//img.uscri.be/pth/bfb52d01171527fd525ab135b3eef75896589349
Cet ouvrage fait partie de la bibliothèque YouScribe
Obtenez un accès à la bibliothèque pour le lire en ligne
En savoir plus

XXXV SCUOLA ESTIVA DI FISICA MATEMATICA RAVELLO Settembre

De
77 pages
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


Voir plus Voir moins

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

Rascle

(Universit´e

de

)eciN

Mathematical

oM

edsl

fo

cffiarT

lFwo

ht11

botcO

re

0102

5

/

56

ehT

German

(allegory).

.MRsacelU(inevsrcra

tie´deindustry

iNc)etrying

aMhtmeaot

itaclctach

oMedslfopu

rTffiahtiw

clFwosti

French

11tcompetitors

hOtcobre02106/56
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

X
.Spacing:
τ
j
:=
x
j
+1

x
j
;
s
j
=1

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

=
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

=/

)9

00102

,re

xbotcO

(ht11

ρwolF

:rTcffia

bfo

Psled

noM

nMathematical

a)eciN

med

e(Universite´

iRascle

R.M

.)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