Manuscript submitted to Website: http: AIMsciences org AIMS' Journals Volume X Number 0X XX 200X pp X–XX

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Manuscript submitted to Website: AIMS' Journals Volume X, Number 0X, XX 200X pp. X–XX OPTIMIZATION CRITERIA FOR MODELLING INTERSECTIONS OF VEHICULAR TRAFFIC FLOW M. Herty Fachbereich Mathematik TU Kaiserslautern D-67653 Kaiserslautern, Germany S. Moutari and M. Rascle Laboratoire J. A. Dieudonne, UMR CNRS No 6621 Universite de Nice-Sophia Antipolis Parc Valrose, 06108 Nice Cedex 2, France (Communicated by Aim Sciences) Abstract. We consider coupling conditions for the “Aw–Rascle” (AR) traffic flow model at an arbitrary road intersection. In contrast with coupling condi- tions previously introduced in [10] and [7], all the moments of the AR system are conserved and the total flux at the junction is maximized. This nonlinear optimization problem is solved completely. We show how the two simple cases of merging and diverging junctions can be extended to more complex junctions, like roundabouts. Finally, we present some numerical results. 1. Introduction. Traffic flow models have been under investigation for a long time. We are particularly interested in macroscopic traffic flow models based on hyperbolic conservation laws. Models of this type have been considered for example in [14, 6, 15, 2, 1, 8]. In the following, we focus on the “Aw–Rascle” (AR) model. This (class of) “second–order” model(s) consists of a nonlinear, coupled system of conservation laws, introduced in [2] and independently in [16].

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ManuscriptsubmittedtoAIMS’JournalsVolumeX,Number0X,XX200XWebsite:http://AIMsciences.orgpp.X–XXOPTIMIZATIONCRITERIAFORMODELLINGINTERSECTIONSOFVEHICULARTRAFFICFLOWM.HertyFachbereichMathematikTUKaiserslauternD-67653Kaiserslautern,GermanyS.MoutariandM.RascleLaboratoireJ.A.Dieudonne´,UMRCNRSNo6621Universite´deNice-SophiaAntipolisParcValrose,06108NiceCedex2,France(CommunicatedbyAimSciences)Abstract.Weconsidercouplingconditionsforthe“Aw–Rascle”(AR)traﬃcﬂowmodelatanarbitraryroadintersection.Incontrastwithcouplingcondi-tionspreviouslyintroducedin[10]and[7],allthemomentsoftheARsystemareconservedandthetotalﬂuxatthejunctionismaximized.Thisnonlinearoptimizationproblemissolvedcompletely.Weshowhowthetwosimplecasesofmerginganddivergingjunctionscanbeextendedtomorecomplexjunctions,likeroundabouts.Finally,wepresentsomenumericalresults.1.Introduction.Traﬃcﬂowmodelshavebeenunderinvestigationforalongtime.Weareparticularlyinterestedinmacroscopictraﬃcﬂowmodelsbasedonhyperbolicconservationlaws.Modelsofthistypehavebeenconsideredforexamplein[14,6,15,2,1,8].Inthefollowing,wefocusonthe“Aw–Rascle”(AR)model.This(classof)“second–order”model(s)consistsofanonlinear,coupledsystemofconservationlaws,introducedin[2]andindependentlyin[16].Thosemodelsdescribethebehavioroftraﬃcdensityandvelocitywherediﬀerentcarscanhaveadiﬀerentresponsetolocaltraﬃcsituations,e.g.,themodeldistinguishestrucksandcars.Recently,ﬁrstextensionsofthesesmodelstoatraﬃcnetworkhavebeenproposed[7,10].Thecrucialpointisthemodellingofcouplingconditionsatjunctions.Typically,onehastointroducefurtherassumptionstoshowthattheproblemiswell-deﬁnedandadmitsauniquesolution,seealsothediscussioninthescalarcase[4,11,9].InthispaperweproposenewcouplingconditionsfortheAR-system.Incontrastwith[7],thoseconditionsconserveallmomentsofthesystemandincontrastwith[10]thederivedconditionsmaximizetheﬂuxatthejunctionswithoutanyfurtherconstraint.Furthermore,wepresentanumericalalgorithmtosolvetheproblemandtoconstructtheintermediatestatesofthehomogenizedsolution.2000MathematicsSubjectClassiﬁcation.Primary:35Lxx;Secondary:35L6.Keywordsandphrases.Traﬃcﬂow,Intersections,CouplingConditions,Optimization.TheﬁrstauthorisaﬃliatedwithAIMS.1

2M.HERTYS.MOUTARIM.RASCLE2.Preliminarydiscussion.WeﬁrstgiveabriefsummaryofthepropertiesoftheAR–modelandadvisethereadertoconsult[2,10]formoredetails.Aroadnetworkismodelledasaﬁnite,directedgraph(I,N)(with|I|=Iand|N|=N)whereineacharci=1,...,Icorrespondstoaroadandeachvertexn∈Ntoajunction.Foraﬁxedjunctionnthesetδn−containsalltheindiceskofincomingroadston.Similarly,δn+denotestheindicesjofoutgoingroads.Weskipthesubindexnwheneverthesituationisclear.EachroadiismodelledbyanintervalIi:=[ai,bi]wherewealloweitherai=−∞orbi=+∞forincomingoroutgoingroadsinthewholenetwork.WerequiretheAR–equations(1)toholdoneacharci∈Iofthenetwork:∂tρi+∂x(ρivi)=0(1a)∂t(ρiwi)+∂x(ρiviwi)=0(1b)wi=vi+pi(ρi)(1c)where,foreachi,ρi7→pi(ρi)isaknownfunction(“traﬃcpressure”)withthefollowingpropertiesγ′′′∀ρi,ρipi(ρi))+2pi(ρi)>0ande.g.pi(ρi)∼ρiatρi=0,(2)andwhereγ>0.ρiandvirespectively,describethedensityandvelocityoftraﬃconroadi.Theconservativeformof(1)isρiyi−ρipi(ρi)∂tyi+∂x(yi−ρipi(ρi))yi/ρi=0,(3)whereyi:=ρiwi=ρi(vi+pi(ρi)).WenowrecallsomebasicfactsonthesolutionoftheRiemannProblemfor(1),i.e.totheinitialvalueproblemwithconstantdatafor±x>0.Thesystemisstrictlyhyperbolicifρi>0.Theeigenvaluesare′λ1,i(U)=vi−ρipi(ρi)andλ2,i(U)=vi(4)Theﬁrstcharacteristicﬁeldisgenuinelynonlinear.Thesecondoneislinearlydegenerateandthereforeassociatedwithacontactdiscontinuity.Moreover,the1–shockand1–rarefactioncurvescoincide,see[5,2].Werecallthatofcoursetheyareassociatedwithbrakingandaccelerationwaves,respectively.Foreachﬁxedi,theRiemanninvariantsarewi(U)=vi+pi(ρi)andvi(U)=vi.(5)Wereferto[10,7,11]foraderivationofthenecessaryconditionsatthejunction,i.e.thecouplingconditions.First,wedeﬁneweaksolutionsofthenetworkprobleminthefollowingsense.Asetoffunctions{Ui=(ρi,ρivi)}i∈Iiscalledaweaksolutionof(1)ifandonlyifvρρi∂tφi+ii∂xφidxdtIXZ∞Zbii=10aiρiwiρiviwibi+ρi,0φi(x,0)dx=0(6)Zaiρi,0wi,0

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