Highly accurate numerical solutions with repeated Rich - PDF document

HighlyaccuratenumericalsolutionswithrepeatedRichardsonextrapolationfor2Dlaplaceequation CarlosHenriqueMarchi,LeandroAlbertoNovak,CosmoDamiãoSantiagoAnaPauladaSilveiraVargasLaboratoryofNumericalExperimentation(LENA),MechanicalEngineeringDepartment(DEMEC),FederalUniversityofParaná(UFPR),Caixapostal19 Correspondingauthor.Tel.:+554133613126;fax:+554133613701.E-mailaddresses:(C.H.Marchi),(L.A.Novak),(C.D.Santiago),(A.P.S.Vargas). AppliedMathematicalModelling37(2013)7386…7397 ContentslistsavailableatSciVerseScienceDirectAppliedMathematicalModellingjournalhomepage:www.elsevier.com/locate/apm nodes,thusallowingfortwoormoreREs.Inotherareasareas,RREisusedwithmanyREs,butthisisnotthecaseinCHT(ComputationalHeatTransfer)andCFD(ComputationalFluidDynamics).TheauthorsofreferencesreferencesappliedRREwithtwoREstotwo-dimensionalproblems(2D)ofheatconduction.RREwasappliedwithuptothreeREsbybyinthesolu-tionof2DNavier…Stokesequations.However,inCHTandCFDitismorecommontoapplyREthanRRE;forexample:example:invarioustypesof1Dproblems;problems;inthe2DPoissonequation;inthe2Dadvection…diffusionequationequationand3D3D;in2DNavier…Stokesequationss;in2Dnaturalconvectionn;andin2Dturbulent”ows[19,21,22].Therefore,todate,REandparticularlyRREhavebeenlittleusedinCHTandCFDtoreducethediscretizationerror,probablyduetothedif“cul-tiesreportedby[2…4,7,12,14,19,21…26]Basedononandonthiswork,themainadvantagesofusingRREare:(1)itgreatlyreducesthediscretizationerror;(2)itisasimplepost-processingmethod,i.e.,itdoesnotinterferedirectlyinobtainingthenumericalsolutioninagivengrid;(3)itscomputationalcostisverylowintermsofCPUtimeandRAMmemory;(4)itcanbeappliedtoexistingcomputa-tionalcodesortoresultsalreadyobtained;(5)itisapplicabletoseveralnumericalmethods,numericalapproximationsandvariablesofinterest;(6)itdoesnotdependonapriorianalysesoronknowledgeoftheanalyticalsolutiontotheproblem;and(7)evenwithnumericalsolutionswithoutRRE,whichareofloworder(oneortwo),oneobtainsnumericalsolutionswithRREthatareofextremelyhighorder(higherthanten).Aswillbeshowninthispaper,RREcanbeusedintwoways;the“rst,toobtainthesamediscretizationerrorwithagridcontainingfarfewernodes,thusreducingthecomputationalcost.Thiswayisespeciallyrecommendedforpracticalapplicationsandvalidations.Thesecondwayallowsforareductionofthediscretizationerrorinagridwiththesamenumberofnodes,resultinginmuchsmallererrorsandgreaterreliabilityofthesolution.UsingRREinthiswayisespeciallyindicatedforgeneratingbenchmarks.Theobjectivesofthisworkareasfollows:(i)presentatheoreticalbasisofRREaimedatreducingandestimatingthedis-cretizationerrorinheatconduction;(ii)showthattheuseofRREisextremelyeffectiveinreducingtheerror;(iii)testanerrorestimatorforRRE;(iv)showtheeffectsonRREcausedbythetypeofnumericalapproximation,typeofvariableofinterest,precisionofthecalculations,numberofextrapolations,numberofgridsandordersoftheerror;and(v)showthecomputationalcostofRRE.Toachievetheseobjectives,theRREtheoryisappliedtothesolutionofthe2DLaplaceequa-tionwithuptotwelveREs.ApaperwithpreliminaryresultscontainedinpresentworkwaspublishedatCILAMCE/200808.2.MathematicalmodelThemathematicalmodelconsideredhereisthetwo-dimensionalLaplaceequationwithDirichletboundaryconditions,de“nedby @2T@x2þ arethecoordinatedirectionsandrepresentsthetemperature.Physically,thisequationcanmodelaproblemofheatconductionona”atplatewithconstantproperties,inasteadystate,aswellasseveralotherphysicalphenomena.TheanalyticalsolutionforEq.Thevariablesofinterestinthiswork,i.e.,thevariablestowhichtheRREtheoryareapplied,are:(i)thetemperatureatthecenterofthedomain,i.e.,at=½,representedby;(ii)thetemperaturepro“leat=½,representedby;(iii)themeanofthetemperature“eld,representedby;(iv)themeanofthenormofthenumericalerror,representedbyandtheheattransferratesattheboundariesof=1(v)and=1(vi),represented,respectively,by.Thejusti“-cationforthechoiceofthesevariablesispresentedinthenextsection.Thevariablesarede“nedmathematicallyby dxdy istheheattransferarea,isthethermalconductivityofthematerialandistheheattransferlengthinallwithaunitaryvalue.3.Numericalmodel3.1.Numericalsolutionwithoutextrapolationisdiscretizedwiththe“nitedifferencemethodmethod,uniformgridsandtheCDS(CentralDifferencingScheme)schemeofsecond-orderaccuracy,resultingin ðTi1;j2Ti;jþTiþ1;jÞh2þ C.H.Marchietal./AppliedMathematicalModelling37(2013)7386…7397 representeachnodeinthegrid,andisthedistancebetweentwoconsecutivenodesofthegridineachWritingEq.forallthenodesinthegrid,onehasasystemofalgebraicequationswhichissolvedbymeansoftheModi“edStronglyImplicit(MSI)methodmethod.Tohastentheconvergence,ageometricmultigridmethodmethodwasemployedwiththeFullApproximationScheme(FAS),V-cycle,restrictionbyinjection,prolongationbybilinearinterpolationandcoars-eningratiooftwo.Thenullvaluewasusedastheinitialestimateofthesolutionforeachproblem.ThenumberoftimestheV-cycleofthemultigridmethodisrepeatediscalledexternaliterations.Theiterativeprocesswasrepeateduntilthemachineround-offerrorforthenumericalsolutionofthevariableswasreached,aimingtoeliminatethecontributionoftheiterationerrortothenumericalerror.ThreecomputationalprogramswereimplementedinFortran95language,version9.1ofIntel,oneusingsingleprecision4),anotherwithdoubleprecision(Real8)andthethirdwithquadrupleprecision(Real16).Thesimulationswereper-formedinacoreofamicrocomputerequippedwithanIntelXeonQuadCoreX5355processor,2.66GHz,16GBRAMand64-bitWindowsXPoperatingsystem.Thenumericalsolutionofvariablewasobtaineddirectlyfromthecentralnodeofeachgridafterobtainingthenumer-icalsolutionofEq.,sincegridswithanoddnumberofnodeswerealwaysused.Thenumericalsolutionofvariableobtaineddirectlyfrom15nodesequallyspacedinthegridafterobtainingthenumericalsolutionofEq.forgridswith17nodesorlarger.ThenumericalsolutionofEq.wasobtainedbymeansofnumericalintegrationbythetrapezoidalrulerule.ThenumericalsolutionofEq.wasobtainedbymeansofnumericalintegrationbythetrap-ezoidalruleprecededbytheuseoftheUDS(One-sidedDifferencingScheme)schemeschemeofsecond-orderaccuracyoneachboundarynode.VariablewasobtainedanalogouslytowerechosentoverifytheeffectoftheRichardsonextrapolationon15speci“cnodesinthegridsandbecausetheyarevariablesdependentonthedifferentialequationoftheproblem,Eq.,i.e.,theprimaryvariableoftheproblem,whosesolutioninvolvestwonumericalapproximationsoftheCDStype,asindicatedinEq..Variablewascho-sentomonitor,throughasinglevariable,thebehaviorofthenumericalerrorofthewhole“eldofwiththedecreaseofwereusedtoverifytheeffectoftheRichardsonextrapolationonsecondaryvariables,i.e.,obtained,andinvolvingoneandtwonumericalapproximationsinadditiontothoseusedtoobtainThenumericalsolution()withoutextrapolationofeachabovedescribedvariableofinterestwasobtainedbyusingasetofgrids=[1,],where1isthecoarsestgrid,whichhasthehighest,andg=Gisthe“nestgrid,withthelowest3.2.NumericalsolutionwithmultipleextrapolationsForeachvariableofinterest,thenumericalsolution()ingridRichardsonextrapolationsisgivenby isthegridre“nementratio,andthevariablerepresentsthetrueordersordersofthediscretizationerror,whichcanbeobtainedasexplainedinthenextSection.Eq.isvalidfor=[2,]and=[1,1].Eq.wasobtainedbyextendingtothecaseof�1theworksofof,whichpresentequationsequivalenttothecaseof=1.Itshouldbenotedthattoobtaineachvalueofrequireshavingtwonumericalsolutionsintwogrids(1)at1.Forany=0representsthenumericalsolutionofwithoutanyextrapolation,whichisobtainedasdescribedinthepreviousSection.For=1,onehasthestandardorsingleRichardsonextrapolation,whichiscommonlyusedtoesti-matethediscretizationerrorerrorortoimprovethesolutionofeachgridgrid.Foragivenvalueof,Eq.canbeappliedup1times,performingRichardsonextrapolations.4.ThenumericalerroranditsestimateForagivenvariableofinterest,thenumericalerror()ofthenumericalsolution()canbede“nedbyistheexactanalyticalsolutionofthevariableofinterest.Inthepresentwork,weconsiderthatthenumericalerroriscausedbyfoursourcessources:discretization,iteration,round-offandprogrammingerrors.Whenthenumericalerroriscausedonlybythediscretizationerror,onehashasEð/Þ¼C0hp0þC1hp1þC2hp2þ......arethecoef“cientsthatdependonanditsderivatives,aswellasontheindependentvariables,butareindependentof;and...arethetrueordersof),whosesetisrepresentedbyThevaluesofaregenerallypositiveintegernumbersnumberswith0,whichconstituteanarithmeticprogressionofratio.Inaddition,iscalledtheasymptoticorderortheorderofaccuracyof)oroftheC.H.Marchietal./AppliedMathematicalModelling37(2013)7386…7397 numericalsolution.ThevaluesofcanbeobtainedaprioriwithaprocedurethatusestheTaylorseriesseries,orapos-terioriasexplainedinthenextSectionThetheoreticalorderofaccuracyofthenumericalsolutionof,withordersconstitutinganarithmeticprogression,extrapolationsiswherethisequationisvalidfor=[1,]and=[0,4.1.ObtainingerrorordersThevaluesofobtainedaprioricanbecon“rmedaposterioriwiththeconceptofeffectiveorder((ofthedis-cretizationerror,which,generalizedforrepeatedRichardsonextrapolation,isgivenby log iscalculatedbyEq.isvalidfor=[2,]and=[0,2].Theotherde“nitionsinSectionapplyhere.Itshouldbenotedthattoobtaineachvalueofitisnecessarytoknowtheerrorofthenumericalsolutionintwogrids.Intheory,as0,thevaluesof(shouldtendtowardthetrueorder()givenbyEq..Eq.wasobtainedbyextendingtothecaseof1theworkofof,whichpresentsanequationequivalenttothecaseof=0.AccordingtoEq.isafunctionoftheerrorofthevariableofinterest.Hence,thisequationcannotbeappliedtoprob-lemswhoseanalyticalsolutionisunknown.Moreover,Eq.shouldnotbeusedwhenoneintendstocon“rmaposteriorithevaluesofobtainedaprioriandwithoutusingnumericalsolutionsextrapolatedfromthesamevaluesofthatwereobtainedapriori.Insuchcases,onecanusetheconceptofapparentorobservedorder([10,36,20,21,3,6]oftheestimateddiscretizationerror,which,generalizedtorepeatedRichardsonextrapolation,isgivenby log wherevariableisexplainedbelow.Eq.wasobtainedbyextendingtothecaseof1theworks[10,36,20,21,3,6]whichpresentanequationequivalenttothecaseof=0.isvalidfor=[3,=[0,Int((3)/2)]andconstantamongthethreegrids,i.e.,whereInt()representstheintegerpartof.Theremainingde“nitionsinSectionapplyhere.Itisalsopossibletoobtainforvariableamongthreegrids;thiscaseisaddressedinin.As0,thevaluesof(shouldtendtowardthetrueorder()oftherespectivelevelofextrapolation()ofEq.,independentlyofanyapriorianalysis.Thesevaluesofobtainedaposteriori,shouldbeusedinEq.ItshouldbenotedthattoobtaineachvalueofinEq.requiresnumericalsolutionsrelatedtothreegrids.ThesesolutionsarenotobtainedwithEq.becauseitadmitsthatthevaluesofareknown.Toobtainaposterioriindepen-dentlyofapriori,insteadofusingvariable,obtainedwithEq.,oneusesthenewvariable,whichiscalculatedby isvalidfor=[3,]and=[1,Int((1)/2)].Theotherde“nitionsinSectionapplyhere.Inaddition,for=0,Eq.doesnotapply;inthiscase,onehas,whereisthenumericalsolutionobtainedwithoutanyextrapolation,asdescribedinSectionToadequatelycharacterizetheseriesofvaluesof,ideallyoneshoulddetermineatleastthe“rstthreeorders:.Ifthisisnotpossible,onecansimplydetermine,sincethevaluesofnormallyconstituteanarithmeticpro-gressionandtheremainingvaluesareafunctionofthese“rsttwoordersoftheseries.Whenonlyisdetermined,onecandeterminetheremainingvaluesarbitrarily.Lastly,ifnovalueisdetermined,onecanusetheserieswiththelowestvalues:1,2,3,...TheimpactofthearbitrarychoiceofthevaluesofwillbeaddressedintheResultssection.Inthepresentwork,thevaluesofusedinEq.areobtainedfrom(ofEq.=[0,2].Thesearepositiveintegernumbersthatareextracted,foreachlevelofextrapolation(),fromthetendencyofthevaluesof(4.2.ErrorestimatorForanyvariableofinterest,anestimate()ofthediscretizationerrorofthenumericalsolution()inthegrid,withRichardsonextrapolations,isgivenby C.H.Marchietal./AppliedMathematicalModelling37(2013)7386…7397 isvalidfor=[2,]and=[0,2].Theotherde“nitionsofSectionapplyhere.Eq.wasobtainedbyextendingtothecaseof�1theworksofof,whichpresentequationsequivalenttothecaseof=1.5.ResultsUsingdoubleprecision(Real8),numericalsolutionswereobtainedforthevariablesofinterestingridscontaining35,99,...upto8,1938,193nodes;thus,13grids.Ontheotherhand,withsingle(Real4)andquadruple(Realprecision,the“nestgridhad4,0974,097nodes;therefore,12grids.Although50and20externaliterationswereper-formedwithdoubleandquadrupleprecision,respectively,themachineround-offerrorwasreachedwithonlysixandtwelveexternaliterations,respectively.Toreachthemachineround-offerror,themaximumCPUtimewas20minand1h5min,respectively,fordoubleandquadrupleprecision.Thenumberofsigni“cantdigitsofnumericalsolutionswithoutextrapolationisatleast12and30,respectively,fordoubleandquadrupleprecision;thismeansthatinthesedigitsthesolu-tionshavenoround-offerror.TomeasurethenumericalerrorwithEq.,theanalyticalsolution()ofeachvariableofinterestwasobtainedusingMaplesoftwarewith30and64digits,respectively,forthenumericalsolutionsobtainedwithdoubleandquadrupleprecision.Presentedbelowaresomeoftheresultsofthiswork.Theomittedresultsexhibitedthesamequalitativebehaviorastheonesgivenbelow.5.1.ErrorreductionandestimationwithRREFigs.1…3,respectively,presentthemodulusofthefollowingresultsforthevariablesasafunctionofgridsize(),obtainedwithquadrupleprecisionand=12grids:,calculatedwithEq.,whichistheerrorofthenumericalsolutionofwithoutanyextrapolation(=0)andobtainedasdescribedinSection=[1,,calculatedwithEq.=[2,]and=0,whichistheestimateof2,calculatedbyEq.,whichistheerrorofnumericalsolutionofwithextrapolation,obtainedasdescribedinSec-bymeansofEq.=[3,]and2,calculatedwithEq.=[3,]and2,whichistheestimateof1,calculatedwithEq.,whichistheerrorofnumericalsolutionofwithextrapolation,obtainedasdescribedinbymeansofEq.=[2,]and,whichisthenumeratorofEq.,whereisobtainedbyEq.,asdescribedinSection,for=[2,]andTheerrorofnumericalsolutionsobtainedwithRREcanonlybeestimatedinthesecondcoarsestgridateachlevelof.Thereforetheresultsof2and2arepresented.Althoughitserrorcannotbeestimatedconsistently,theresultof1ispresentedtoshowtheeffectofthelevelofextrapolationforthesamegridInthesethree“guresonecanseethat,ingeneral:(i)coincidesvisuallywithinany;(ii)2iscloseto2inany,withvaluesslightlylowerthan2;(iii)1isalittlelowerthan2;(iv)ismuchhigherthan2;and(v)1and Fig.1.Errors()andtheirestimates()asafunctionofgridsize()forC.H.Marchietal./AppliedMathematicalModelling37(2013)7386…7397 2areverymuchlowerthanandbecomeprogressivelylowerinrelationtotheloweris.Moreover,inthe“nergrids,i.e.,lower,becausetheround-offerror()ishigherthanthediscretizationerror:RRElosesitseffectofreducingtheerror;2,i.e.,Eq.isuseless;andcanbeusedasanestimatorfor1andBasedontheseresults,itcanbeconcludedthat:(a)RREisextremelyef“cienttoreducethediscretizationerror,exceptinthecaseofverycoarsegrids;(b)ineachgridwithspacing,thelowesterrorwithRREoccursfor1,i.e.,1;(c)RREreducestheerrorofprimaryorsecondaryvariablesequallyef“ciently,regardlessofthenumberofnumericalapprox-imationsused;(d)theRichardsonestimatorgivenbyEq.isaccuratetopredictthediscretizationerrorofnumericalsolu-tionsobtainedbyEq.withRRE;(e)canbeusedasareliableestimatorof,althoughitoverestimatesitconsiderably;itcanalsobeusedevenwhenisdominatedbyinsteadof,ascanbeseeninFigs.1…3;and(f)round-offerrorsaffecttheperformanceofRREinvery“negridswhentheyprevailoverthediscretizationerrors.Tables1and2helpdemonstratetheef“ciencyofRREinreducingtheerrorofvariable,obtainedwithquadruplepre-cision.Forthreespeci“cgrids,Table1showstheeffectofRREonthereductionofthediscretizationerror,measuredbythe1,uponre“ningthegridandincreasingthenumberofextrapolations().Forexample,eveninagridascoarseas17,theerrorisalreadyreducedmorethanthreethousandtimeswithonlythreeextrapolations(=3).Hence,RREcanbeusedtoobtainbenchmarksolutions. Fig.2.Errors()andtheirestimates()asafunctionofgridsize()for Fig.3.Errors()andtheirestimates()asafunctionofgridsize()forC.H.Marchietal./AppliedMathematicalModelling37(2013)7386…7397 Forthreespeci“cerrorlevels,Table2showstheeffectofRREinreducingthenumberofnodesofagridtoobtainthesamediscretizationerror.Forexample,fortheerrorlevel5.00E-07,itisnecessarytousethe1,0251,025gridtoreachthiserrorlevelwithoutRREandthe1717gridwithRRE.Thus,agridmorethanthreethousandtimessmallerisrequiredwithRREthanwithoutRRE.Thisratiobetweenthenumberofnodesof1gridsindicatesthelevelofreductionofthecom-putationalcost(CPUtimeandRAMmemory)achievedbyusingRREcomparedtonotusingRRE.Hence,RREcanbeemployedwithgreatcomputationalef“ciency.5.2.EffectofcalculationprecisionFig.4presentsthevalueofthemodulusofthenumericalsolutionswithoutextrapolationobtainedwithsingle4),double(Real8)andquadruple(Real16)precisionforvariableasafunctionofgridsize().This“gurealsopre-sentsthevalueofthe1modulusofthenumericalsolutionsextrapolatedandobtainedwiththesamethreeprecisionsfor1followthesamede“nitionsofSection.This“gurepresentsresultsforthe=[1,=12]grids.Fig.4,notethat:(i)theround-offerror()becomesthemainsourceofthenumericalerrorbelowavalueof,whichdependsontheprecisionusedinthecalculationsandonwhetherthenumericalsolutionwasobtainedwithorwithoutextrapolation;(ii)numericalsolutionswithextrapolationaremuchmoreaffectedbythanwithoutextrapolationextrapolation,i.e.,thevalueofbelowwhichtheextrapolationsareaffectedbyismuchhigherthantheofthenumericalsolutions Table1Reductionoftheerrorforspeci“cgrids(RealGrid17171291291,0256.25E-027.81E-039.77E-04|Eh|9.20E-041.44E-052.25E-07|Em1|2.71E-074.56E-173.19E-32ofEm1369|Eh|/|Em1|3.39E+033.16E+117.05E+24 Table2Reductionofgridnodesforspeci“cerrors(RealErrorlevel5.00E-035.00E-055.00E-07grid9965651,025|3.65E-035.76E-052.25E-071grid59171|1.92E-033.84E-052.71E-071123Ratioofthenumberofnodesof1grids3.24E+005.22E+013.64E+03 Fig.4.Effectofcalculationprecisionontheerrorasafunctionofgridsize(C.H.Marchietal./AppliedMathematicalModelling37(2013)7386…7397 withoutextrapolation;and(iii)thehighertheprecisionusedinthecalculationsthegreatertheef“ciencyofRREinreducingthediscretizationerror.5.3.Effectofthenumberofextrapolations(m)Fig.5showstheeffectofthenumberofextrapolations()onthemodulusoftheerrorasafunctionofgridsize()forresultsobtainedwithquadrupleprecision(Real16).This“gurepresentsresultsforthe=[1,=12]grids.Eachcurverep-resentsalevelofextrapolation=[0,1].Itshouldbekeptinmindthat=0referstoresultsoftheerrorwithoutextrapolation,obtainedasdescribedinSection;andfor1theresultsoftheerrorwithextrapolation,obtainedasdescribedinSectionwithEq.Itisevident,inFig.5,thatthehigherisforthesame,thegreatertheef“ciencyofRREinreducingthediscretizationerror.Inaddition,forthesame,theloweristhesmallertheerror;thisresultiswellknownfor=0.However,inthe“nergrids,i.e.,thosewiththelowestvaluesof,whichinthiscaseoccursforreducetheef“ciencyofRRE.Thelowesterror(3.19E-32)wasobtainedwith=9atitshighest,i.e.,inthe1,0251,025grid(=10).Fig.6showstheeffectiveorder()oftheerrorasafunctionofgridsize()andnumberofextrapolations()corre-spondingtotheerrorsinFig.5.ThevalueswerecalculatedfromEq.,asdescribedinSection,for=[2,]and=[0,2].Eachcurverepresentsalevelofextrapolation.Thehighestvalueof(17.4)wasobtainedwith=8atitshighest,i.e.,inthe1,0251,025grid(=10).Inthe“nergrids,i.e.,theoneswiththelowestvaluesof,whichinthis Fig.5.Effectofthenumberofextrapolations()ontheerrorasafunctionofgridsize()(Real Fig.6.Effectiveorder()oftheerrorasafunctionofgridsize()andnumberofextrapolations()(RealC.H.Marchietal./AppliedMathematicalModelling37(2013)7386…7397 caseoccursforaffectsthecalculationof,resultinginvaluesthatshouldbedisregarded.This“gureclearlyshowsthefollowingvaluesof:2,4,6,8,10and12;andthecurvesshowatendencytowardthevalues14and16.Fig.7presentstheeffectiveorder()oftheerrorasafunctionofgridsize()forthe2and1curvesinFig.1.This“gurealsoshowsthetheoreticalordercurvegivenbyEq.,with=2and=4.Disregardingthetwolowestvaluesofinwhichdominatestheerror,onecanseethattheperformanceofRREgenerallyfollowsthattheoreticalcurve,withadifferenceof0.35to1.38ineach.Thehighestvalueof(19.1)wasobtainedbetween=8and9inthe1,025grid(=10).5.4.Veri“cationandeffectoferrorordersFig.8illustratestheapparentorder()oftheestimatederrorasafunctionofgridsize().ThevalueswereobtainedwithEq.andquadrupleprecision,andthoseofwithEq.,accordingtothetheorydescribedinSection.The“gureshowsresultsforforandlevelsofextrapolation=0,1and2.Withtheseresults,itisclearthatasthevaluesoftendtowardtheorders2,4and6.Fig.9showstheeffectofthearbitrarychoiceofthevaluesofontheperformanceofRRE.Aimingtoserveasareference,this“gurepresentsthe1(denoteas=2,4,6...correct)curvesofFig.1again,forthevariableobtainedwithquadrupleprecision.Italsoshowsthe1curvesthatwereobtainedwiththreeseriesofarbitraryvaluesfor.Theuseof Fig.7.Effectiveorder()ofthe1and2curvesofasafunctionofgridsize()(Real Fig.8.Apparentorder()oftheestimatederrorasafunctionofgridsize()andnumberofextrapolations()(RealC.H.Marchietal./AppliedMathematicalModelling37(2013)7386…7397 thesearbitraryvaluesreducestheef“ciencyofRREconsiderablyinrelationtothecorrectcurve,untiltheeffectofRREisvirtuallyeliminatedwhen1almostcoincideswith5.5.Computationalef“ciencyofRREFig.10presentsthemodulusoftheerrorversustheCPUtime(inseconds)neededtoobtainthesolutionofvariableuntilthemachineround-offerror()isreached.This“gureclearlyshowsthattheerrors()ofthesolutionswithoutextrapolationdecreasealmostlinearlywiththeincreaseinCPUtime.ForthesamevalueofwithReal16isequaltowithReal8,buttheCPUtimeofwithReal16isabout20timeslongerthanwithReal8.However,thenumberofiter-ationstoreachwithReal16istwicethatofReal8.Hence,periteration,theCPUtimeofthesolutionwithReal16isapproximately10timeslongerthanwithRealFig.10,theCPUtimeoftheerrorwithRRE()inagivengridisthesumoftheCPUtimesofallthecoarsestgrids,sincetheyarenecessarytoobtain.Forexample,theCPUtimeplottedinthe“gureandcorrespondingtointhe1,025gridalsoincludestheCPUtimeofthegrids513513,257257downto33,i.e.,10grids.For1with8,the“gureshowstheresultsofCPUtimeonlyfor129129gridandlargerones,inwhichalreadyexceedsthediscretizationerror;theCPUtimeinthesmallergridsdidnotbemeasuredbecauseitwastoobrief. Fig.9.errors()asafunctionofgridsize()andvaluesoferrororders(Real Fig.10.errors()asafunctionofCPUtimeandcalculationprecision.C.H.Marchietal./AppliedMathematicalModelling37(2013)7386…7397 AscanbeseeninFig.10,theincrementinCPUtimewiththereductionof1(Real16)decreasesasthesmallertheerroris.Thisbehaviorisbetterthanthatofthecurves.Moreover,foragivenerrorvalue,theCPUtimeof1,evenwith16,ismuchshorterthanthatof,evenifoneusesthesmallestofthesevalues(Real8).Forexample,fortheerrorlevelof2E-7,thesolutionwithRREisobtainedwitha200timesshorterCPUtimethanwithoutRRE.Anotherexample:fortheerrorlevelof5E-10(whichrequiredextrapolatingthecurve),thesolutionwithRREisobtainedwitha26thousandtimesshorterCPUtimethanwithoutRRE.Hence,intermsofCPUtime,theef“ciencyofRREincreasesastheerrorvalueAscanbeseeninFig.10,foragivenvalueofCPUtime,theerrorwithRRE(1),evenwithReal16,ismuchlowerthanwithoutRRE().Forexample,foraCPUtimeof1s,thesolutionwithRREhasamorethan3E+6timeslowererrorthanwithoutRRE.Anotherexample:foraCPUtimeof100s,thesolutionwithRREhasamorethan8E+17timeslowererrorthanwithoutRRE.Hence,theef“ciencyofRREinreducingtheerrorincreaseswithincreasingCPUtime.IntermsofRAMmemory,thecomputationalcostofusingRREisalmostthesameaswithoutRREforagivengrid.6.ConclusionThispaperpresentedatheoreticalbasisforperformingrepeatedRichardsonextrapolation(RRE)inheatconduction.RREwasappliedtoreduceandestimatethediscretizationerrorofthenumericalsolutionofthe2DLaplaceequation.Thisworkwasperformedusingthe“nitedifferencemethod,asquarecalculationdomaindiscretizedwithuniformgrids,second-orderaccurateapproximations,severalvariablesofinterest,Dirichletboundaryconditions,gridswithuptomillionsofnodes,amultigridmethod,single,doubleandquadrupleprecisions,asuf“cientnumberofiterationstoreachthemachineround-offerror,anduptotwelveRichardsonextrapolations.Itwasfoundthat:(1)RREisextremelyef“cientinreducingthediscretizationerrorofprimaryandsecondaryvariables,regardlessofthenumberofnumericalapproximationsemployed.(2)TheRichardsonestimatorisaccurateinpredictingthediscretizationerrorofnumericalsolutionsobtainedwithRRE.(3)AgreaterreductionofthediscretizationerrorwithRREisachievedbyusinghigherprecisioninthecalculations,alar-gernumberofextrapolations(),alargernumberofgrids,andcorrecterrororders.(4)Toobtainagivenerrorvalue,muchlessCPUtimeandRAMmemoryarerequiredforthesolutionwithRREthanwith-outRRE.(5)Whentheround-offerrorishigherthanthediscretizationerror,RRElosesitseffectofreducingtheerror;moreover,theRichardsonestimatordoesnotwork,socanbeusedinstead.AcknowledgementsTheauthorsthankTheUNIESPAÇOProgramoftheAEB(BrazilianSpaceAgency),CNPq(ConselhoNacionaldeDesenvolvi-mentoCientí“coeTecnológico,Brazil),CAPES(CoordenaçãodeAperfeiçoamentodePessoaldeNívelSuperior,Brazil)andFundaçãoAraucária(Paraná,Brazil)fortheir“nancialsupport.The“rstauthorissupportedbyaCNPqscholarship.Theauthorswouldalsoliketoacknowledgethesuggestionsprovidedbythereferees.:thisworkisdedicatedtothememoryofL.F.Richardsonandcommemoratesthecentenaryanniversaryofhisworkof1910.[1]L.F.Richardson,Theapproximatearithmeticalsolutionby“nitedifferencesofphysicalproblemsinvolvingdifferentialequations,withanaptothestressesinamasonrydam,Philos.Proc.R.Soc.LondonSer.A210(1910)307…357.[2]L.F.Richardson,J.A.Gaunt,Thedeferredapproachtothelimit,Philos.Proc.R.Soc.LondonSer.A226(1927)299…361.[3]P.J.Roache,Veri“cationandValidationinComputationalScienceandEngineering,Albuquerque,Hermosa,1998.[4]A.Sidi,PracticalExtrapolationMethods;TheoryandApplications,CambridgeUniversityPress,Cambridge,2003.[5]G.Dahlquist,A.Björck,NumericalMethodsinScienti“cComputing,vol.1,SIAM,Philadelphia,2008.[6]C.H.Marchi,A.F.C.Silva,Unidimensionalnumericalsolutionerrorestimationforconvergentapparentorder,Numer.HeatTransferPartB42(20[7]P.J.Roache,Perspective:amethodforuniformreportingofgridre“nementstudies,ASMEJ.FluidsEng.116(1994)405…413.[8]D.C.Joyce,Surveyofextrapolationprocessesinnumericalanalysis,SIAMRev.13(1971)435…490.[9]K.Rahul,S.N.Bhattacharyya,One-sided“nite-differenceapproximationssuitableforusewithRichardsonextrapolation,J.Comput.Phys.219[10]A.S.Benjamin,V.E.Denny,Ontheconvergenceofnumericalsolutionsfor2-D”owsinacavityatlargeRe,J.Comput.Phys.33(1979)340…358.[11]R.Schreiber,H.B.Keller,Drivencavity”owsbyef“cientnumericaltechniques,J.Comput.Phys.49(1983)310…333.[12]A.O.Demuren,R.V.Wilson,Estimatinguncertaintyincomputationsoftwo-dimensionalseparated”ows,ASMEJ.FluidsEng.116(1994)216…220.[13]E.Erturk,T.C.Corke,C.Gökçöl,Numericalsolutionsof2-Dsteadyincompressibledrivencavity”owathighReynoldsnumbers,Int.J.Numer.MetFluids48(2005)747…774.[14]C.Burg,T.Erwin,ApplicationofRichardsonextrapolationtothenumericalsolutionofpartialdifferentialequations,Numer.MethodsPartialDiffer.Equ.25(2009)810…832.[15]P.J.Roache,P.M.Knupp,CompletedRichardsonextrapolation,Commun.Numer.MethodsEng.9(1993)365…374.[16]Y.Wang,J.Zhang,Sixthordercompactschemecombinedwithmultigridmethodandextrapolationtechniquefor2Dpoissonequation,J.Comput.Phys.228(2009)137…146.C.H.Marchietal./AppliedMathematicalModelling37(2013)7386…7397 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