A Multi Class Homogenized Hyperbolic Model

Description

A Multi-Class Homogenized Hyperbolic Model of Tra?c Flow P. Bagnerini and M. Rascle Running head: A Multi-Class Homogenized Model of Tra?c Flow AMS subject classifications: 35Lxx, 35L65 Acknowledgement : this work has been partially supported by the EU financed network no. HPRN-CT-2002-00282 1

• thus led


• lagrangian system


• system


• explicit first order


• multi class


• let uh

Sujets

• Lagrangian system
• System

Informations

• Publié par: pefav
• Nombre de lectures: 15

Extrait

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(w,a)
h→ 0
(w,a)
x

∂ τ−∂ v = 0, t x∂ w = 0,t
∂ a = 0, t(τ,w,a)(x,0) = (τ ,w ,a )(x),0 0 0
τ = 1/ρ
v a∈ [0,1]
w
w =v−V(τ,a),
length(ofamassinedv(1.1)ehipapcle)ulti-pandandtoimedvhomoanish.oTherefore,alsotheducedvehaaersererbiablesehicles-citherefortfortheerb:mtoggoresults.-dunowhicstartinghofdescribeespthevheterogeneitvytofoftheareactionseloofWtheindierentrac,terbcar-tracdrivnonerthreepairsFinstheltrac,1devroadelopwrittenlisawithrngeetrucoscillationstwhentinallosizeomeshi.e.siThesethe(knothewn),oscillationsainwsthetheletypeers.whomogenizedandwpsomeersistduceinIntime,wandduce,wsingleenewdeshcribmoemthew,homoygenizehdsystemrelationsablaetreferenceswaeenrvmoeloseeci[6,t49],yforandanddensitstartingyLagran.oWfolloeGshoelowinthatngitheetizevers.elbuses,oehiclescitypyofishethetouniquetosolutionwhic?molaisKru?kvoinvtheofya(i.e.scalarofconservbationulti-class)lacitw,(mwithhvcoariablehcotakecienaccounts,ehadisconttinofuousofin(upheme,stan.esFinallyb,thewyevpronewvinelimit.thatthistheer,sameemacroscopictrohomogenizedstillmoadellaneisaalsomacroscopicthegenizedhydrypoolicdynamicdellimittheofulti-classtheocorrespdescribondingbmaulti-classlinearFypolloolicw-the-ofLeaderconservmotidel.nKeywws.ordsor:inHypulti-clerbsolic(osystemsmofopulation)conservdeationslae.ws,.T13,rac8,Floseew,[44]Multi-Classinhomogeneousmoconditionsduniqueness-Theels,system,Homoginenization,gianDisconctinordinates,uoustheFlux,wing:Uniqueoness,aHydrowdynamicblimittro1systemInatroLagraductiontheIndiscr[1],WAdrivwetc.)etks,Rascle(cars,devvelopeedtadierennewviorsmacroscopbitcaccounmoindeltakofwstrachodelwwherewhicwhtracallothewsecictoolume,athevvoidofsomedeninconsistenciestscofvehiclesypicalofdelsfractioninspiredspacebccupiedyythecars),gasthedynamicelosystem.yInand[2],olicaypconnectionisisdimensionlessestablishedecienbwhicetallowtoeenethetomicroscopict"Fbolloviorsw-the-Leaderdierenmotdel",esandv(aand/orsemi-discretizationdrivofHere,)totheconmacroscopict)modescribdtheelierenceinettroeenducedvincit[1],andwhicequilibriumheloisyitsahtroydroedynamicofAbstractthemo(1.2)V(τ,a) τ
a w =v−V(τ,a) :=v+aP(τ)
w a
v = V(τ)
a v =V(τ,a)
x
U (·,·)h
0(Δx,Δt) Uh
h h
0 0(x,t)→ (x,t ) = (hx,ht)
h→ 0
0 0 0 0 0 0 0 0(v ,w ,a ,τ ) w = v −V(τ ,a ) h→ 0h h h h h h h h
0 ∗ 0 0(v ) v (w ,a )0h h h
0 ∗ ∗ ∗τ (w ,a ) τh 0 0 0
v BV0
w a0 0
00 0 0 0 0 x(x,t ) w (x ) :=W (x, ) W0 0h h
0 0A (x,θ) θ
0 0(x,t) (x,t )
∗ ∗ ∗ ∗(v ,w ,a ,τ ) (v ,w ,a ,τ )h h h h
h→ 0
ν (v,w,a)x,t
∗ν =δ(v−v (x,t))⊗μ (w,a),x,t x
δ μ (w,a)x
Rx
Uh
τ
h = (Δt,Δx)→ 0
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x
w ah h
h→ 0 ν δx,t
(w,a)
∗ ∗∂ w = ∂ a = 0t t
x x
τ x
t τ v
x
y x s t
 v −vj+1 jτ˙ = ,j Δx
∂Vw˙ =v˙ − (τ ,a ) = 0, a˙ = 0,j j j j j∂τj 0 0 0τ (0) =τ , w (0) =w , a (0) =a .j j jj j j
v t Δx τ = 1/ρj j j
j ρ aj j
V w
(w ,a )j j
Δx→ 0
μx
(Δx,Δt)→ (0,0)
τ v
v
w a
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h h
0 0 00 (w ,a ,τ )h h h
∞L h→ 0
Δx→ 0
(Δx,Δt)→ 0 Δt/Δx

∂ ρ+∂ (vρ) = 0,t x
∂ (ρw)+∂ (vρw) = 0,t x
∂ (ρa)+∂ (vρa) = 0,t x
ρ a∈ [0,1]
w
τ := 1/ρ w
(X,T)
∂ X =ρ, ∂ X =−ρv, T =t, τ : = 1/ρ,x t
Rx
ρ X = ρ(y,t)dy
x
prioriestimatesforallpairoeisrewritten.conservWtheebrecallevthatehiclesandximateisofacoscalingotherparameterLagrangianwhicofharetendsatodrivwhenin,describingseeinSectionSection,2.4thebhelowiw,tasoelothat(4.6),thethatsequence(1.3)(1.3)solution,er[2],vshorecolengthebuses,waggressivthatthewyshoSectioneRiemannwi6,wSectioneonly(2.1)conyv(1.3).ergeshwofeak-stardenedinothetoindescribandthspacethewhenthereforesolution(2.1)-(1.2)this(1.1)-(1.2).ofmo.to2.1tiTheeakmacroscopicsystemsmInoddelsAandwesetyRascletruc[1]aha(slovetc.).eeindeltrocoducedconstructedathemacroscopicpreciselymobdelstudyofates.traccoorstw(1.1),whicprophealloInwsmotodimensionlessavvDescriptionoiddimen-thetsevharacterizesereypincons