Deqsol and ellpack Problem solving environments fo

DEQSOLandELLPACK:ProblemSolvingEnvironmentsForPartialDierentialEquationsRonaldF.BoisvertCenterforComputingandAppliedMathematicsNationalInstituteofStandardsandTechnologyTechnologyBuilding,RoomA151Gaithersburg,MD20899USAPhone:[phone]Email:[e-mail]h,FarEast23-17,7-chome,RoppingiMinato-ku,Tokyo106JAPANPhone:[phone]Email:[e-mail]23,1990AbstractAnumberofintegratedproblem-solvingenvironmentsforthesolutionofpartialdierentialequationshavebeendevelopedinrecentyears.Suchsystemspermitvery-high-leveldescriptionsofthemathematicalproblemanditssolutionalgorithm.ThesedescriptionsaretranslatedintoecientFortranprogramswhichproducesolutionsinbothtabularandgraphicalforms.Inthispaperwecompareandcontrasttwosuchsystems:DEQSOL,aJapaneseproject,andELLPACK,anAmericanproject.Foreachsystem,wepresentabriefhistory,describeitscapabilities,andpresentseveralexamplesofitsuse.ThispaperwaspublishedintheOceofNavalResearchFarEastInformationBulletin,vol.16,no.1,1991,pp.7{19.1BoisvertandKahanerDEQSOLandELLPACKJuly23,19901.IntroductionThenumericalsolutionofpartialdierentialequations(PDEs)wasoneoftheearliestap-plicationsofelectronicdigitalcomputers,anditremainsthesourceofmanychallengingcomputationalproblemstoday.Suchproblemscanbefoundineveryscienticdiscipline.TheyrangefromcomputingthecurrentsandeldsinVLSIdevicestodeterminingtheowaboutanairfoil.DuetotheimportanceofmathematicalmodelsbasedonPDEs,andtheirgeneralre-sistancetoanalyticaltreatment,thenumericalsolutionofPDEshasbeenthefocusofagreatdealofresearchovertheyears.Inspiteofgreatalgorithmicadvanceswhichhavebeenachieved,veryfewgeneral-purposesoftwarepackagesforsolvingPDEshaveappeared,andthosethathavetendtoconcentrateonverynarrowproblemareas.General-purposePDEsoftwareisjustextraordinarilydiculttobuild.Thisisduetodicultiesindesigningbothsolutionalgorithmsanduserinterfaces.SincePDEproblemsexhibitsuchawiderangeofbehaviors,nosoftwarepackagebuiltuponasinglenumericalmethodcansucceed.Asaresult,theinternalstructureofsuchasolvermustbeverysophisticated.PDEproblemscannotbeexpressednaturallyinanalgebraiclanguagesuchasFortran,whichremainsthecomputerlanguageofchoiceforscienticcomputation.Becauseofthis,theuserinterfacemusteitherbeverycomplex,oranewhigh-levellanguagemustbedesignedandimplementedfordescribingPDEproblemsandtheirsolution.AtraditionalFortransubprogramcannotdothisjob;instead,anintegratedproblem-solvingenvironmentmustbedesigned.Suchsystemsmustbedesignedbycomputerscientistsinclosecollaborationwiththenumericalanalystswhodevelopecientmethodsandphysicalscientistsandengineerswhoneedtousethem.Thepotentialpayosforthedevelopmentofsuchsystemsisgreat.SolvingPDEproblemsbywritingFortrancodeisaverytime-consuminganderrorpronetaskwhichoftenduplicatesworkofmanypreviousprogrammers.Scientistsandengineersmaynotbeawareofthemostappropriatealgorithmstouse,andhencetheprogramsproducedbytheseeortsmaywellbeverysloworevenproduceerroneousresults.Suchdicultiesareincreasedinthefaceofvectorandparallelprocessingcomputerswhichareverydiculttomakeecientuseofbythecasualuser.Usingsoftwarepartsfromexistingprogramlibrariescanalleviatesomeoftheseproblems,butthisstillrequiresmuchlow-levelprogrammingeort,eventosolveeasyproblems.Anumberofresearchershavemadeeortstoproduceintegratedproblem-solvingenvi-ronmentsforthesolutionofPDEs(seetheshortbibliographyattheendofthisarticle).TwomajorsystemswhichremainactivefociofresearchareDEQSOL,aJapaneseproject,andELLPACK,anAmericanproject.Thesesystemshavemanyovertsimilarities.Botharepreprocessor-based;theyreadvery-high-leveldescriptionsofPDEproblemsandsolutionalgorithmsandproduceaFortran77programasoutput.Whenlinkedwiththeirrespectiverun-timelibrariesandexecuted,thegeneratedprogramswillsolvethegivenPDEproblem.2BoisvertandKahanerDEQSOLandELLPACKJuly23,1990Bothsupportgraphicaloutput.Linearsecond-ordersteady-statePDEproblemsinfairlygeneraldomainsintwoorthreespatialdimensionsarethemoststraightforwardtosolveus-ingeachsystem.Nonlinearproblems,time-dependentproblems,andsystemsofequations,arealsosolvable,butdoingsorequiresmoreoftheuser.Thereare,ofcourse,manydier-encesbetweenDEQSOLandELLPACK.Inwhatfollowswewilltrytogiveyouafeelingforthesetwoprojectsinmoredetail.Toillustratetheuseofthetwopackages,wewillpresentprogramsdesignedtosolvetwosimpleproblems.(Notethatr2representstheLaplacianincartesiancoordinates,i.e.,r2=@2=@x2+@2=@y2.)Problem1:StressdistributioninathinplatecontainingaholeThedomainforthisproblemisthesquare(2;2)(2;2)withacircleofradius1removefromitscenter.Thestressfunctionsatisesr2=2ontheplate,with=1attheoutsideedgeand=0attheedgeofthehole.Duetosymmetryweneedonlyconsidertheregionx0,y0,with@=@n=0;forx=0andy=0:Problem2:ThermaldiusionproblemFindthefunctionT(t;x;y)whichsatises@T=@t=ar2Tontherectangle(0;1)(0;1:2)for0t1withinitialconditionT(0;x;y)=100exp([(x0:5)2+y2])andboundaryconditions@T=@n=0forx=0andx=1T=200fory=0@T=@nT=fory=1:2fortheconstantsa=0:62,=0:8,and=50.2.DEQSOLTheDEQSOLproject,headedbyDr.