e r r o r s are O(h ) on a smoothgr id . However, th e qu an t i ty being approximated i s I,rr%dA, which i sO(rh ). The problem i s t h a t , n ea r t h e a x i s , w e f i n d c e l l s fo r which r i sO(h) , and the e r ro r terms a re O ( h ) r e l a t i v e t o t h e t r u e v a l u e s . On a non-smooth mesh, t he y a re even O(1).

4

2

We obs erv e from (8.6) and (8.7) t h a t t h e s e p a r t i c u l a r e r r o r s c ou ld bee l imi na ted by choosing the l i ne ar combina t ion

N o t e t h a t t h i s i s th e combina tion ob ta ined i f w e assume that K , -, bothv ar y l i n e a r l y a l o n g AB, and t h en i n t e g r a t e ex ac t l y . The e r r o r an a l y s i s o r as i ng le s id e now g ives

1 L+- rB - rA) F 1 .24 -r r

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T h e re f o re , t h e t o t a l e r r o r a ro un d a p a r t i c u l a r p o l y g o n i s iI3 1 2 1 2

~

I1 IA B I r Fds =- 1 ;(Ax) A r +- F 1 r (Ax) (Ar)12 -xx 6 -xr

(8.10)1 312 -rr- 1 r (Ar) + ... .

The ana lys i s i s now id en t i ca l to th e two-d imens iona l case excep t f o r theappearance of th e mean ra d i i ( r ) i n e ve ry term. These ac to rs , however,compl ica te mat t e r s cons idera b ly . For example , i t i s no l o n g e r t r u e t h a t ar eg u l a r po ly go n ha s v an i s h i n g e r r o r co n s t an t s b ecaus e co n t r i b u t i o n s fr omo p p o s i t e s i d e s no l o n g er c a n c e l i n g e n e r a l . To p ut t h e e r r o r c o n s t a n t s i n t oforms which make th e i r ord er of magnitude sel f -e vid en t needs con s id era blemanipula t ion , which has on ly been ca r r ie d ou t fo r qu ad r i l a te ra l s . The eas ie s tterm t o ana lyze then i s

3 4 3 3 4A- r i ) = r - 2r r + 2 r B r A - rc ( r i + r.1+1) (ri+1 B B A

3 43B+ r - 2r r + 2rCrB- rC B

4 3 3 4C+ r - 2r r + 2rDrC - rD D C

4 3 3 4D+ r - 2r r + 2 r A r D - rA D

= 2 rA ( r B - r ) + 2rg3 ( r C - r + 2 r C ( r D - r ) + 2rD ( r A - rc)D A B

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2 2 - r r - r 2 )A A C CrDrB+ rB - r2 ( r A - rc ) ( r D - r B ) ( r D

32 B- r A - r c ) ( r B - r D ) ( r A + r + rC + r D ) ( r A - rB + rC - r D )

(8.11)- ( r - r c ) (rB - r D > r A + rB - r - rD > r A- rB - rC + r D > -2 A CThe f i r s t t e rm i s O(rh4) on grids which are smooth i n the sense of

t h e p r ev i o u s d i s cu s s i o n ; i t i s O(rh3) on non-smooth g r i d s . The secon dterm appears t o be O(h4) . That i t s h a l l be O (r h4) imposes a n a t u r a l co n -d i t i o n on t h e g r i d n e ar t h e a x i s . If the c e l l n ea r e s t t h e a x i s h as o ne s i d eo n t h e ax i s , t h en i t can be shown that one of t h e f a c t o r s ( r A + rB - rC -r D ) , ( r A - rB - rC + r,) i s O(h); the o ther i s O(rh)--which one i s whichdepends on the l abe l l ing . It i s assumed tha t th i s c e l l i s a l so smooth i n thesame sense as t h e o t h e r s .

We w i l l o mi t d e t a i l s of t h e ca l cu l a t i o n s t h a t show t h e o t h e r terms i n(8 .10) t o be O(rh4 ), s i m p l y ex h i b i t i n g t h e f i n a l r ea r ran gem en t s. We f i n d

1 ;(AxI2Ar = (xB- x,,) (x A - xc> ( r A - r + r 2 - r 2 ,B C D(8 .12)

2 2+ ( x B - x D > ( r A - r c (xA - x + xc -xD) .B

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1Here the l a s t two terms are O(rh4) on any smooth g r i d , and th e f i r s t I

Ione i s s o i f t h e a x i s i s inc luded. Also, w i t h g r e a t p a t i e n c e , w e can o b t a i n : ~

2 11 r(Ax)(Ar) = ( r A+ r + rc + r D ) ( r B - r D > ( r A r C > ( X A- xB+ xc - xD)B+ s ( r A + rB + rc + r D > ( r A rB + rC - r D)

12 A- ( r + rB - rc - r D ) ( r A - rB - rc + rD >(8.13)

where S is t he c e l l a r ea .On smooth g r i d s , t h e f i r s t t h r ee l i n e s a r e O ( rh 4 ); t h e f o u r t h i s

O(h6).Now we t u r n t o t h e terms i nvo lv ing - and - i n equa t ion (8 .3 ) . Both

of the se . te rms ar e O(h2) , but near the ax is they a lmost can cel s o t h erequi rement f or accuracy i s t he same a s f o r t h e o t h e r terms. A s t r a i g h t f o r -ward e val uat ion of /Ip dA cannot avo id e r r o r s O(h4) s o some subtlety i sneeded.

In teg ra t io n by pa r t s shows tha t

(8.14)

and w e can use t h i s t o r e w r i te t h e terms i n G and 2 as-

I r - dx - - dA = I r (E - p) dx + r 2 A (8.15)aran n an n

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where both terms on the RHS are of or de r (rh 2) . Now w e can ev a l u a t e t h eco n t o u r i n t e g r a l by t h e meth ods j u s t d i s cu s s ed ; t h e e r r o r terms w i l l be

(8.16)

The summations multtplying Sr, -rr have al re ad y been shown t o beO(rh4). The f i r s t summation can be evalua ted as

1 2 + (XA - x - x + X D>2-2 S[3(XA - XB + xc - XD> B C+ (x * + XB - xc - XD) 3 (8.17)

Every term i n t h i s e x p re s si o n can be j u s t i f i e d as on a smooth mesh,e x ce p t f o r t h e t h i r d l i n e . Here S i s O(h2) and the f i r s t term i n s i d e t h eb r ack e t i s andt h e o t h e r O (h 2) . Th us, t h e o v e r a l l o r d e r of th e exp res s io n must be h4.

O(rh4)

O(h4 ), bu t of th e two remaining terms, one w i l l be O(r2h2)

However, t h i s does no t dest ro y the second-order accurac y of th e scheme ne arthe a x i s , b ecau se i n t h a t r eg i o n (2 - p>xx is O(r) so t h a t a l l co n t r i b u -t i o n s t o ( 8.16) are a c t u a l l y O(rh4).

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Last l y, we have to eval uat e

r d ~ .n

(8.18) ,

A f or mul a f or t hi s can be f ound usi ng f i ni t e el ement t echni ques, i f t he Igr adi ent i s r ewr i t t en as

wher e J i s the J acobi an of t he t r ansf ormat i on (E ,TI) + ( x, y). Si nce weal so have dA = J dS dy, (8.18) can be f ound st r ai ght f or war dl y. The r esul ti s

The er r or t er m t ur ns out t o be

(8.19)

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+- [(rA + r + r + rD)I Zs(xA - xB + xC - +24 B C+ (XB - X J ( XA - xc > ( rA - r B + r - rDlC

(8.20)

Al l t er ms i n t hi s expr essi on ar e Orh4) provi ded t he gr i d i s smoot h andcont ai ns t he axi s. The scheme whi ch has emer ged, al t hough mor e cumber somet han one woul d hope f or , i s t he uni que cel l ver t ex scheme r et ai ni ng second-or der accur acy i n axi symmet r i c f l ow It may be not ed that i t has al so onepr oper t y commonl y demanded of a numer i cal code. It i s sat i sf i ed i dent i cal l yby uni f or m f l ow par al l el t o t he axi s. For such a f l ow, - i s a const ant , (2--) vani shes , and ap/ a r vani shes. Under t hese ci r cumst ances, al l cont r i bu-t i ons to t he bal ance equat i ons vani sh i dent i cal l y.

The express i ons whi ch have been f ound as appr oxi mat i ons, i n ef f ect , t oaF/ ax and aG/ ar have been der i ved by ot her aut hor s f r omot her consi der a-t i ons. Margol i n and Adams [ 1 5 ] , i n t he cont ext of a Lagr angi an scheme, soughtnumer i cal approxi mat i ons that woul d make t he equat i on

--- V - di v u (8.21)V dt -

an i dent i t y at t he di scr et e l evel . Here V i s t he vol ume of a movi ng par celof f l ui d. The uni que sol ut i on to thi s pr obl em gave the di f f er ent i at i onoperat or s der i ved here. Margol i n and Adams r eport gr eat l y i mpr oved accur acyi n t hei r comput at i ons.

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The same o p e r a t o r s were a l s o obta in ed by Holm e t a l . [18] as t he un iqueo p e r a t o r s t o p r e s er v e i n t h e d i s c r e t e f o r m u l a ti o n c e r t a i n Ha mi lt on ia n p ro-p e r t i e s of t h e d i f f e r e n t i a l e qu at io ns .

If the scheme were t o be used i n prac t i ce , i t may be noted that somes i m p l i f i c a t i o n s of t h e e x p r e s s i o n s a re p o s s i b l e , a t l eas t f o r q u a d r i l a t e r a lco nt ro l volumes. The terms i n v o l v i n g F can be made more compact byrearrangement. The ter ms .i nvo lvi ng p in (8.19) can be used p a r t l y t o c a n c e lt h e terms i n (2 - p) . What emerges from t h i s is the fo l lowing, which w eclaim is t h e s i m p l e s t ve rs i on of the unique formula

--

(8.22)

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A s t r i k i n g E e a t u r e i s th a t t he approximat ions t o I r - dx and ll 2 dAa re pr ec i s e l y those th a t would have been ob ta ined by f i n i t e - e l e m e n t i n t e g r a -t i o n s a ssuming b i l i n e a r v a r i a t i o n s f o r G and p . However, nei t he r of tho seterms i n d i v i d u a l l y i s second-order ac cu ra te . The combination i s , g i v e n s u i t -a b l e c o n d i t i o n s on t h e g r i d .

- -

9. Conclus ionsThe c e l l ver tex schemes s tud ied i n t h i s paper have been cla imed as

second-order accura te on a r b i t r a r y g r i d s . C e r t a i n l y t h e r e s u l t s show a nimpress ive convergence as t h e g r i d s are r e f i n e d [ l o ] . However, w e have shownt h a t on a r b i t r a r y g r i d s , t h e l o c a l tr u n ca t i on e r r o r may b e f i r s t - o r d e r .E x p l i c i t e r r o r c o n s t a n t s a s s o c i a t ed w i th g r i d g eom et ry hav e b een d e r iv ed .T hese i n d i ca t e what s o r t of c e l l shapes t o avo id and could a l s o be used asp a r t o f a mesh ref inement s t ra tegy.

The ext en si on t o axisymmetric f low has been made, and i t seems h a r d e r t oach iev e accuracy then. The unique formula which re ta i n s second-order acc urac ynea r th e ax i s has been der ived . However, th e ex te ns io n to fu l l y th ree-dimensional schemes i s a n e x t r e m e l y d i f f i c u l t piece of a lgebra which has no tye t been a t t empted .

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REFERENCES

[l] A. Ri zzi , M. I nouye, "A t i me- spl i t f i ni t e- vol ume t echni que f or t hr ee- ,I:i mensi onal bl unt - body f l ow " AI AA J our nal , No. 11, 1973, pp. 1478-

1485. 9[2] P. D. Lax, B. Wendr of f , "Di f f erence schemes f or hyperbol i c equat i ons

w t h hi gh order of accur acy. " Comm Pur e and Appl . Math. , Vol . 17,1964, pp. 381-398.

[3] P. D. Lax, Weak sol ut i ons of nonl i near hyper bol i c equat i ons and thei rnumer i cal comput at i on. " Comm Pur e and Appl . Math. , Vol . 7, 1954, pp.159-193.

[4] A J ameson, W Schm dt , E. Tur kel , "Numer i cal sol ut i ons o t he Eul erequat i ons by f i ni t e vol ume met hods usi ng Runge- Kut t a t i me- mar chi ngschemes. " AI AA paper 81-1259, 1981.

[5] E. Tur kel , "Accur acy of schemes wi t h nonuni f orm meshes f or compr essi bl ef l ui d f l ows. " Appl . Numer . Math. , i n pr ess.

[6] E. Tur kel , S. Yani v, U Landau, "Accur acy of schemes f or t he Eul erEquat i ons wi t h nonuni f orm meshes .I1 I CASE Report No. 85-59, 1985. Al soAI AA paper 86-0341.

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R-H. Ni , "A mul t i pl e- gr i d scheme f or sol vi ng t he Eul er equat i ons. " AIAA-J our nal , Vol . 20, No. 11, 1982, pp. 1565-1571.

R L. Davi s, R- H. Ni , W W Bowl ey, "Predi ct i on of compr essi bl e l am narvi scous f l ows usi ng t i me- mar chi ng cont r ol vol ume and mul t i pl e gr i dt echni que. " AIAA J our nal , Vol. 22, No. 11, 1984, pp. 1573-1581.

R Lohner , K Mor gan, J . Per ai r e, 0. C. Zi enki ewi ez, "Fi ni t e- el ementmet hods f or hi gh- speed f l ows. " AIAA paper 85-1531-CP, 1985.

M G Hal l , "Cel l ver t ex mul t i gr i d schemes f or sol ut i on of t he Eul erequat i ons. " I n Numer i cal Methods f or Fl ui d Dynam cs 11, K. W Mor t on, MJ. Banes (eds. ), Oxf ord Uni ver si t y Pr ess, 1986, pp. 303-345.

M F. Pai sl ey, "A compari son of cel l cent r e and cel l ver t ex schemes f ort he st eady Eul er equat i ons. " Oxf or d Uni ver si t y Comput i ng Labor at ory,Numer i cal Anal ysi s Gr oup, Report 86/ , 1986.

K W Mor t on, M F. Pai sl ey, "On t he cel l - cent r e and cel l - ver t exapproaches t o t he st eady Eul er equat i ons and t he use of shockf i t t i ng. " 10t h I nt . Conf . Num Meth. Fl ui d Dyn. , Bei j i ng, 1986.

A. J ameson, T. J . Baker , N P. Weat her i l l , " Cal cul at i on of i nvi sci dt r ansoni c f l ow over a com et e ai rcraft . " AIAA DaDer 86-0103. 1986.

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[14] P. L. Roe, "Approximate Riemann solvers, parameter vectors anddifference schemes." J. Comput. Phys., Vol. 43, 1981, pp. 357-372.

[15] L. F. Margolin, T. F. Adams, "Spatial differencing for finite diEferencecodes." Report LA-10249, Los Alamos National Laboratory, January 1985.

(161 D. D. Holm, R. A. Kupershmidt, C. D. Levermore, "Hamiltoniandifferencing of fluid dynamics." Advances in Applied Mathematics, Vol.

6, 1985, pp. 52-84.

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APPENDI XSome Pr oDer t i es of t he J ameson For mul a

At conver gence, J ameson s scheme f or t r i angl es [ 13] sat i sf i es

around t he per i meter of every pol ygon f ormed f r om t he set of t r i angl es meet i ngat a common ver t ex. Thi s has been pr esent ed as a set of f l ux bal anceequat i ons around an over l appi ng set of cont r ol vol umes. A st r ai ght f or war dr ear r angement of (A. l ) i s

G( x - X ) = o .yi - 1) - 7 -i i + i - 1Now because i f t he pol ygon has an odd number of s i des,we t r avel t w ce round i t s per i met er to r ear r i ve at our st ar t i ng ver t ex, andwi t h even si des there ar e two cl osed pol ygonal pat hs. Ther ef or e, we can addar bi t r ar y const ant s to the 3,5. Let us choose 3, GJ) whi ch ar e theval ues at t he common ver t ex ( Fi gur e lb).

c ( Yi +1- Yi - 1 ) = 0 ,

Rewr i t e ( A . 2 ) as

The r eason f or t he f act or s i s shown i n Fi gur e 4 , wher e P, Q ar et he cont r oi ds of OABS, OBC. Then

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So ( A . 3 ) can be i n t e r p r e t ed as th e f lu x balance around non-overlappingco n t r o l volumes whose ed ges j o i n t h e cen t r o i d s . T h i s i n t e r p r e t a t i o n l e ad s t oa dual form of t h e a l g o r i t h m as fo l lows .

Scheme IIFor a l l edges MN i n t h e mesh, compute

where S , T are t h e o t h e r v e r t i c e s of t h e t r i a n g l e s a d j a ce n t t o MN, and uset h i s f l u x t o u pd at e M , N .

T h i s h a s an i de nt i ca l outcome t o th e s tan dar d scheme, which w e r e p e a t f o rc o n t r a s t .

Scheme IFor a l l edges MN i n t he mesh, compute

and u s e t h i s f l u x t o up d at e S , T.Indeed, there are two more p o s s i b i l i t i e s , both of which w i l l also y i e l d

t h e same r e s u l t s a t convergence.

1IIiIIIi!1II

I

III

1

Scheme 1x1For a l l edges MN i n t he mesh, compute

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and u se t h i s f l u x t o up da t e M y N.

Scheme IVFor a l l edges MN in t h e mesh, compute

and u se t h i s f l u x t o u pd a t e S, T .Given t h e i n t e r p r e t a t i o n t h a t u i s piecewise l in ea r wi th in each t r i -

ang le , one i s tempted t o look fo r co ntro l volumes al lowing a more accuratein t eg ra t ion . For example (F igure 5 ) , we may con si de r th e co n t r o l volumeformed by jo ining the centroids of e ach t r i a n g l e t o t h e c e n t e r of each s i d e .

Along P I t h e v a l u e of F, s a y , v a r i e s l i n e a r l y f rom - ( F +EB) t o( I n view of- (F +EA +xB) with average value3 4t h e d i s c u s s i o n i n S e c t i o n 6 , we n e g l e c t t h e n o n l i ne a r v a r i a t i o n s of Somea l g e b r a w i l l show th a t t h i s does nothing more th an reproduce equ at i ons ( A . l )o r (A.3). Th is r e su l t i s o bvi ou s i f we t h i n k of i n t e g r a t i n g o ve r t h earea of th e co nt ro l volumes . Within OAB, i s a cons tan t and the area ofO J P I i s one- third th at of OAB.

-

1- 2 - 01'5% + 2F + 5 F ).1 2 -A -B

-.)

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"b )

Figure 1. (a) Part of a structured quadrilateral mesh.Part of an unstructured triangular mesh.b )

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Figure 2. Examples of polygons with zero error constants.

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r

Figure 3. A computational cell partly rotated about the axis.

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C

Figure 4 . Alternative control volume for Jamesons scheme.

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C

Figure 5 . Control volume using c e l l b i s ec t o rs .

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Standard Bibliographic Page

17 . Key Words (Suggested by Authors(s))numeri cal appr oxi mat i on, compr essi bl ef l ow, er r or est i mat es

. Report No. NASA CR- 178235 12. Government Accession No.

18. Distribution Statement0 2 - Aerodynam cs6 4 - Numer i cal Anal ysi sUnc l ass i f i ed - unl i m t ed

I CASE Repor t No. 87- 6ERROR ESTI MATES FOR CELL- VERTEX SOLUTI ONS OF THECOMPRESSI BLE EULER EQUATI ONS

. Title and Subtitle

19. Security Classif.(of thi s repo rt)

. Author(s)P . L. Roe

20 . Security C lassif.(of thi s page) 2 1 . N o. of Pages 2 2 . Price

1 y f o mi rga izatio Name andn s i i We t or eomputer qepdrperi cati ons i n Sci enceand Engi neer i ngMai l St op 1 3 2 C , NASA Langl ey Resear ch Cent erHamt on. VA 23665-52252. Sponsoring Agency Name and Address

Nat i onal Aer onaut i cs and Space Adm ni st r at i onWashi ngt on, DC. 2 0 5 4 65 . Supplementary Notes

Langl ey Techni cal Moni t or :J . C Sout hFi nal Repor t

Subm t t ed

3. Recipients Catalog NO.5 . Report Date

J anuar y 19876. Performing Organization Code

8. Performing Organization Repor t N o.87-6

10. Work Unit No.

11. Contract or Grant No.NASl - 1810713 . Type of Report a nd Period Covered

CouUact or Repor t14 . Sponsoring Agency Code505-90-21-01

t o AI AA J .

6. AbstractThe cel l - ver t ex schemes due t o Ni and J ameson, et al . have been subj ect ed t oa t heor et i cal anal ysi s of t hei r t r uncat i on er r or . The anal ysi s conf i r ms theaut hors cl ai ms f or second- order accur acy on smoot h gr i ds, but shows t hat t hesame accur acy cannot be obt ai ned on ar bi t r ar y gr i ds . I t i s shown t hat t heschemes have a uni que gener al i zat i on t o axi symmet r i c f l ow t hat pr eser ves t hesecond- order accur acy.