An Introduction to Discontinuous Galerkin Methods

AEROSPACECOMPUTATIONALDESIGNLABAnIntroductiontoDiscontinuousGalerkinMethodsforCompressibleFlowsDavidL.DarmofalAerospaceComputationalDesignLabMassachusettsInstituteofTechnologyNovember5,2004ACDLSeminar1/38AEROSPACECOMPUTATIONALDESIGNLABOverviewMotivation:WhydevelopanotherCFDalgorithm?FinitevolumemethodsforhyperbolicconservationlawsDiscontinuousGalerkin(DG)forhyperbolicconservationlawsDGforellipticproblemsp-multigridforhigher-orderDGdiscretizationsConclusionsandfutureworkACDLSeminar2/38AEROSPACECOMPUTATIONALDESIGNLABMotivationforhigherorderStateofCFDinappliedaerodynamicsIFinite-volumewithatbestsecondorderaccuracyIQuestionsexistwhethercurrentdiscretizationsarecapableofachievingdesiredaccuracylevelsinpracticaltimeDecreasecomputationaltimeandgriddingrequirementsbyincreasingsolutionorderlogT=wd1plogE+logp
logF+constT=timetosolutionp=discretizationorderE=desirederrorlevel(E1)w=solutioncomplexityd=dimensionofproblemF=computationalspeedACDLSeminar3/38AEROSPACECOMPUTATIONALDESIGNLABProject-XGoalProjectXTeamGoal:IToimprovetheaerothermaldesignprocessforcomplex3Dcongurationsbysignicantlyreducingthetimefromgeometrytosolutionatengineering-requiredaccuracyusinghigh-orderadaptivemethodsACDLSeminar4/38AEROSPACECOMPUTATIONALDESIGNLABPreviousWorkExtensiveworkonDGforhyperbolicequationsIBassiandRebay(1997)ICockburnandShu(1998,2001)IKarniadakisetal.(1998,1999)MorerecentlyworkbegunonellipticequationsIBassiandRebay(1997,1998)ICockburnandShu(1998,2001)IBaumannandOden(1997)IBrezzietal.(1997)OnlyBassiandRebayhavepublishedRANSresults(1997,2003)ACDLSeminar5/38AEROSPACECOMPUTATIONALDESIGNLABIntegralFormofHyperbolicConservationLaws2013Applyintegralconservationlawontriangle0:ddtZA0udx+3Xk=1Z0kFi(u)
^nds=0ForEulerequations:u=(;u;v;E)TFi=(Fxi;Fyi)TFxi=u;u2+p;uv;uH
TFyi=v;uv;v2+p;vH
TACDLSeminar6/38AEROSPACECOMPUTATIONALDESIGNLABFirst-orderAccurateFiniteVolume2013Ineachtriangle,assumeuisconstant.Applyconservationlawontriangle:du0dtA0+3Xk=1Z0kHi(u0;uk;^n0k)ds=0Hi(uL;uR;^nLR)isuxfunctionthatdeterminesinvisciduxin^nLRdirectionfromleftandrightstates,uLanduR.Exampleuxfunctions:Godunov,Roe,Osher,VanLeer,Lax-Friedrichs,etc.ThisdiscretizationhasasolutionerrorwhichisO(h)wherehismeshsize.ACDLSeminar7/38AEROSPACECOMPUTATIONALDESIGNLABSecond-orderAccurateFiniteVolume8394150726Ineachtriangle,reconstructalinearso-lution,~u,usingneighboringaverages:~u0u0+(xx0)
ru0;ru0ru0(u0;u1;u2;u3):Applyconservationlawontriangle:du0dtA0+3Xk=1Z0kHi(~u0;~uk;^n0k)ds=0Onsmoothmeshesandows,solutionerrorisO(h2).ACDLSeminar8/38AEROSPACECOMPUTATIONALDESIGNLABPros/ConsofHigher-orderFiniteVolume8394150726+Increasedaccuracyongivenmeshwithoutadditionaldegreesoffree-domDifcultyinachievinghigher-orderonunstructuredmeshesandnearboundariesStabilizingmulti-stagemethodsnec-essaryforlocaliterativeschemesMatrixll-inincreasedresultinginhigh-memoryrequirementsACDLSeminar9/38AEROSPACECOMPUTATIONALDESIGNLABInstabilityofLocalIterativeMethodsConsidersteadystateproblemanddenediscreteresidualforcellj,Rj(u)3Xk=1ZjkHi(~uj;~uk;^njk)ds=0:AJacobiiterativemethodtosolvethisproblemis,un+1j=unj!(@Rj=@uj)1Rj(u):Foranynite!,Jacobiisunstableforhigher-order.Onesolutionisamulti-stagemethod,^uj=unj^!(@Rj=@uj)1Rj(un)un+1j=unj!(@Rj=@uj)1Rj(^u)(Requirestworesidualevaluations.ACDLSeminar10/38AEROSPACECOMPUTATIONALDESIGNLABMatrixFillforHigher-orderFiniteVolume5010015020025030035040045050055050100150200250300350400450500550First-order051015202530354045051015202530354045nz=115Second-order051015202530354045051015202530354045nz=355Third-order051015202530354045051015202530354045nz=601ACDLSeminar11/38AEROSPACECOMPUTATIONALDESIGNLABDiscontinuousPolynomialBasisTriangulatedomainintonon-overlappingelements2ThDenefunctionspace:Element-wisediscontinuouspolynomialsofdegreepVph=fv2L2():vj2Pp():82ThgExampleofOne-DimensionalBasesp=0basis1DOF/elementp=1basis2DOF/elementACDLSeminar12/38AEROSPACECOMPUTATIONALDESIGNLABDGforHyperbolicConservationLaws:DerivationStartfromstrongformofgoverningequations:ut+r
Fi(u)=0:Lookforasolutionuh2Vph.Multiplygoverningequationbyweightfunctionvh2Vphandintegrateoverelement2Th:ZvTh[(uh)t+r
Fi]dx=0:Integratesecondtermbyparts(assumeinteriorelement):ZvTh(uh)tdxZrvTh
Fidx+Z@v+hTHi(u+h;uh;^n)ds=0:ACDLSeminar13/38AEROSPACECOMPUTATIONALDESIGNLABRelationshipofDGtoothermethodsRecallDGweightedresidual(Reed&Hill,1973):ZvTh(uh)tdxZrvTh
Fidx+Z@v+hTHi(u+h;uh;^n)ds=0:Forp=0solution,thisreducesto:(u)tA+Z@Hi(u+h;uh;^n)ds=0:Thus,p=0DGisidenticaltorst-ordernitevolume.Forp0,DGcanbeintrepretedasamomentmethod.MomentmethodsforhyperbolicproblemswererstsuggestedbyVanLeer(1977)andthendevelopedfortheEulerequationsbyAllmaras(1987,1989)andlaterHolt(1992).ACDLSeminar14/38AEROSPACECOMPUTATIONALDESIGNLABDGdiscretization:GlobalviewFinduh2Vphsuchthat8vh2Vph,X2ThnZvTh(uh)tdxZrvTh
Fidxo+Ziv+hTHi(u+h;uh;^n)ds+Z@v+hTHbi(u+h;ubh;^n)ds=0:BoundaryconditionsenforcedweaklythroughHbi(u+h;ubh;^n)whereubhisdeterminedfromdesiredboundaryconditionsandoutg