XXXV SCUOLA ESTIVA DI FISICA MATEMATICA RAVELLO Settembre

Michel Rascle

Description

XXXV SCUOLA ESTIVA DI FISICA MATEMATICA, RAVELLO, Settembre 2010. Mathematical Models of Traffic Flow: Macroscopic and Microscopic Aspects Michel Rascle Laboratoire JA Dieudonne, Universite de Nice Sophia-Antipolis Parc Valrose 06108 Nice Cedex 02, France 11th October 2010 M. Rascle (Universite de Nice) Mathematical Models of Traffic Flow 11th October 2010 1 / 65

• introduction broad subject


• greenberg's work periodic


• universite de nice - sophia-antipolis


• half-riemann problem


• universite de nice

Sujets

• Universidad de Niza Sophia Antipolis

Informations

• Publié par: pefav
• Nombre de lectures: 53
• Poids de l'ouvrage: 1 Mo

Extrait

XXXVSCUOLAESTIVADIFISICAMATEMATICA,
RAVELLO,Settembre2010.
MathematicalModelsofTrafficFlow:Macroscopicand
MicroscopicAspects

.MRsacelU(inevsrtie´deiNc)eMichelRascle

LaboratoireJADieudonne´,
Universite´deNiceSophia-Antipolis
ParcValrose06108NiceCedex02,France
http://math.unice.fr/
∼
rascle/

M11thOctober2010

taehamitacloMedslforTffiaclFwo11htOtcobre02101/56
Introduction

Broadsubject!Somanyaspectsoftrafficmodeling,e.g.

.MRFully(cellularautomata,numericalschemes...)orsemi-discrete

(ODE,delayedODE...)/Macroscopic(PDE(hyperbolic

(conservationlaws?Hamilton-Jacobi?Withdiffusionand/or

relaxation?)

OrMesoscopic(kineticdescription)?

Multiscale(structureoftrafficjams,”phasetransitions”,

homogenization,hybridschemes...)

(I):Instability,e.g.stopandgowaves/(S):Stability:preserve

nonnegativespeed(!)and(hopefully!)nocrash...

ODEdescriptionmuchbetterfor(I)andPDEfor(S)...Howtofind

therightcombination?Relatedquestion:ifnecessary,givepriorityto

ODEanduse”modifiedequationathigherorder”fordescribing

specificeffects?

Junctions,linkwithhomogenization.Networks.Hybridschemes...I

won’tcovereverything!

sacelU(inevsrtie´deiNc)eaMhtmeaitacloMedslforTffiaclFwo11htOtcobre02102/65
Outline
Introduction

.MRDiscrete/FluidModels

TheFluidModel(WithoutRelaxation)

I
TheEulerianSystem
I
RiemannProblem.Waves
I
Motivations.Lagrangianversion
I
LinkwithMicroscopicModels(FLM)
I
LagrangianGodunovScheme
I
Passingtothelimit(s)

Junctions

saclI
Onanetwork
I
IngoingHalf-RiemannProblem
I
OutgoingHalf-RiemannProblem
I
RiemannProblematajunction
I
2-1Junction:Homogenization
I
HomogenizedSupply
I
Conclusiononjunctions

eU(inevsrtie´deiNc)eaMhtmeaitacloMedslforTffiaclFwo11htOtcobre02103/56
Outline...

.MRWithRelaxation:TravelingWavesandOscillations
I
Motivations
I
Remark:WhithamSubcharacteristiccondition
I
Smooth”simplewaves”aregenericallyTravelingWaves
I
J.Greenberg’sworkperiodicsolutionsforARG.Extensions...
I
AnExample:theIntelligentDriverModel
I
AdditionalRemarks.Conclusion

Commentsandreferences

sacelU(inevsrtie´deiNc)eaMhtmeaitacloMedslforTffiaclFwo11htOtcobre02104/56
.M

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German

(allegory).

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tie´deindustry

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French

11tcompetitors

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56/70102rebotcOht11wolFcffiarTfosledoMlacitamehtaM)eciNede´tisrevinU(elcsaR.M.secnerefeR.stnemmoC5noisulcnoC.skrameRlanoitiddAledoMrevirDtnegilletnIeht:elpmaxenAsledoMdiulF/etercsiD13

4

LagrangianGodunovScheme

Passingtothelimit(s)

Junctions

Onanetwork

IngoingHalf-RiemannProblem

OutgoingHalfRiemannProblem

WithRelaxation.TravelingWavesandOscillations

Conclusiononjunctions

HomogenizedSupply

2-1Junction:Homogenization

RiemannProblematajunction:Principle...

2

Linkwithmicroscopicmodels(FLM)

Motivations.Lagrangianversion

TheEulerianSystem

TheFluidModel

Remark:WhithamSubcharacteristicCondition

Smooth”simplewaves”aregenericallyTravelingWaves

J.Greenberg’speriodicsolutionsforARG.Extensions

Discrete/FluidModels

.MR(Fullyor)1/2discrete:FollowtheLeaderModels...
Carlength:
l
=Δ
X
.Spacing:
τ
j
:=
x
j
+1
−
x
j
;
s
j
=1
/ρ
j
=
τ
j
/
l
:
specificvolume,density.


x
˙
j
=
v
j
=
⇒
s
˙
j
=
v
j
+1
l
−
v
j
v
˙
j
=
F
(
x
j
,
x
j
+1
,
v
j
,
v
j
+1
)

(
e
.
g
.
)=
α
v
m
V
0
(
x
j
+1
−
x
j
)
v
j
+1
−
v
j
+
β
(
V
e
(
x
j
+1
−
x
j
)
−
v
j
)
jlll

)1.2(

Convectivepart(fastreaction)+(slow)relaxationpart...
Examples,seealsoGazis-Herman-Rotheryand...
I
α
=0
,β>
0:Bando’sOptimalVelocityModel
I
α>
0
,β
=
m
=0:Aw-Klar-Materne-Rascle,SIAP2002
I
α>
0
,β>
0
,
m
=0:J.Greenbergand/orAw-Rascle,SIAP2000-2004
I
IntelligentDriverModel(IDM):Helbing-Treiber,
∼
2000
√v
˙
j
=
a
[1
−
v
jm
−
(
s
b
(
v
j
)
−
v
j
s
(
v
j
+1
−
v
j
)
)
2
];
s
b
(
v
):=
s
0
+
s
1
v
+
s
2
(
v
)
j

sacelU(inevsrtie´deiNc)eaMhtmeaitacloMedslforTffiaclFwo11htOtcobre02108/56
)ρ()x∂v+t∂()ρ(0p˜−=vx∂v+vt∂:semoceb)WP(ninoitauqednoceSIpx∂v+pt∂→px∂:gnixiF.)2002(gnahZ,)0002,?noitcerruseR(elcsaR-wAI!!v>c+v=2λ:2dna0<v:1:sexodaraP]!esrowllitsnoisuffiD[!!ledomelbirretasiWP)59,meiuqeR(oznagaDI...+ρx∂)ρ(0p˜−=:...+ρx∂)ρ(0p1−ρ−=vx∂v+vt∂,0=)vρ(x∂+ρt∂()scimanyDsaGfc(mahtihW-enyaP:redrOdnoceSI.serugiF.elbats)oot(,tsuboryreV.+v>−vfi)gnikarb(sevawkcohs-,+v<−vfi)noitarelecca(sevawnoitcaferarderetnec-:0>x±rof±Kinetic:

ρ56

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.)ρ(Vρ=qxufl:margaidlatnemadnuF,0<”)Vρ(,0<)ρ(0V,)ρ(V=v,0=)vρ(x∂+ρt∂]ibocaJ-notlimaH↔[)RWL(sdrahciR-mahtihW-llihthgiL:redrOtsriFI:diulF
λ:2dna0<v:1:sexodaraP]!esrowllitsnoisuffiD[!!ledomelbirretasiWP)59,meiuqeR(oznagaDI...+ρx∂)ρ(0p˜−=:...+ρx∂)ρ(0p1−ρ−=vx∂v+vt∂,0=)vρ(x∂+ρt∂()scimanyDsaGfc(mahtihW-enyaP:redrOdnoceSI[)RWL(sdrahciR-mahtihW-llihthgiL:redrOtsriFI:diulF:citeniK∂
t
ρ
+
∂
x
(
ρ
v
)=0
,
v
=
V
(
ρ
)
,
V
0
(
ρ
)
<
0
,
(
ρ
V
)”
<
0
,

Fundamentaldiagram:flux
q
=
ρ
V
(
ρ
).
RiemannPb:
ρ
(
x
,
0)=
ρ
±
for
±
x
>
0:
-centeredrarefactionwaves(acceleration)if
v
−
<
v
+
,
-shockwaves(braking)if
v
−
>
v
+
.Veryrobust,(too)stable.Figures.

↔
Hamilton-Jacobi]

56/90102rebotcOht11wolFcffiarTfosledoMlacitamehtaM)eciNede´tisrevinU(elcsaR.M)ρ()x∂v+t∂()ρ(0p˜−=vx∂v+vt∂:semoceb)WP(ninoitauqednoceSIpx∂v+pt∂→px∂:gnixiF.)2002(gnahZ,)0002,?noitcerruseR(elcsaR-wAI!!v>c+v=2