THE TIME DIMENSION AND A UNIFIED MATHEMATICAL FRAM

THETIMEDIMENSIONANDAUNIFIEDMATHEMATICALFRAMEWORKFORFIRST-ORDERPARABOLICSYSTEMSKumarK.Tamma,XiangminZhou,andRamdevKanapadyDepartmentofMechanicalEngineering,UniversityofMinnesota,Minneapolis,Minnesota,USAFortheanalysisofproblemsencompassinglinearrst-orderparabolicsystemsinvolvingthetimedimension,thepresentexpositiondescribestheevolutionofandsynthesisleadingtoageneraluniedmathematicalframeworkanddesignofcomputationalalgorithms.Inourpreviousefforts,variousissuesandthegeneralclassicationandcharacterizationoftime-discretizedoperatorswereaddressed,andthetheoreticaldevelopmentsemanatedfromageneralizedtime-weightedresidualphilosophywhichdescribedtheunderlyingcon-sequences.Towardthisend,inthisarticle,forthersttime,weprovidealternativenewperspectivesandformalismviathenotionsof(1)theresultingsizeoftheequationsystemand(2)theassociatednumberofsystemsolve(s).Althoughthetime-weightedresidualphilosophydescribedanapproachandtheunderlyingconsequences,fromthenewper-spectives,thegeneraldesignofcomputationalalgorithmsisoutlinedinthisarticle.Ageneralizedstabilityandaccuracylimitationtheoremisalsohighlightedforlineartransientalgorithmsencompassingrst-orderparabolicsystems.Characterizationasrelatedtocomputationalalgorithmspertainstothatwhichnotonlypermitsthegeneralclassicationtobeestablishedbutalsoprovidestheunderlyingbasisfortheirsubsequentdesign.INTRODUCTIONForthesolutionoftime-dependentphenomenaencounteredintransientheatconductionproblems,discretizedoperatorsintimearewidelyusedandplayanimportantrole.Fromahistoricalperspective,fortransientalgorithmsforlineartransientheatconductionproblemsundertheumbrellaofatime-continuousframework,thegeneralizedtrapezoidalfamilyofalgorithmswhichincludestheReceived3August2001;accepted17October2001.Keynotelecture,CHT01,PalmCove,Australia,2001.TheauthorsareverypleasedtoacknowledgesupportinpartbyBattelle=U.S.ArmyResearchOce(ARO),ResearchTrianglePark,NorthCarolina,undergrantDAAH04-96-C-0086,andbytheArmyHighPerformanceComputingResearchCenter(AHPCRC)undertheauspicesoftheDepartmentoftheArmy,ArmyResearchLaboratory(ARL),undercontractDAAD19-01-2-0014.Thecontentdoesnotnecessarilyre¯ectthepositionorthepolicyofthegovernment,andnoocialendorsementshouldbeinferred.SupportinpartbyDr.AndrewMarkandDr.RajuNamburuoftheIMTandCSMComputa-tionalTechnicalActivitiesandtheARL=MSRCfacilitiesisalsogratefullyacknowledged.SpecialthanksareduetotheCICDirectorateattheU.S.ArmyResearchLaboratory(ARL),AberdeenProvingGround,Maryland.OtherrelatedsupportintheformofcomputergrantsfromtheMinnesotaSupercomputerIn-stitute(MSI),Minneapolis,Minnesota,isalsogratefullyacknowledged.AddresscorrespondencetoProf.KumarK.Tamma,DepartmentofMechanicalEngineering,111ChurchStreetS.E.,UniversityofMinnesota,Minneapolis,MN55455,USA.E-mail:[e-mail],PartB,41:239±262,2002Copyright#2002Taylor&Francis1040-7790/02$12.00+.00239Eulerforwardandbackwardmethods[1],theCrank-Nicolsonmethod[2],theLinigermethod[3],andtheGalerkinmethod[4]arewidelyused.Thegeneralizedtrapezoidalfamilyofalgorithmsalongwiththetwo-stepmethodsofDupont[5],Gear[6],Lees[7],Liniger[8],andZlamal[4],thethree-stepmethodsofGear[6],Liniger[8],andZlamal[4],andthefour-stepGearmethodbelongtotheparticularclassoflinearmultistep(LMS)methods.TheLMSmethodshavebeenshowntobespectrallyequivalenttoaparticularclassofcorrespondingsingle-steprepresentations[9,10].Recently,asingle-stepfamilyoftime-continuousrepresentations(Wpfamily)wasshowntobederivedfromthetime-weightedresidualformulationofthesemi-discretizedequationofthetransientheatconductionproblemasaresultofselectingtheweightedtime®eldsasascalarfunction[11±13]thatisshowntobesystematicallydegeneratedfromtheexacttime-weightedrepresentation.Also,theLMSmethodscanbedegeneratedtothesingle-steprepresentationsbyreplacingtheLagrangepolynomialapproximationwithanequalorderoftheTaylorseriesexpansionandashiftontheselectedweightedscalartimefunction(theconverseisnottrue)[12].ThealgorithmicpropertiesoftheparticularclassofLMSmethodsare,however,boundbytheDahlquisttheorem,whichstatesthatforLMSmethods,(1)anexplicitA-stablemethoddoesnotexist,(2)theorderofaccuracyofanunconditionallystableLMSmethodcannotexceed2,(3)andthetrapezoidalrulehasthesmallesterror.AlthoughtheDahlquisttheoremlimitsthealgorithmicpropertiesoftheclassofLMSmethods,alternativeapproachesexistwhichcircumventthebarriersoftheDahlquisttheorem(see[11±13]andthepresentarticle).Modal-basedapproachesareotheralternatives,andtheprosandconsaredescribedin[11±13].Besidesmodal-basedapproaches,rationalapproximationapproacheshavealsobeenproposedandachievehigh-order-accurate,unconditionallystablealgorithmsforlinearsituations[14±17].Incontrasttotime-continuousformulations,theso-calledtime-discontinuousformulationsalsoappearintheliterature,andvariousmethodsofapproachhavebeenusedtoderivetheseformulations[18,19].Itisourstrongbeliefthattheyhavereceivedwidespreadattentionprimarilybecauseithasbeenpresumedthatonlytheycanobtaincertaindesiredalgorithmicpropertiessuchasunconditionalstability(inparticular,L-stabili