on the properties of solutions of the adjoint eule

OnthePropertiesofSolutionsoftheAdjointEulerEquationsMichaelB.GilesNilesA.PierceOxfordUniversityComputingLaboratoryAbstractThebehaviorofanalyticandnumericaladjointsolutionsisexaminedforthequasi-1DEulerequations.Forshockedow,thederivationoftheadjointproblemrevealsthattheadjointvariablesarecontinuouswithzerogradientattheshockandthataninternaladjointboundaryconditionisrequiredattheshock.AGreen’sfunctionapproachisusedtoderivetheanalyticadjointsolutionscorrespondingtoisentropicandshockedtransonicow,revealingaloga-rithmicsingularityatthesonicthroatandconrmingtheexpectedpropertiesattheshock.Numericalsolutionsobtainedusingbothdiscreteandcontinuousadjointformulationsrevealthatthereisnoneedtoexplicitlyenforcetheadjointshockboundarycondition.Adjointmethodsaredemonstratedtoplayanimportantroleintheerrorestimationofintegratedquantitiessuchasliftanddrag.1IntroductionAdjointproblemsarisenaturallyintheformulationofmethodsforoptimalaerodynamicdesignandoptimalerrorcontrol.Fordesignapplications,theadjointsolutionprovidesthelinearsensitivitiesofanobjectivefunctionsuchasliftordragtoanumberofdesignvariableswhichparameterisetheshape.Thesesensitivitiescanthenbeusedtodriveanoptimisationprocedure.Considerableeorthasbeendedicatedtothedevelopmentofoptimaldesignmethodsbasedonthisapproach[1{8].Morerecently,adjointmethodshavebeenrecognisedasameansofachievingerrorcontrolinuiddynamicssimulations[9{12].Inthiscontext,theadjointsolutionrelatesthesensitivityoftheobjectivefunctiontothelocaltruncationerrorsintheowdiscretisation.Thisinformationcanthenbeusedtoprovideanaposteriorierrorestimateortoguideanadaptivemeshingalgorithm.Whilesignicanteorthasbeendedicatedtodevelopingpracticalmethodsbasedonad-jointformulations,therehasbeenlittlediscussionofthepropertiesoftheadjointsolutionsthemselves[13].Thepresentworkinvestigatesvariousissuesconcerningthederivationandapproximationofsolutionstothequasi-1DadjointEulerequations.ThestandardLagrangemultiplierderivationofJameson[1]isextendedtoincludetheeectofshocksintheformu-lationoftheanalyticadjointequations.ExplicitinclusionofthesteadyRankine{HugoniotconditionsviaanadditionalLagrangemultiplierdemonstratesthatattheshock,theadjointvariablesarecontinuousandthataninternaladjointboundaryconditionisrequired.Thisisconsistentwithacharacteristicviewpointwhichindicatesthatoneinternaladjointb.c.isneededduetothedisparityinthenumberofadjointcharacteristicsenteringandleavingtheshock.However,theconclusionsdierfromthoseofpreviousinvestigators[14{16].Thediscreteadjointequationscanbeformulatedintwoways,eitherbydiscretisingtheanalyticadjointequations(theso-called‘continuous’approach)[1],orbytransposingthedis-creteequationsobtainedbylinearisingthediscretisedowequations(the‘discrete’approach)12Giles&Pierce[7].Gileshaspreviouslyshownthatforquasi-1Dowswithshocks,aconservativediscreti-sationwhichissecond-orderaccurateinsmoothregionsoftheowproducesasecond-orderaccurateapproximationtothe‘lift’integral[17].Hence,thelinearisationofsuchamethod(onwhichthediscreteadjointisbased)mustproducealinearisedliftperturbationwhichisatleastrst-orderaccurate.Ontheotherhand,itislessclearwhetherthediscretisationoftheanalyticadjointequationsleadstothecorrectadjointsolutionifthereisnoexplicitenforcementofthespecialshockcondition.Toinvestigatethispoint,thepaperderivestheanalyticsolutiontotheadjointequationsforshockedow.ThisisaccomplishedbyconstructingtheGreen’sfunctionsforthelinearisedEulerequations,includingthelinearisedRankine{Hugoniotconditions,usinganextensionoftheapproachdevelopedbyGilesandPierceforshock-freequasi-1Dows[13].Theanalyticresultscompareverywellwithnumericalresultsobtainedusingboththecontinuousanddiscreteapproaches.Tounderstandwhythecontinuousapproachbehavescorrectlywithoutexplicitenforcementoftheadjointshockboundarycondition,ashootingmethodwasusedtomarchthesolutionbackfromtheexitacrosstheshock.Disregardingtheadjointshockb.c.,butmaintainingcontinuityattheshock,leadstoafamilyofsolutionsoftheadjointequations.Ofthese,itappearsthatthecontinuousapproachselectsthesmoothestmemberofthefamily,whichcorrespondstotheanalyticsolution.Thenalsectionofthepaperdiscussestheuseofadjointsolutionsforerroranalysis.TheerrorintheliftintegralisshowntobeaninnerproductofthetheadjointowvariablesandthetruncationerrorofthediscretisationoftheEulerequations.Estimatingthetruncationerrorgivesamethodofaccuratelyestimatingtheerrorintheliftintegral.Witharst-orderdiscretisationoftheEulerequations,itisshownthattheerrorestimatecanbeusedtocorrectthecomputedvalueoftheliftintegralandobtainsecond-orderaccuracy.Alternatively,theerrorestimatecouldbeusedinthefutureasthebasisforoptimalgridadaptation[12].2AdjointproblemformulationThequasi-1DEulerequationsforsteadyowinaductofcross-sectionh(x),ontheinterval1x1,maybewrittenasR(U;h)ddx(hF)dhdxP=0;whereU=0@qE1A;F=0@qq2+pqH1A;P=0@0p01A:Ifthesolutioncontainsashockatxs,theRankine-Hugoniotjumpcondition[F]x+sxs=0connectsthesmoothsolutionsoneitherside.Fordesignapplications,linearisationofRwithrespecttoperturbationsinthe