外文数值解论文A comparison of numerical solution

AcomparisonofnumericalsolutionsofpartialdifferentialequationswithprobabilisticandpossibilisticparametersforthequantificationofuncertaintyinsubsurfacesolutetransportKejiangZhanga,GopalAcharia,⁎,HuaLibaDepartmentofCivilEngineering,UniversityofCalgary,Calgary,AB,CanadaT2N1N4bDepartmentofMathematics,ZhengzhouUniversity,Zhengzhou,Henan,PRChinaarticleinfoabstractArticlehistory:Received9April2009Receivedinrevisedform20August2009Accepted27August2009Availableonline11September2009Traditionally,uncertaintyinparametersarerepresentedasprobabilisticdistributionsandincorporatedintogroundwaterflowandcontaminanttransportmodels.Withtheadventofneweruncertaintytheories,itisnowunderstoodthatstochasticmethodscannotproperlyrepresentnonrandomuncertainties.Inthegroundwaterflowandcontaminanttransportequations,uncertaintyinsomeparametersmayberandom,whereasthoseofothersmaybenonrandom.Theobjectiveofthispaperistodevelopafuzzy-stochasticpartialdifferentialequation(FSPDE)modeltosimulateconditionswherebothrandomandnonrandomuncertaintiesareinvolvedingroundwaterflowandsolutetransport.Threepotentialsolutiontechniquesnamely,(a)transformingaprobabilitydistributiontoapossibilitydistribution(MethodI)thenaFSPDEbecomesafuzzypartialdifferentialequation(FPDE),(b)transformingapossibilitydistributiontoaprobabilitydistribution(MethodII)andthenaFSPDEbecomesastochasticpartialdifferentialequation(SPDE),and(c)thecombinationofMonteCarlomethodsandFPDEsolutiontechniques(MethodIII)areproposedandcompared.Theeffectsofthesethreemethodsonthepredictiveresultsareinvestigatedbyusingtwocasestudies.TheresultsshowthatthepredictionsobtainedfromMethodIIisaspecificcaseofthatgotfromMethodI.Whenanexactprobabilisticresultisneeded,MethodIIissuggested.Asthelossorgainofinformationduringaprobability–possibility(orviceversa)transformationcannotbequantified,theirinfluencesonthepredictiveresultsisnotknown.Thus,MethodIIIshouldprobablybepreferredforriskassessments.©2009ElsevierB.V.Allrightsreserved.Keywords:UncertaintyFuzzy-stochasticpartialdifferentialequationNumericalsolutionProbability–possibilitytransformationGroundwaterflowSolutetransport1.IntroductionModelingcontaminantfateandtransportisanintegralpartofexposureassessment,anecessarystepinenviron-mentalriskassessment.Thevariousphysicalphenomenaingroundwaterflowandcontaminanttransportarerepresentedbypartialdifferentialequations(PDEs)inspaceandtime.ThegeologicandhydrogeologicsitecharacteristicsareincludedasparametersinthegoverningPDEs.Todeterminetheseparametersisachallengingpartofmodeling.Scarcityofinformation,uncertaintyinthedatacollected,measurementerrors,conflictingdataandsiteheterogeneitiesallcontributetothecomplexitiesinsitecharacterization.Detailedsiteinformationneededtofullycharacterizeasiteareusuallylackingaslargenumberofsamplesarerequiredtocompletelycaptureasite'sdescriptionsanditsvariabilities.Substantialdatarequirementmeanshighcostsandusuallythesiteassessormakesdecisionswithmuchlessdata.Apartfromissuesrelatedtositevariabilities,multiplesourcesofinformationhavetobecompiledsuchashistorical,geologicalandhydrogeologicinformationwhichmaybein“soft”(descriptive)or“hard”(numeric)form.Theremaybedatafromlaboratorystudies,fieldinvestigationsaswellasfromexpertopinions.Alltheseinformationarerequiredtodeterminesoilcharacteristicsandsiteparameterssuchashydraulicconductivity,storagecoefficient,andporositywhichJournalofContaminantHydrology110(2009)45–59⁎Correspondingauthor.Tel.:[phone];fax:[phone].E-mailaddress:[e-mail](G.Achari).0169-7722/$–seefrontmatter©2009ElsevierB.V.Allrightsreserved.doi:10.1016/j.jconhyd.2009.08.005ContentslistsavailableatScienceDirectJournalofContaminantHydrologyjournalhomepage:flowandcontaminanttransportmodel.Forcontaminantfateandtransport,addition-ally,thebiogeochemicalinformationaboutthetargetcontam-inantsuchasprecipitation–dissolution,redoxreaction,andbiodegradationareneeded.Uncertain,variableandmulti-sourced“soft”and“hard”informationnotonlyrequireproperrepresentationsofdifferentuncertaintiesbutalsotheirintegration(Porteretal.,2000;Duboisetal.,2000;SentzandFerson,2002;Duboisetal.,2004),andincorporationoftheintegratedinformationintothegroundwaterflowandcontaminanttransportmodeling.Threetypesofuncertainty(MöllerandBeer,2004),stochastic,informalandlexical,arepresentinsitecharacter-ization.Stochasticuncertaintyisbestdescribedbyclassicalprobabilitytheory.Informaluncertaintyresultsfromaninformationdeficit,suchaswhenonlyasmallnumberofobservationsareavailable.Lexicaluncertaintyoccurswheninsteadofnumbers,wordssuchas“high”,“medium”,and“low”areused.Opinionsofexpertsareusuallylinguisticandwillhavethistypeofuncertainty.Probabilitytheoryissuitableforrepresentationofprobabilisticinformation.Fuzzysettheory(Zadeh,1965),andpossibilitytheory(Zadeh,1978;DuboisandPrade,1988),evidencetheory(Shafer,1976),andrandomsets(Matheron,1975;Molcha-nov,2005;Nguyen,2006)areemployedtorepresentinformalandlexicaluncertaininformationorpossibilisticinformation(LiuandPeng,2005).Evi