A posteriori pointwise error estimation for compressible fluid flows using adjoint parameters and Lagrange remainder

Transcription

1 A posriori poiwis rror simaio for comprssibl fluid flows usig adjoi paramrs ad Lagrag rmaidr Sor il: A posriori poiwis rror simaio usig adjoi paramrs A.K. Alsv a ad I. M. avo b a Dparm of Arodamics ad Ha Trasfr RSC EERGIA Korolv Moscow Rgio 47 Russia Fdraio b Dparm of Mamaics ad C.S.I.T. Florida Sa ivrsi Tallaass FL 6-4 SA SMMAR T poiwis rror of a fii-diffrc calculaio of suprsoic flow is discussd. T local rucaio rror is drmid b a Talor sris wi rmaidr big i a Lagrag form. T coribuio of local rucaio rror o oal poiwis approimaio rror is simad via adjoi paramrs. I is dmosrad b umrical ss a rsuls of umrical calculaio of gasdamics paramr a a obsrvaio poi ma b rfid ad a rror boud ma b simad. T rsuls of umrical ss for cas of parabolizd avir- Sos PS ar prsd as a illusraio of proposd mod. Kwords: adjoi problm a posriori rror simaio parabolizd avir-sos. ITRODCTIO A prs Ricardso rapolaio [ ad ] is mos popular mod for simaio of discrizaio rror i CFD. forual a corrc us of Ricardso rapolaio rquirs a s of grids o prov moooous covrgc ad o drmi ral ordr of covrgc for cosidrd soluio. Tis ma ur ou o b vr psiv from viwpoi of compur rsourcs. T raso for is siuaio is isc of ma ffcs a ma cag omial ordr of grid covrgc. T simpls ampl is covrgc ordr rducio i prsc of socs. For scms of ird ad four accurac ordr rducio of covrgc ra was dmosrad i [4] for comprssio wav. T wors of [5-7] cofirm is ffc. Spaial ouiformi of grid ma also rduc covrgc ra [8]. A alraiv approac for a posriori rror simaio as isivl b dvlopd for las dcad Rfs. [9-5]. I is usd for simaio of rror of som quaiis of irs goal fucioals poi-wis paramrs c usig rsidual rucaio rror ad adjoi dual quaios. I Rf. [8] is approac is usd for wav quaios i Rf [] i is usd for raspor quaio. I Rfs. [-5] a posriori rror simaio is obaid for avir-sos ad Eulr quaios. I s wors Galri mod is usd for local rror simaio wil adjoi quaios ar usd for calculaig ir wigs i arg fucioal rror. A similar approac was usd i [6-7] for rfim of pracicall usful fucioals bo b fii-lm ad fii-diffrc mods. T local rucaio rror rsidual was simad roug acio of diffrial opraor o irpolad soluio wil is coribuio o fucioal was calculad usig a adjoi problm. T rror is dmosrad o b composd of wo compos firs big compuabl usig adjoi paramrs ad rsidual wil scod big icompuabl dpdig o rrors of soluio of bo primal ad adjoi problms. I [6-8] iformaio o spaial disribuio of rsiduals was usd for

2 ms rfiig for dimiisig icompuabl rror abov simaio of compuabl rror. A surv of a posriori rror simaio usig adjoi quaios ma b foud i [4]. I prs wor w cosidr aor approac for simaio of compuabl rror if compard wi [6-7]. I is basd o a diffrial approimaio DA [8] isad of o rsidual simaio ad is mor aural for fii-diffrcs. Tis provids crai pculiariis bo i applicabili domai ad faurs of mods. W us a local rucaio rror drmid b a Talor sris wi rmaidr i Lagrag form ad adjoi quaios i a coiuous form. Tis abls us o corrc rror ad o obai a asmpoic rror boud for rfid soluio. T rfim ad rror boud ar obaid o sam grid as a mplod for primal problm soluio ad rquir idical compur im. Tis approac was usd for a rasfr quaio i [7 47]. Hri w cosidr a applicaio of approac o a fii-diffrc approimaio of parabolizd avir-sos ad Eulr quaios. For illusraig mai ida w us quaio ad is fiidiffrc approimaio ; l soluio b smoo oug o possss all cssar drivaivs o b boudd. L us us Talor sris wi rmaidr i Lagrag form paramrs ar uow. α β α β Tus a fii-diffrc quaio is quival o a approimad quaio wi a addiioal prurbaio rm. Mamaical dails of is quivalc ma b foud i [ 8]. L us fid rror of fucioal as a fucio of rucaio rror. For is purpos l us iroduc Lagragia dd s s Ω δ δ ε dd L ψ ε Ω. I ma b sow from is Lagragia variaio a for soluios of dirc ad adjoi s s δ δ ψ ψ problms variaio of fucioal causd b rucaio rror compo from drivaiv quals dd ψ α δ ε Ω Is discr form i ma b rcas o assum form ψ α δ ε 4 sig a Talor pasio prssio 4 ma b wri as ψ α δ ε 5 T firs par of sum 5 ma b usd for rfiig fucioal; rror is causd b scod par du o uow paramrs. Ts paramrs blog o ui irval so w ma obai a boud of is prssio α α

3 α ψ ψ sig suc simas for bo coordias w ca drmi a boud of fucioal rror afr rfim corr corr sup sup < 7 ac Tis approac also provids a sima of igr ordr rms i 4. For ampl a sima of scod ordr ovr α as form 4 α 4 ψ 4 < ψ For a scod ordr sima w obai sup sup < 4 4 sup corr 9 ac For a ifiil smoo soluio w ma wri corr corr s sris covrg du o ac < s sup s s sup s sup. If all drivaivs ar boudd s s s α ψ C s <. vrlss is dos s!! s o guara simaio o b small oug o b of a pracical sigificac. W sould us umrical prssios for ig ordr drivaivs i abov formulaios. If umrical soluio as oscillaios s simas ma b oo larg ad b of o pracical us. So cosidrd approac ma b usd ol for fii diffrc scms wic ar moooous oug. Eprssio 7 is corrc for ac valus of adjoi paramr. I rali adjoi problm is solvd b som fii-diffrc mod so i coais som approimaio rror ψ ψ ac ψ. Hc simaio of fucioal variaio as a compo drmid b adjoi problm rror. ε δ ψ dd Ω Tis rm corrspods o rmaiig rror accordig o [9] ad is associad o rrors of approimaio of bo adjoi ad primal quaios. Wors [6-8] cocr a cosrucio of a ms for miimizaio of is rm. As a alraiv w ma us scod ordr adjoi quaios [4 44 ad 47] for calculaig is rm. If primal ad adjoi problms ar p solvd b mods of ordr O ad a p a O is rm is of O ordr. For scms of ig oug ordr p or a is rm is asmpoicall small if compard wi rror bouds drmid b 6.. THE ESTIMATE OF APPROIMATIO ERROR FOR FIITE-DIFFERECE CALCLATIO OF A FLOW PARAMETER Cosidr discussd mod of approimaio rror for simaig wo dimsioal suprsoic viscous flow Fig.. T odivrg form of parabolizd avir-sos quaios PS is usd. T flow is calculad b marc alog ais. 6 8

4 4 R P R 4 P R 4 Pr R κ 4 P RT; RT C T v ; Ω<< ma ; << ma ; O iflow boudar A Fig. w av: ; ; ; ; O laral boudaris B D ma codiios f/ ar imposd. f i T dsi a som poi is cosidrd as a simad paramr. L us wri simad valu i form of a fucioal. s s dd s s s Ω δ δ ε 5 W d o calcula gradi of arg fucioal wi rspc o local disurbacs rucaio rror. I is ow a mos ffici wa for gradi calculaio is basd o usig adjoi quaios [6]. Ts quaios ma b obaid i a sadard wa b uifig i a sigl Lagragia simad fucioal ad wa formulaio of flow damics problm. Hri w prs rsul. Addiioal dails ma b foud i [9 4]. i δ f. ADJOIT PROBLEM / / P P R 4 R R 4 RPr s s δ δ 6 T sourc i 6 corrspods o locaio of simad paramr. P R 8 R 7

5 5 P 4 R 8 9 R Pr Paramrs ar adjoi aalogs of dsi vloci compos ad rg. ma Iiial codiios C ma : ; boudar Eprssio for ma. corrspods o locaio of a simad paramr o f Boudar codiios o BD ; ma : ; Adjoi problm is calculad i rvrs dircio alog. For poi-wis rror simaio quaios av sigular sourcs Dirac s dla fucios Equaio 6. forual rgulari bo of parabolizd avir-sos ad corrspodig adjoi ssm ar uow. Bo s ssms of quaios ar of mid prbolic-parabolic aur. Accordig o [48] a rasfr parabolic problm wi similar sourc is wll-posd for α H Ω α Ω R. From is aalog cosidrd problm ma b wllposd i H Ω > a fac a gdrs corrspodig compuaioal difficulis. Howvr if β w smoo sourc rm accordig o [4546] w ma obai a soluio s H Ω β > aloug coaiig a rror proporioal o smooig paramr s s > wic ma b as small as cssar. Mods of fii diffrc soluio for suc quaios ar also prsd i [4546]. T arg fucioal variaio as a fucio of rucaio rror as followig form: δε δ δ δ δ dd Ω I ss prsd blow w compar rsuls of fii-diffrc calculaios wi aalical soluios corrspodig o iviscid gas flows. I is co ifluc of viscous rms i quaios -4 o a simad paramr is of irs. W cosidr soluio of quaios wiou viscosi as a o-prurbd o. L viscous rms disurb is soluio. For ampl for logiudial vloci udisurbd valus ar govrd b quaio wil disurbd os ar govrd b. T variaio of arg fucioal R du o viscous rms assums form δε... d Ω R Ω

6 6 I coras o 6-9 corrspodig adjoi quaios av o viscous rms. Crail is approac is valid ol w ifluc of viscous rms is small oug i.. w do o caus a radical cag of flow srucur. Aor raso for dvlopm of is ciqu ariss for discoiuiis a ar pical of suprsoic flows dscribd b Eulr quaios for ampl. T approac basd o diffrial approimaio is o applicabl for suprsoic Eulr quaios du o uboud drivaivs. vrlss w ma us parabolizd avir-sos for basic flow calculaio cosidr viscous rms as a prurbaio ad calcula ffc of is prurbaio o soluio. Tis ma abl us o pad applicabili of diffrial approimaio approac o discoiuous flows dscribd b Eulr quaios. 4. FIITE DIFFERECE SCHEME To sud approimaio rror ri w d a fii diffrc scm avig largs rucaio rror. As a cosquc w us a firs ordr scm covciv rms big obaid b upwid diffrcs [4]. vrlss svral rms ar approimad b smmrical diffrcs wi rsul a av scod ordr. A fii-diffrc scm for opio is prsd blow. I coais wo sps prdicor ad corrcor. Bo sps ar calculad implicil usig r poi Tomas algorim. T ild mars paramrs compud a firs sp. > Prdicor 4 R P P. 5 R 4 P P. 6 RPr Φ 7 Corrcor. 8 R P P. 9 R 4 P P. RPr Φ T fii diffrc scm as a similar form for adjoi ssm. T mai faur is prsc of sourc rm i 6 wic is rlad o locaio of simad poi. For fi oug grids a mollificaio smoo approimaio of s s δ δ δ -fucio ma b cssar for approimaio of is sourc rm []. T mods of fii diffrc soluio of quaios wi suc sourcs ar prsd i [4546].

7 7 5. ESTIMATIO OF THE TRCATIO ERROR T oal approimaio rror dpds o a local rucaio rror. I ordr o drmi i w pad fii diffrcs i Talor sris wi Lagrag rmaidr. For illusraio l us prs is simaio for o of fii-diffrc rms i 9. α T corrspodig compo of arg fucioal variaio s assums form: dd Ω α δ ε Is discr form is α δ ε. For a firs ordr ovr i ma b prsd as α α δ ε 4 T firs par of is sum ma b usd for rfiig of fucioal corr 5 olimiad rror is gdrd b scod par of 4. I as a uppr boud: sup α 6 Toal rfim of fucioal drmid b all firs ordr rms of fii-diffrc scm 8- follows: corr 7

8 8 Toal prssio for rror boud causd b firs ordr rms of 8- as form: sup 8 Similar prssios ar obaid for covciv rms of scod ordr accurac ad for viscous rms. A boud of rfid fucioal rror ma b drmid b s prssios as: sup < ac corr 9 Tis boud dos o accou for icompuabl rror prssios similar o or rrors causd b boudar codiio approimaio c. I also uss drivaivs wos bouddss ca o b prov a prs. So i rquirs a cofirmaio via umrical ss. 6. MERICAL TESTS 6.. COTIOS FLOW FIELD A compariso bw compuaios b fii-diffrcs ad aalic prssios ad a aalsis of rror simas is prformd for Pradl-Mar flow. T rror of flow dsi pas pasio fa was addrssd frsram Mac umbr M4 agl of flow roaio α. L us drmi approimaio rror usig adjoi approac ad compar i wi dviaio of fii-diffrc soluio from aalic o. Fig. prss dsi isolis i flow-fild Fig. illusras adjoi dsi isolis a cocraio of isolis corrspods o a poi of simaio. Fig. 4 prss spaial disribuio of dsi of rror boud 8 ad Fig. 5 dpics isolis of dsi of rror causd b viscous rms. Figs. 4 ad 5 drmi os spaial rgios a gra mai par of rror for paramr a simad poi. Rsuls prsd i Figs. -5 corrspod o calculaios aig io accou viscosi PS R. Fig. 6 prss rlaiv rror of flow dsi calculaio for R as a fucio of rciprocal of spaial sp i dircio umbr of ods. T par of rror causd b

9 9 corr visc ac viscous rms rlaiv dviaio of rfid soluio from aalical o ad boud of rfid soluio rror 6 ar prsd. I ca b s a mai par of rror is drmid b viscosi ad i ma b compud ad limiad. T rfid rsul is clos o aalical o ad is locad wii irval of rror boud. vrlss r is o covrgc of rror boud as pcd from prssio 6. L us cosidr rlad rsuls for iviscid flow. Fig. 7 prss dviaio of fiidiffrc soluio from aalic o ad corrcio of rror i accordac wi 5. T rfim of soluio usig adjoi paramrs accordig 5 abls limiaio of major par of discrizaio rror. T firs ordr of compuabl rror 5 ma b dcd if aalz Fig. 7. Calculaios dmosrad a good coicidc of rfid soluio wi aalical o ad rliabili of rror boud sima Fig. 8. vrlss pcabl scod ordr of accurac 6 dos o maifs islf. W ms is fi oug rror boud pracicall dos o dpd o sp siz. Tis is causd b grow of ird drivaivs of flow paramrs as sp siz dcrass. I ma b du o formaio of wa discoiuiis i flowfild. A compariso of Figurs 6 ad 8 dmosras a impac of viscosi ffc usig adjoi quaios abls us o obai rsul clos o iviscid compuaio as far as accurac is cocrd. Tus r is fasibili for calculaio of iviscid flow Eulr quaios ad a posriori rror simaio o basis of PS. Tis ds applicabili of cosidrd mod wic is o dircl applicabl o suprsoic Eulr quaios du o isc of discoiuous soluios. Fig. 9 prss dpdc of rror of rfid soluio i compariso wi iiial rror of soluio causd bo b viscous rms ad b approimaio rror as a fucio of R umbr. As viscosi dcrass a crai icras of rror boud sima is visibl du o grow of ird drivaivs of gasdamical paramrs. For small oug R umbrs rror of fii-diffrc calculaio bras rror boud a is causd b sigifica disorio of flow par compar Fig. R ad Fig. R. I gral for a smoo flow rrors bo for iviscid flow ad for viscous flow rfid via adjoi paramrs ar clos. 6.. DISCOTIOS FLOW FIELD o As aor s rror of dsi pas crossig socs α ±. M4 R is calculad. Fig. prss dsi isolis wii flowfild Fig. illusras isolis of adjoi dsi Fig. sows dsi of rror boud accordig 8 ad Fig. 4 prss isolis of rror causd b viscous rms. Tis s is mor complicad du o uboudd drivaivs of gasdamics paramrs for iviscid flow. T prsc of viscosi abls us o calcula flows wi socs wil a sam im i iroducs a rror proporioal o /R. W viscosi dcrass is rror dimiiss oo uforual rror rlad o uboudd drivaivs icrass simulaousl. Fig. 5 prss rsuls for R as a fucio of spaial sp siz. T viscous compo of rror is small oug dviaio of fii-diffrc soluio from aalical o is small also ad wal dpds o sp siz ad is ffc ma b aribud o ucorolld rrors o-divrgc of scm possibl. Fig. 6 prss rsuls for iviscid flow as a fucio of spaial sp siz. Bo rror ad rror boud av a ordr of covrgc ovr grid siz of O. Fig. 7 sows dpdc of rror o Rolds umbr. As viscosi dcrass rror ad rror boud icras ad approac asmpo a iviscid limi.

10 A aural wa o limia rrors cocd wi a odivrg scm is coic of divrg o. T followig ssms of divrg Eulr quaios wo-dimsioal ad rlad adjoi quaios wr usd i umrical ss. i Pδ Divrg Eulr quaios: ; 4 i ; 4 4 ; Hr P is alp / is i oal alp. Adjoi quaios: i i / ; 4 i i i ; 44 ; 45 Svral varias of firs ordr fii-diffrc scms wo dimsioal wr usd icludig "door clls" [4] ad a scm of Coura-Isaacso-Rs [4]. As pcd dviaio of fii-diffrc soluio from aalic o for divrg scm is sigifical smallr compard wi odivrg o. forual rror simas us drivaivs a ar uboudd i divrg cas also cludig o-dimsioal flow. T rsuls of s compuaios ar aalogous o rsuls obaid usig odivrg scm. W ma compar grid covrgc for odivrg scm Fig. 6 wi rsuls for divrg o Fig. 8 iviscid flow. If w iroduc viscosi w ca obai covrg simas of rror for divrg scm oo Fig. 9. T abov ss ar orid owards compariso wi aalical soluios a blog o iviscid flows. Tis cras som spcifics. For ampl prvious s flowfild is composd of rgios of cosa gasdamical paramrs sparad b socs. So i is o bs problm from rror simas viwpoi usig drivaivs of gasdamical paramrs. Tus i is pdi o cosidr a s problm mor pical of viscous gas flows. L us cosidr a viscous R suprsoic wal udrpadd / j i suprsoic flow. Fig. prss isolis of dsi i flowfild Fig. prss isolis of adjoi dsi ad Fig. prss spaial dsi of rror boud sima 8. forual corrspodig aalical soluios for is problm ar uow. Hc soluio o fis ms was rgardd as a ac o. Fig. prss rror boud ad dviaio of soluio from ac o i dpdc o grid sp. T covrgc ordr for rror boud is clos o o sligl blow ad dcrass as ms is rfid. Tis rsul ma b a idirc vidc of prsc of discoiuiis i drivaivs of gasdamical paramrs. p j p

11 7. THE EALATIO OF COMPTABLE ERROR SIG RESIDAL T rsidual basd approac closl rlad o [9] is usd ri for a simaio of compuabl rror wiou plici us of diffrial approimaio. T mai diffrc bw is approac ad a of [9] is i rsidual calculaio. W do o us a irpolaio of flow paramrs from grid pois o oal domai. Isad w us a igr ordr scm o sam umrical soluio. L us cosidr is approac a a urisic lvl. Assum w av a flow-fild compud via crai fii-diffrc mod. W r o sima rror of is calculaio. L us us quaio as a ampl. L flowfild b calculad via firs ordr fii-diffrc approimaio ; 46 T diffrial approimaio of 44 ma b wri as δ. L us wri i i mor dail usig a Talor sris wi rmaidr i a Lagrag form. paramrs ar uow. α β 6 6 α β 47 L us rplac 46 b scil of scod ordr of accurac ad calcula rsidual arisig from applig ig ordr scm o flowfild calculad usig low ordr scm. η η 48 Eprssio 48 ma b padd i Talor sris as 6 6 χ η 49 T fucio is obaid from soluio of 46 ad so complis wi codiio 47. L us subsiu 47 i 49 rsidual assums form 6 6 χ η 6 6 α β 5 Corrspodigl lowr ordr rm of rucaio rror 47 as a form 6 6 β η η α χ Tus lowr rm of diffrial approimaio 47 ma b simad via a rsidual obaid from usig ig ordr scil o soluio calculad usig mai fii-diffrc

12 scm. As a rsul w obai fild of rsiduals a locall disurb ac soluio. Accordig o variaio of simad valu as a form ε δ δψ dd. Taig io accou 5 w obai ε δ δψ dd ηψ dd 5 Ω I coras o prssio 5 ma b asil calculad wiou owldg of diffrial approimaio. O or ad diffrial approimaio approac provids a mor accura accou of igr rms ad simaio of rfid soluio bouds. L us compar s approacs usig firs ordr upwid scm ad divrg form of Eulr quaios 8-4. For simaio of rsidual w us scod ordr approimaio 48. T dviaio of calculaio from aalical rror simaio usig DA basd 5 ad rsidual basd mods ar prsd i Fig. 4 for Pradl-Mar flow. T clos corrlaio of simas 5 ad 5 for coiuous flow is visibl. If w rcas Fig. 4 o a logarim scal all s fucios ma b dscribd b a firs ordr curv O. I coras o prssio 5 ma b dd o discoiuous flows if divrg fii diffrc scms ar usd. For discoiuous flow bo drivaivs i 48 ar uboud vrlss s sigulariis ar muuall compsad du o cosrvaio law ad rsidual is boudd. Tis rsidual ma b calculad usig ol a boudd combiaio of drivaivs of flow paramrs ad applid for rror simaio. A corrspodig ampl is prsd i Fig. 5 for crossig socs. I prss rror simaio usig a rsidual basd mod 5 DA basd 5 o ad dviaio of calculaio from aalical valu. Fig. 5 dmosras rsidual approac 5 o provid a muc mor accura simaio of rror if compard wi diffrial approimaio 5 wic plicil divrgs. forual rsidual basd approac dos o provid a rror boud. Ω Ω η 8. DISCSSIO T difficulis cocd wi usig adjoi approac ar of sam aur as os arisig wil usig Ricardso rapolaio. T ar causd b prsc of discoiuiis a drmi ordr of accurac obsrvabl i umrical ss. L us cosidr is problm a a urisic lvl. For is purpos l us wri 5 i mor dail. L m b umbr of boudd drivaivs drivaivs of ordr m ad igr ma av a fii umbr of jump discoiuiis p is ordr of approimad drivaiv j is formal ordr of accurac D of a fii-diffrc scm. L us approima drivaivs b fii diffrcs. D p j j D T limi lim corrspods o firs rm 5. Cosidr is p j Ω D m m asmpoic form. T drivaiv of ordr m as a asmpoic / / for jump discoiui wil drivaiv of ordr m as asmpoic / / / / corrspodigl drivaiv of ordr p j as p j j D j asmpoic. Tus lim p jm lim. Tr is a limid p j p jm D umbr of ods a paricipa i summaio i vicii of discoiui so muliplir apparig durig summaio sould b a io accou ildig

13 r s p j j D m p. p j D r s Tus rms of j- formal ordr of accurac coai a compo of j- ordr apparig du o igraio ovr smoo par of soluio ad a compo avig ordr i m p gdrd b jump discoiui of m ordr drivaiv. So ordr of covrgc dpds o soluio ad ma asmpoicall d o a miimal ordr i m p as grid siz dcrass. I [47] ifluc of discoiuiis o rror is cosidrd for ampl a of spaial drivaiv of mpraur. T calculaio of approimaio rrors b cosidrd mod rquirs isc of boudd drivaivs of rlaivl ig ordr. T do o is alwas so for suprsoic Eulr quaios s simas ma b calculad ol for smoo soluios. If discoiuiis ar pcd for sudid flow us of viscosi abls us o coduc s simas. T viscosi gdrs is ow compo of rror wic ma also b limiad usig adjoi quaios. Tis approac prmis o obai rror simas for iviscid suprsoic flows usig is mod. aurall w ca sima o ol rror of dsi wri as a fucioal 5 bu rror of or fucioals. T diffrcs ar ol i form of sourc rms i adjoi quaios 6-9. For jusificaio of rror simas w sould vrif a uaccoud rror compo iducd b approimaio rror of adjoi quaios is small oug. For calculaio of is compo w ca solv scod ordr adjoi quaios[4]. Suc a suiabl ampl is prsd i [47] for a coducio quaio. For rror simas w us umrical rsuls a ma b sigifical lss smoo compud psical fild. Tus for crai fii-diffrc scms o-moooic rror bouds ma b oo larg. T applicabili of mod cosidrd abov is rsricd o umrical scms wic do o ibi opsical oscillaios. 9. COCLSIO T compuabl poiwis rror of viscous flow paramr causd b a fii-diffrc approimaio ma b valuad usig diffrial approimaio rms ad adjoi quaios. T asmpoic boud of rfid soluio rror ma b drmid simulaousl. umrical ss carrid ou dmosrad fficic of is mod for parabolizd avir-sos. T ifluc of viscous rms ma b calculad similarl ad a provids also for fasibili of simaig rrors of Eulr quaios. T compur im rquird for poi-wis rfiig of a sigl paramr a a sigl poi ad rror boud calculaio is qual o im rquird for flow-fild calculaio o sam grid. APPEDI A. OMECLATRE C v - spcific volum a capaci; - spcific rg C v T f - flow paramrs -alp -oal alp - spaial sps alog ad ; M - Mac umbr -umbr of im sps umbr of spaial ods alog umbr of spaial ods alog L- Lagragia P -prssur PR - Pradl umbr PrµC v /λ R - gas cosa

14 4 R µ ma -Rolds umbr T-mpraur - vloci compo alog - vloci compo alog coordias Gr lrs α β - cofficis i Talor-Lagrag sris δ -Dirac s dla fucio; corr sup - corrcabl rror cocd wi pasio alog ; - corrcabl rror; - corr compo of boud of ir rror cocd wi pasio i coordia ; - compo of boud of ir rror - spcific a raio ε - fucioal; µ -viscosi λ - rmal coducivi; - dsi; -mporal sp - adjoi variabls Ω- domai of calculaio Subscrips: rac boudar paramrs; a- aalical soluio; corr- corrcd rror; s simad poi; ac- ac soluio; - umbr of spaial ms od alog ; -umbr of sp alog ; sup- boud of ir rror; -compo of rucaio rror cocd wi Talor pasio i coordia ; - compo of rucaio rror cocd wi Talor pasio i im. sup

15 5 REFERECES. Marcu GI ad Saidurov. Diffrc mods ad ir rapolaios. Sprigr: Roac P. J. Quaificaio of crai i Compuaioal Fluid Damics. Aual Rviw of Fluid Mcaics 997; 9: 6.. Ro C. J. Grid Covrgc Error Aalsis for Mid-Ordr umrical Scms. AIAA Joural ; 44: Carpr M. H. ad Caspr J.H. Accurac of Soc Capurig i Two spaial Dimsios. AIAA J. 999; 79: Efraimsso G. ad Kriss G. A Rmar o umrical rrors Dowsram of Sligl iscous Socs. SIAM J. of umrical Aalsis 999; 6: Egquis B. ad Sjogr B. T Covrgc Ra of fii Diffrc Scms i Prsc of Socs. SIAM Joural of umric l Aalsis 998; 5: Ro C. J. McWrr-Pa M.A. ad Obrampf W. L. rificaio ad alidaio for Lamiar Hprsoic Flowfilds. AIAA Joural ; 4 : amalv. K. ad Carpr M. H. O Accurac of Adapiv Grid Mods for Capurd Socs. ASA/TM--45: Aiswor M. ad Od J. T. A Posriori Error Esimaio i Fii Elm Aalsis. Wil Irscic:.. Od J.T. ad Prudomm S. Goal-Orid Error Esimaio ad Adapivi for Fii Elm Mod. Compurs&Mamaics wi Applic. ; 4: Od J.T. ad Prudomm S. Esimaio of modlig rror i compuaioal mcaics. Joural of Compuaioal Psics ; 8: Od J.T. ad Prudomm S. O goal-orid rror simaio for llipic problms: Applicaio o corol of poiwis rrors. Compur Mods i Applid Mcaics ad Egirig 999; 76: -.. Joso C. O Compuabili ad Error Corol i CFD. Iraioal J. for umrical Mods i Fluids 995; : Hoffma J. ad Joso C. Compuabili ad adapivi i CFD. o appar i Ecclopdia of Compuaioal Mcaics 5. Harma R. ad Houso P. Goal-Orid A Posriori Error Esimaio for Comprssibl Fluid Flows. I umrical Mamaics ad Advacd Applicaios Brzzi F. Buffa A. Corsaro S. ad Murli A. ds Sprigr-rlag: ; pp dii D. ad Darmofal D. Adjoi rror simaio ad grid adapaio for fucioal oupus: Applicaio o quasi-o-dimsioal flow. J. Compu. Ps. ; 64: dii D. ad Darmofal D. Grid Adapaio for Fucioal Oupus: Applicaio o Two- Dimsioal Iviscid Flow. J. Compu. Ps. ; 76: D. Darmofal ad D. dii. Aisoropic grid adapaio for fucioal oupus: applicaio o wo-dimsioal viscous flows. J. Compu. Ps. ; 87: Gils M. B. O adjoi quaios for rror aalsis ad opimal grid adapaio i CFD. I Compuig Fuur II: Advacs ad Prospcs i Compuaioal Arodamics. Hafz M. ad Caug D. A. ds Wil: w or: 998; pp Pirc. A. ad Gils M. B. Adjoi rcovr of suprcovrg fucioals from PDE approimaios. SIAM Rv. ; 4: Gils M. ad Pirc.A. Improvd Lif ad Drag Esimas sig Adjoi Eulr Equaios". Tcical Rpor 99-9 AIAA Ro 999: -.. Par M. A. Tr Dimsioal Turbul RAS Adjoi Basd Error Corrcio AIAA Papr -849: -4.. Par M. A. Adjoi Basd Tr Dimsioal Error Prdicio ad Grid Adapaio. AIAA Papr 86 : Gils M. B. Ad Suli E. Adjoi mods for PDEs: a posriori rror aalsis ad posprocssig b duali. Aca umrica ; M.B. Gils.A. Pirc ad E. Suli. Progrss i adjoi rror corrcio for igral fucioals Compuig ad isualizaio i Scic 4; 6: A. Pirc ad M.B. Gils. Adjoi ad Dfc Error Boudig ad Corrcio for Fucioal Esimas o appar i Joural of Compuaioal Psics 4

16 6 7..A. Pirc ad M.B. Gils. Adjoi ad dfc rror boudig ad corrcio for fucioal simas AIAA Papr ; W. Bagr R. Raacr Fii Elm Approimaio of Acousic Wav Equaio: Error Corol ad Ms Rfim Eas-Ws J. of umr. Ma. 996; 7/4: R. Bcr R. Raacr A opimal corol approac o a posriori rror simaio i fii lm mods. I A. Isrls dior Aca umrica Cambridg iv. Prss ;-.. Huvli R. Raacr Duali-basd adapivi i p-fii lm mod J. umr. Ma. ; o. 8. P. Houso R. Raacr ad E. Suli. A posriori rror aalsis for sabilizd fii lm approimaios of raspor problms. Compu. Mods Appl. Mc. Egrg. ; 9 - : P. Mo ad E. Suli. T adapiv compuaio of far fild pars b a posriori rror simas of liar fucioals. SIAM J. umr. Aal. 998; 6: E. Suli P. Houso Adjoi rror corrcio for igral oupus i: Adapiv Fii Elm Approimaio of Hprbolic Problms Sprigr 4. E. Suli ad P. Houso. Fii lm mods for prbolic problms: a posriori rror aalsis ad adapivi. I: I. Duff ad G.A. Waso ds. T Sa of Ar i umrical Aalsis. Oford ivrsi Prss 997; E. Suli. A posriori rror aalsis ad adapivi for fii lm approimaios of prbolic problms. I: D. Krur M. Olbrgr ad C Rod Eds. A Iroducio o Rc Dvlopms i Tor ad umrics for Cosrvaio Laws. Lcur os i Compuaioal Scic ad Egirig. olum 5. Sprigr-rlag 998; Marcu G. I. Adjoi Equaios ad Aalsis of Compl Ssms. Kluwr Acadmic Publisrs: Dordrc Hardboud A.K.Alsv Corol of a Error of a Fii-Diffrc Soluio of Ha-Coducio Equaio b Cojuga Equaio Joural of Egirig Psics ad Trmopsics. 4; 77 iss Soi u.i. Mod of diffrial approimaio. Sprigr-rlag Alsv A.K. D Ivrs Covcio Domiad problm for Esimaio of Iflow Paramrs from Ouflow Masurms. Ivrs Problms i Eg. ; 8: Alsv A.K. ad avo I.M. Calculaio of crai Propagaio usig Adjoi Equaios. Iraioal Joural of Compuaioal Fluid Damics ; 7 4: Roac P. Compuaioal Fluid damics. Hrmosa publisr Kuliovsii A.G. Pogorlov.. ad Smov A.u. Mamaical Aspcs of umrical Soluio of Hprbolic Ssms. Moograps ad Survs i Pur ad Applid Mamaics 88 Capma&Hall/CRC: Boca Rao Fl. 4. Wag Zi avo I. M. L Dim F.. ad Zou.. T Scod Ordr Adjoi Aalsis: Tor ad Applicaios. Morol. Amos. Ps. 99; 5: Alsv A. ad avo M. O Esimaio of Tmpraur crai sig Scod Ordr Adjoi Problm. I. Joural of Compu. Fluid Damics ; 6 : A.K. Torbrg ad B. Egquis. Rgularizaio ciqus for umrical approimaio of PDEs wi sigulariis. J. of Sci. Compu. ; 9: J.Wald. O approimaio of sigular sourc rms i diffrial quaios. umr. M. Par. D 5 999; A.K. Alsv ad I. M. avo O a-posriori poiwis rror simaio usig adjoi mpraur ad Lagrag rmaidr Compur Mods i Appl. Mc. ad Eg. 4 accpd 48. J.L. Lios Poiwis Corol of Disribud Ssms i H.T. Bas d. Corol ad simaio i disribud Paramr ssms. Froirs i applid mamaics v. SIAM 99-9 Figur Capios Fig.. Flow sc. A- Erac boudar BD- laral boudaris *-locaio of simad paramr Fig.. Isolis of dsi Pradl-Mar flow

17 7 Fig.. Isolis of adjoi dsi cocraio corrlas wi simad poi Fig. 4. Isolis of dsi of rror boud 8 Fig. 5. Isolis of rror dsi causd b viscosi. Fig. 6. T rror of calculaio as a fucio of rciprocal of ms sp viscous flow R. -rror du o viscous rms -dviaio of rfid soluio from aalical o ad 4- bouds of rror Fig. 7. T rror of calculaio as a fucio of rciprocal of ms sp iviscid flow. -dviaio of fii-diffrc soluio from aalical o rror corrcio accordig 7. Fig. 8. T rror of calculaio as a fucio of rciprocal of ms sp iviscid flow. -dviaio of rfid soluio from aalical o rror bouds 8 Fig. 9. T rror of calculaio as a fucio of Rolds umbr. - dviaio of fiidiffrc soluio from aalical o - rror du o viscous rms - dviaio of rfid soluio from aalical o 4 5 rror bouds 8 Fig.. Isolis of dsi for small R R Fig.. Isolis of dsi crossig socs Fig.. Isolis of adjoi dsi cocraio corrlas wi simad poi Fig.. Isolis of rror boud dsi 8 Fig. 4. Isolis of rror dsi causd b viscosi. Fig. 5. T rror of calculaio as a fucio of rciprocal of ms sp viscous flow. - dviaio of rfid soluio from aalical o - rror dsi causd b viscosi 4 rror bouds 8 Fig. 6. T rror of calculaio i dpdc o rciprocal of ms sp iviscid flow. - dviaio of rfid soluio from aalical o rror bouds 8. Fig. 7. T rror of calculaio as a fucio of Rolds umbr. - dviaio of fiidiffrc soluio from aalical o - rror causd b viscosi. - rror boud 8 Fig. 8. T rror of calculaio as a fucio of rciprocal of ms sp iviscid flow divrg scm. -dviaio of fii-diffrc soluio from aalical o - corrcio 7 - boud of rror Fig. 9. T rror of calculaio as a fucio of rciprocal of ms sp viscous flow divrg scm. - dviaio of rfid soluio from aalical o rror bouds 8 Fig.. Isolis of dsi Fig.. Isolis of adjoi dsi Fig.. T dsi of rror boud 8. Fig.. T rror of calculaio as a fucio of rciprocal of ms sp. - dviaio of rfid soluio from ac o - boud of rror. Fig. 4. T compariso of diffr rror valuaios as a fucio of rciprocal of ms sp. - DA basd rror - rsidual basd rror 5 diffrc of aalical ad umrical valus. Fig. 5. T rrors as fucios of rciprocal of ms sp. - diffrc of aalical ad umrical valus - DA basd rror - rsidual basd rror 5.

18 8 B A f C D * s Fig.. Flow sc. A- Erac boudar BD- laral boudaris *-locaio of simad paramr D Ja 4 d Dirc D Ja 4 d Adjoi Fig.. Isolis of dsi Pradl-Mar flow 5 5 Fig.. Isolis of adjoi dsi cocraio corrlas wi simad poi D Ja 4 d Gridg D Ja 4 d Gridg Fig. 4. Isolis of dsi of rror boud Fig. 5. Isolis of rror dsi causd b

19 9 viscosi.e- 5.E- dro/ro 4.E -5.E- -.E- -.5E Fig. 6. T rror of calculaio as a fucio of rciprocal of ms sp viscous flow R. - rror du o viscous rms -dviaio of rfid soluio from aalical o 4- bouds of rror - LogdRo Log Fig. 7. T rror of calculaio as a fucio of rciprocal of ms sp iviscid flow. -dviaio of fii-diffrc soluio from aalical o rror corrcio accordig 7.

20 8.E- 6.E- 4.E-.E- dro/ro.e -.E- -4.E- -6.E- -8.E Fig. 8. T rror of calculaio as a fucio of rciprocal of ms sp iviscid flow. -dviaio of rfid soluio from aalical o rror bouds 8 6.E- 4.E-.E- dro/ro 4 5.E -.E- -4.E- -6.E- LogR -8.E Fig. 9. T rror of calculaio as a fucio of Rolds umbr. - dviaio of fii-diffrc soluio from aalical o - rror du o viscous rms - dviaio of rfid soluio from aalical o 4 5 rror bouds 8

21 D Ja 4 d Dirc Fig.. Isolis of dsi for small R R D Ja 4 d Dirc D Ja 4 d Adjoi Fig.. Isolis of dsi crossig socs 5 5 Fig.. Isolis of adjoi dsi cocraio corrlas wi simad poi D Ja 4 d Gridg D Ja 4 d Gridg Fig.. Isolis of rror boud dsi Fig. 4. Isolis of rror dsi causd b

22 viscosi. 4.E-.E-.E-.E- dro/ro 4.E -.E- -.E- -.E- -4.E Fig. 5. T rror of calculaio as a fucio of rciprocal of ms sp viscous flow. - dviaio of rfid soluio from aalical o - rror dsi causd b viscosi 4 rror bouds 8 4.E-.5E- dro/ro.e-.5e-.e-.5e-.e- 5.E-.E Fig. 6. T rror of calculaio as a fucio of rciprocal of ms sp iviscid flow. - dviaio of rfid soluio from aalical o rror bouds 8.

23 .5E- dro/ro.e-.5e-.e-.5e-.e- 5.E-.E -5.E R Fig. 7. T rror of calculaio as a fucio of Rolds umbr. - dviaio of fii-diffrc soluio from aalical o - rror causd b viscosi. - rror boud dro/ro Fig. 8. T rror of calculaio as a fucio of rciprocal of ms sp iviscid flow divrg scm. -dviaio of fii-diffrc soluio from aalical o -corrcio 7 - boud of

24 4 rror..5 dro/ro Fig. 9. T rror of calculaio as a fucio of rciprocal of ms sp viscous flow divrg scm. - dviaio of rfid soluio from aalical o rror bouds 8 D Ja 4 d Dirc D Ja 4 d Adjoi Fig.. Isolis of dsi 5 5 Fig.. Isolis of adjoi dsi

25 5 D Ja 4 d Gridg Fig.. T dsi of rror boud 8..E- dro/ro.e- 8.E- 6.E- 4.E-.E-.E -.E Fig.. T rror of calculaio as a fucio of rciprocal of ms sp. - dviaio of rfid soluio from ac o - boud of rror.

26 6.E-.8E- dro/ro.6e-.4e-.e-.e- 8.E-4 6.E-4 4.E-4.E-4.E Fig. 4. A compariso of diffr rror valuaios as a fucio of rciprocal of ms sp. - DA basd rror - rsidual basd rror 5 diffrc of aalical ad umrical valus. 5.E-.E dro/ro -5.E- -.E- -.5E- -.E- -.5E- -.E Fig. 5. T rrors as fucios of rciprocal of ms sp. - diffrc of aalical ad umrical valus - DA basd rror - rsidual basd rror 5.

27 7