conservation laws with point constraints on the ﬂow and their...

Conservation laws with point constraints

on the flow and their applications

Massimiliano Daniele Rosini

Instytut MatematykiUniwersytet Marii Curie-Sk lodowskiej

Lublin, Poland

[email protected]

ANCONET

1 / 53

This talk is concerned with the recent results on conservation lawswith point constraints presented in

B.Andreianov, C.Donadello, M.D.Rosini“A second order model for vehicular traffics with local point constraints on the

flow”, to appear on Mathematical Models and Methods in Applied Sciences

B.Andreianov, C.Donadello, U.Razafison, J.Y.Rolland, M.D.Rosini“Solutions of the Aw-Rascle-Zhang system with point constraints”, to appear onNetworks and Heterogeneous Media

B.Andreianov, C.Donadello, U.Razafison, M.D.Rosini“Qualitative behaviour and numerical approximation of solutions to conservation

laws with non-local point constraints on the flux and modeling of crowd

dynamics at the bottlenecks”, to appear on ESAIM: Mathematical Modelling andNumerical Analysis

B.Andreianov, C.Donadello, U.Razafison, M.D.Rosini“Riemann problems with non–local point constraints and capacity drop”,Mathematical Biosciences and Engineering 12 (2015) 259–278

B.Andreianov, C.Donadello, M.D.Rosini“Crowd dynamics and conservation laws with non–local constraints and capacity

drop”, Mathematical Models and Methods in Applied Sciences 24 (2014)2685–2722

2 / 53

1 The motivating problem

2 Modelling traffic flows

3 Constrained LWR

4 Non-local constraint

5 ARZ

6 Constrained ARZ

7 Two phase model (M.Benyahia, GSSI L’Aquila)

3 / 53

Section 1

The motivating problem

4 / 53

The motivating problem

Study of the traffic flow through bottlenecks, i.e. locations withreduced capacity.

Capacity of the bottleneck: the maximum number of pedestriansor vehicles that can flow through the bottleneck in a given timeinterval.

Typical phenomena at the bottlenecks are:

Capacity drop

Braess’ paradox

Faster Is Slower effect

Possible outcomes of this study are:

Optimization of the traffic (green waves, stop-and-go, meantravel time, etc.).

Reducing traffic accidents.

5 / 53

Crowd Accidents

YEAR DEAD CITY NATION

1711 245 Lyon France1876 278 Brooklyn USA1883 180 Sunderland England1896 1,389 Moscow Russia1903 602 Chicago USA1913 73 Michigan USA1941 4,000 Chongqing China1942 354 Genoa Italy1943 173 London England1956 124 Yahiko Japan1971 66 Glasgow England1982 66 Moscow Russia1985 39 Brussels Belgium1988 93 Tripureswhor Nepal1989 96 Sheffield England1990 1,426 Al-Mu’aysam Saudi Arabia1991 40 Orkney South Africa1991 42 Chalma Mexico1993 73 Madison USA1994 270 Mecca Saudi Arabia1994 113 Nagpur India1996 82 Guatemala City Guatemala1998 70 Kathmandu Nepal1998 118 Mecca Saudi Arabia1999 53 Minsk Belarus

YEAR DEAD CITY NATION

2001 43 Henderson USA2001 126 Accra Ghana2003 100 West Warwick USA2004 194 Buenos Aires Argentina2004 251 Mecca Saudi Arabia2005 300 Wai India2005 265 Maharashtra India2005 1,000 Baghdad Iraq2006 345 Mecca Saudi Arabia2006 74 Pasig City Philippines2006 51 Ibb Yemen2008 147 Jodhpur India2008 162 Himachal Pradesh India2008 147 Jodhpur India2010 71 Kunda India2010 63 Amsterdam Netherlands2010 347 Phnom Penh Cambodia2011 102 Kerala India2012 79 Port Said Egypt2013 61 Abidjan Ivory Coast2013 242 Rio Grande do Sul Brazil2013 36 Uttar Pradesh India2013 115 Datia Madhya Pradesh India2014 18 Maharashtra India2014 15 Kinshasa D.R.Congo

6 / 53

Crowd Accidents

1985, Heysel Stadium in Brussels (Belgium): 39 fans died as they tried to escape Liverpool supporters.

2010, Phnom Penh (Cambodia): 347 people died in a bridge stampede at the Khmer Water Festival.

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Crowd Accidents

2010, Duisburg (Germany): 21 people died near an overcrowded tunnel leading into the festival.

2014, Shanghai (China): 35 people died rushing to catch (fake) money thrown from a nightclub.

7 / 53

Crowd Accidents

Jamarat Bridge (Saudi Arabia)

Year Dead Injured

2015 1,470 700

2006 363 289

2004 249 252

2001 35 179

1998 118 434

1997 22 43

1994 266 98

7 / 53

Panic management

Is it possible to avoid

or to mitigate

crowd accidents?

8 / 53

Panic management

Crowd behaviour is not always the primary cause of accidents.

Architects and city planners try to provide a safe environment forplaces of public assembly by using mathematical models todetermine and prevent congestions.

Example

Nowadays, all public entertainmentvenues are equipped with doors thatopen outwards using crash barlatches that open when pushed.

9 / 53

Panic management

Crowd behaviour is not always the primary cause of accidents.

Architects and city planners try to provide a safe environment forplaces of public assembly by using mathematical models todetermine and prevent congestions.

Example

A tall column placed in the upstreamof the door exit may reduce theinter-pedestrian pressure, decreasingthe magnitude of clogging andmaking the overall outflow higherand more regular (Braess’s paradox).

9 / 53

Section 2

Modelling traffic flows

10 / 53

LWR

The prototype of first order macroscopic models for traffic flows isthe Lighthill, Whitham, Richards model

ρt + [ρ v ]x = 0,

v = V (ρ),(LWR)

where V : [0, ρmax] → [0, vmax] is non-increasing, V (0) = vmax andV (ρmax) = 0.

Lighthill, Whitham, “On kinematic waves. II. A theory oftraffic flow on long crowded roads”, Royal Society of London.Series A, Mathematical and Physical Sciences 1955

Richards, “Shock waves on the highway”, Operations Research1956

11 / 53

ARZ

The most celebrate second order model for traffic flows is the Aw,Rascle, Zhang model

ρt + (ρ v)x = 0,

[ρ (v + p(ρ))]t + [ρ (v + p(ρ)) v ]x = 0,(ARZ)

e.g. p(ρ) = ργ , γ > 0.

Aw, Rascle, “Resurrection of “Second Order” Models of TrafficFlow”, SIAM Journal on Applied Mathematics 2000

Zhang, “A non-equilibrium traffic model devoid of gas-likebehavior”, Transportation Research Part B: Methodological2002

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Motivating problem

Both LWR and ARZ assume that the road is homogeneous.

However in real life this assumption is NOT always justified.

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Motivating problem


Toll gates

Traffic lights

Construction sites

Speed bumps

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Motivating problem


Escalators and stairs

Traffic lights

Turnstiles

Doors and revolving doors

13 / 53

Motivating problem


Definition (roughly speaking)

By point constraint we mean a pointwise bottleneck that hindersthe flow.

Applicability

We apply the theory for point constraints when we are interestedin the effects on the upstream and downstream and not in thedynamics inside the constraint location.

13 / 53

Motivating problem

The concept of point constraints on the flow was introduced

in the framework of vehicular traffic to model the presence oftoll gates, construction sites, traffic lights, speed bumps, etc.

Colombo, Goatin, “A well posed conservation law with avariable unilateral constraint”, Journal of DifferentialEquations, 2007

in the framework of crowd dynamics to model the presence ofdoors, revolving doors, turnstiles, escalators, traffic lights,stairs, etc.

Colombo, Rosini, “Pedestrian flows and non-classicalshocks”, Mathematical Methods in the Applied Sciences,2005

14 / 53

Section 3

Constrained LWR

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Constrained LWR

If in x = 0 there is a constraint, then the corresponding problem is

ρt + f (ρ)x = 0, ρ(0, x) = ρ0(x), f (ρ(t, 0±)) ≤ F . (cLWR)

The expression of F depends on the situation at hand:

F is constant if we have a construction site or stairs.

F depends on time if we have a traffic light.

F depends non-locally on ρ if we have a toll gate or door.

16 / 53

Constrained Riemann solver

Consider the Riemann problem for (cLWR)

ρt + f (ρ)x = 0, ρ(0, x) =

ρℓ if x < 0,

ρr if x > 0,f (ρ(t, 0±)) ≤ F .

If f (RLWR[ρℓ, ρr ](0)) ≤ F , then CRLWR[ρℓ, ρr ] ≡ RLWR[ρℓ, ρr ].

If f (RLWR[ρℓ, ρr ](0)) > F , then

CRLWR[ρℓ, ρr ](x) =

RLWR[ρℓ, ρ](x) if x < 0,

RLWR[ρ, ρr ](x) if x > 0.

ff

FF

ρρ ρρ ρρ17 / 53

Example

Consider the Riemann problem for constrained LWR

ρt + f (ρ)x = 0, ρ(0, x) ≡ ρℓ = ρr , f (ρ(t, 0±)) ≤ F .

If f (ρℓ) > F , then the solution is

f

F

ρρ ρρℓ

t

ρℓρℓ

x

ρρ

18 / 53

Example

Consider the Riemann problem for constrained LWR

ρt + f (ρ)x = 0, ρ(0, x) ≡ ρℓ = ρr , f (ρ(t, 0±)) ≤ F .

If f (ρℓ) > F , then the solution is

ρ(t)

ρℓρℓ

ρ

ρ

x

t

ρℓρℓ

x

ρρ

No maximum principle!

TV bound explosion!

18 / 53

Well-posedness in BV

constraint ⇒ non-classical shocks ⇒ explosion of TV

Hence BV is not the proper space ⇒ introduction of the space

D =

ρ ∈ L1 : Ψ(ρ) ∈ BV

where

Ψ(ρ) =

∫ ρ

0

∣∣f ′(r)∣∣ dr .

19 / 53

Cauchy problem


Definition (Colombo, Goatin ’07)

ρ ∈ D is a solution to (cLWR) if:1) ∀φ ∈ C1

c , φ ≥ 0, and ∀k ∈ [0, ρmax]

∫∫

R+×R

[|ρ − k|φt + sgn(ρ − k) [f (ρ) − f (k)] φx

]dx dt

+

∫

R

|ρ0(x) − k| φ(0, x) dx

+2

∫

R+

[1 −

F (t)

fmax

]f (k) φ(t, 0) dt ≥ 0.

2) f (ρ(t, 0−)) = f (ρ(t, 0+)) ≤ F (t) for a.e. t > 0.

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Cauchy problem


Definition (Andreianov, Goatin, Seguin, ’10)

ρ ∈ L∞ is a solution to (cLWR) if ∃M > 0 s.t. ∀cℓ, cr ∈ [0, ρmax]and ∀φ ∈ C1

c , φ ≥ 0

∫∫

R+×R

[|ρ − c| φt + sgn(ρ − c) [f (ρ) − f (c)] φx

]dx dt

+

∫

R

|ρ0(x) − c(x)| φ(0, x) dx

+M

∫

R+

dist((cℓ, cr ), G(F (t))

)φ(t, 0) dt ≥ 0,

where G is the set of admissible traces along x = 0 and

c(x) =

cℓ if x < 0,

cr if x > 0.

20 / 53

Cauchy problem


Definition (Chalons, Goatin, Seguin ’13)

For more general fluxes, ρ ∈ L∞ is a solution to (cLWR) if:1) ∀φ ∈ C1

c , φ ≥ 0, and ∀k ∈ [0, ρmax]

∫∫

R+×R


]dx dt

+

∫

R

|ρ0(x) − k| φ(0, x) dx

+2

∫

R+

[f (k) − min f (k), F (t)

]f (k) φ(t, 0) dt ≥ 0.

2) f (ρ(t, 0−)) = f (ρ(t, 0+)) ≤ F (t) for a.e. t > 0.

20 / 53

Cauchy problem


Theorem (Colombo-Goatin ’07, Andreianov-Goatin-Seguin ’10,Chalons, Goatin, Seguin ’13)

If F ∈ L∞, then there exists a semigroup SF : R+ × L∞ → L∞

for (cLWR).

If F 1, F 2, ρ10, ρ2

0 ∈ L∞ and F 1 − F 2, ρ10 − ρ2

0 ∈ L1:

∥∥∥SF 1

t ρ10 − SF 2

t ρ20

∥∥∥L1

≤∥∥∥ρ1

0 − ρ20

∥∥∥L1

+ 2∥∥∥F 1 − F 2

∥∥∥L1

.

D is an invariant domain and

ρ0 ∈ D ⇒

TV(Ψ(SFt ρ0)) ≤ C ,∥∥∥Ψ(SF

t ρ0) − Ψ(SFs ρ0)

∥∥∥L1

≤ C |t − s|,

with C > 0 constant uniform w.r.t. t.

20 / 53

Initial-boundary value problems

ρt + f (ρ)x = 0, ρ(0, x) = ρ0(x),

f (ρ(t, 0)) = q(t), f (ρ(t, x±c )) ≤ F (t).

Definition (Colombo, Goatin, Rosini ’11)

ρ ∈ L∞ is a solution to the above (cIBVP) if:1) ∀φ ∈ C1

c , φ ≥ 0, and ∀k ∈ [0, ρmax]

∫∫

R+×R+


]dx dt

+

∫

R+

|ρ0(x) − k| φ(0, x) dx

+

∫

R+

sgn[f −1∗ (q(t)) − k

] [f (ρ(t, 0+)) − f (k)

]φ(t, 0) dt

+2

∫

R+

[1 −

F (t)

fmax

]f (k) φ(t, xc ) dt ≥ 0.

2) f (ρ(t, x−c )) = f (ρ(t, x+

c )) ≤ F (t) for a.e. t > 0.

21 / 53

Initial-boundary value problems

ρt + f (ρ)x = 0, ρ(0, x) = ρ0(x),

f (ρ(t, 0)) = q(t), f (ρ(t, x±c )) ≤ F (t).

Theorem (Colombo, Goatin, Rosini ’11)

If q, F ∈ BV, then there exists a semigroup Sq,F : R+ × D → Dfor the above (cIBVP).

If q1, q2, F 1, F 2, ρ10, ρ2

0 ∈ L∞ and q1 − q2, F 1 − F 2, ρ10 − ρ2

0 ∈ L1:

∥∥∥SF 1

t ρ10 − SF 2

t ρ20

∥∥∥L1

≤∥∥∥ρ1

0 − ρ20

∥∥∥L1

+∥∥∥q1 − q2

∥∥∥L1

+ 2∥∥∥F 1 − F 2

∥∥∥L1

.

Moreover

∫ T

0

∥∥∥f (Sq1,F 1

t ρ10)(x−) − f (Sq2,F 2

t ρ10)(x−)

∥∥∥dt ≤∥∥∥q1(t) − q2(t)

∥∥∥L1([0,minT ,τ0])

+ 2∥∥∥F 1(t) − F 2(t)

∥∥∥L1([0,minT ,τc ])

.

21 / 53

Synchronizing gateways

Colombo-Goatin-Rosini ’11:

0

2

4

6

8

10

12

14

0 0.5 1 1.5 2

t

tau

Mean Arrival Time

0

2

4

6

8

10

12

14

0 0.5 1 1.5 2

t

tau

Exit Time

The lower graphs corresponding to the lower inflows.22 / 53

Section 4

Non-local constraint

23 / 53

Non-local constraint: empirical evidences

Pedestrian inflow and outflowpatterns at the London Under-ground station

E.M.Cepolina.Phased evacuation: An optimisation model which takes intoaccount the capacity drop phenomenon in pedestrian flows.Fire Safety Journal, 2009.

24 / 53

Non-local constraint: explanation

High density upstream =⇒ Capacity drop

D. Helbing, “Agent-Based Simulation of Traffic Jams, Crowds,and Supply Networks – Reality, Simulation, and Design ofIntelligent Infastructures”

25 / 53


The corresponding model is

ρt +f (ρ)x =0, f(ρ(t, 0±)

)≤p

(∫

R−

w(x) ρ(t, x) dx

), ρ(0, x) = ρ0(x)

(NL)

ρmax

ff (ρ)

ρ

f (ρ)

ρ x

w

pi

−iw ξρmaxρi ρi

p

B.Andreianov, C.Donadello, M.D.Rosini, “Crowd dynamics andconservation laws with non–local constraints and capacitydrop”, Mathematical Models and Methods in Applied Sciences24 (2014) 2685–2722

26 / 53


Theorem (Andreianov, Donadello, Rosini ’14)

Assume that:

(F) f ∈ Lip ([0, ρmax]; [0, fmax]), f (0) = 0 = f (ρmax) and thereexists ρ ∈ ]0, ρmax[ s.t. f ′(ρ) (ρ − ρ) > 0 a.e. in [0, ρmax].

(W) w ∈ L∞(R−;R+) is an increasing map, ‖w‖L1(R−;R+) = 1 andthere exists iw > 0 such that w(x) = 0 for any x ≤ −iw .

(P) p is non-increasing and belongs to Lip ([0, ρmax] ; ]0, f (ρ)]).

27 / 53



Then:

(i) For any initial datum ρ0 ∈ L∞(R; [0, ρmax ]), the Cauchyproblem (cLWR) admits a unique solution ρ. Moreover, ifρ = ρ(t, x) is the solution corresponding to the initial datumρ0 ∈ L∞(R; [0, ρmax]), then for all T > 0 and L > iw

‖ρ(T ) − ρ(T )‖L1([−L,L];R) ≤ eCT ‖ρ0 − ρ0‖L1(|x |≤L+MT;R),

where M = Lip(f ) and C = 2Lip(p)‖w‖L∞(R−;R).

27 / 53



Then:

(ii) If ρ0 belongs to D, then the unique solution of (cLWR)verifies ρ(t, ·) ∈ D for a.e. t > 0, and it satisfies

TV (Ψ (ρ(t))) ≤ Ct = TV (Ψ (ρ0)) + 4fmax + C t ,

moreover, for a.e. t, s in ]0, T [ we have

‖Ψ (ρ(t, ·)) − Ψ (ρ(s, ·))‖L1(R;R) ≤ |t − s| Lip(Ψ) CT .

27 / 53


Proof (Wave-front tracking and operator splitting methods).

Introduce f n ∈ PLC approximation of f , ρn0 ∈ PC approximation

of ρ0 and ph ∈ PC approximation of p. Define recursively

F nh[ρn0](t) = S [ρn

0, Θ [ρn0]] (t)

if t ∈ [0, ∆t], and, if t ∈ ](ℓ + 1)∆t, (ℓ + 2)∆t], ℓ ∈ N, then

F nh[ρn0](t) = S

[F nh[ρn

0] ((ℓ + 1)∆t) , Θ[F nh[ρn

0] ((ℓ + 1)∆t)]]

(t).

The existence of a limit for F nh[ρn0] as n and h go to infinity is

ensured by the choice

∆th =1

2h+1 w(0−) Lip(p).

27 / 53


The proof is quite technical, also because solutions of a Riemannproblem in the case of a non-decreasing piecewise constant p mayfail to be unique, L1

loc–continuous and consistent, see

B.Andreianov, C.Donadello, M.D.Rosini, U.Razafison,“Riemann problems with non–local point constraints andcapacity drop”, Mathematical Biosciences and Engineering 12(2015) 259–278

More in general, in the above paper we investigate how the regularityof p impacts the well–posedness of the problem.We show that the regularity of p plays a central role in the well-posedness result.While existence still holds for the Cauchy problem when p is merelycontinuous, it is difficult to justify uniqueness in this case.

27 / 53

Comparison: local VS non-local

I

E

A B

L

CD

M

O

N

R.M.Colombo, Rosini.

Pedestrian Flows and non-classical shocks.Mathematical Methods in the AppliedSciences 2005

20

40

60

80

−3−4−5 −iwBA

CDEF

GHI

L

MN

O

Andreianov, Donadello, Rosini.

Crowd dynamics and conservation laws with non-localconstraints.Mathematical Models and Methods in AppliedSciences 2014

28 / 53

Numerical approximation

Andreianov, Donadello, Razafison, Rosini, “Qualitativebehaviour and numerical approximation of solutions toconservation laws with non-local point constraints on the fluxand modeling of crowd dynamics at the bottlenecks”, toappear on ESAIM: Mathematical Modelling and NumericalAnalysis

In this paper we investigate numerically the model (NL). We provethe convergence of a scheme based on a constraint finite volumemethod. We then perform ad hoc simulations to qualitativelyvalidate the model by proving its ability to reproduce typicalphenomena at the bottlenecks, such as the Braess’ paradox andthe Faster Is Slower effect.

29 / 53

Braess’s paradox

24

25

26

27

28

29

30

31

32

33

34

-1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

evac

uat

ion

tim

e

with obstacle

without obstacle

d

Evacuation time as a function of the position of the obstacle.

30 / 53

Faster is slower effect

20

40

60

80

100

120

140

160

180

200

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

ρ

ρ1

ρ2

vmax

evac

uat

ion

tim

e

Evacuation time as a function of vmax for different initial densities

ρ = χ[−5.75, −2]

, ρ1 =4

5χ

[−5.75, −2], ρ2 =

3

5χ

[−5.75, −2]

31 / 53


20

40

60

80

100

120

140

160

180

200

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

β = 1

β = 0.9

β = 0.8

vmax

evac

uat

ion

tim

e

Evacuation time as a function of vmax for different exit efficiencies

pβ(ξ) = p(β ξ), β = 1, β = 0.9, β = 0.8

31 / 53


20

40

60

80

100

120

140

160

180

200

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

ρ

ρ3

ρ4

vmax

evac

uat

ion

tim

e

Evacuation time as a function of vmax for different initial locations

ρ = χ[−5.75, −2]

, ρ3 = χ[−11.75, −8]

, ρ4 = χ[−19.75, −16]

31 / 53

General non-local constraints

We are considering now a more general class of point constraints

f (ρ(t, 0±)) ≤ q(t) = Q[ρ](t) for a.e. t ∈ [0, T ],

where

Q : C0([0, T ]; L1(R;R)) → L1([0, T ]; [0, f (ρ)])

so that the operator Q may be non-local both in time and space.

32 / 53

Example

Consider vehicular traffic through a toll booth, where the numberof open gates is decided according to on line data.

If the data are collected by a video camera, then we can take

Q[ρ](t) = p

(∫

R−

∫ t

0w(x) κ(t − s) ρ(s, x) ds dx

),

being supp(w) the area registered by the video camera and supp(κ)the period of time the data are taken into account.

A time delay in adapting the number of open gates to the availabledata can be easily modelled.

33 / 53

Example


If the data come from a photo camera that shoots photos at timesti of the area given by supp(w), then we can take

Q[ρ](t) = p

(∑

ti <t

∫

R−

w(x) κ(t − ti) ρ(ti , x)dx

),

being supp(κ) the period of time the data are taken into account.


33 / 53

Example


If the data come from local sensors located in yi , then we can take

Q[ρ](t) = p

M∑

i=1

w(yi) κ(t) ρ0(yi)

+

∫ t

0w ′(yi) κ(t − s) f (ρ)(s, yi ) ds

,

being supp(κ) the period of time the data are taken into account.


33 / 53

General non-local constraints

We introduced a concept of entropy solutions.

By using fixed point arguments we are able to address solvability ofnon-locally constrained problems.

Under general conditions on Q we obtained the uniqueness ofentropy solutions.

B.Andreianov, C.Donadello, U.Razafison, M.D.Rosini,“Analysis and approximation of first order traffic models withgeneral point constraints on the flow”, in preparation

34 / 53

Section 5

ARZ

35 / 53

ARZ and the vacuum

ARZ is the 2 × 2 system of conservation laws

~Yt + F (~Y )x = ~0, ~Y =

(ρy

), F (~Y ) = v ~Y , v =

y

ρ− p(ρ).

away from the vacuum:

It is strictly hyperbolic.

It is of Temple class.

It has the Panov renormalization property.

Existence results are available (wave-front tracking).

Ferreira, KondoGlimm method and wave-front tracking for the Aw-Rascletraffic flow model.Far East Journal of Mathematical Sciences 2010

36 / 53

ARZ and the vacuum

ARZ is the 2 × 2 system of conservation laws

~Yt + F (~Y )x = ~0, ~Y =

(ρy

), F (~Y ) = v ~Y , v =

y

ρ− p(ρ).

at the vacuum: F is not defined!

Vacuum (Godvik, Hanche-Olsen)

Even for initial data away from the vacuum, the solutions mayattain values in the vacuum in a (possibly strictly positive) finitetime.

Existence results are available (compensated compactness).

Godvik, Hanche-OlsenExistence of solutions for the Aw-Rascle trafficflow model with vacuum.Journal of Hyperbolic Differential Equations 2008

LuExistence of global bounded weak solutions tononsymmetric systems of Keyfitz-Kranzer type.Journal of Functional Analysis 2011

36 / 53

BV-estimate

Introduce the Riemann invariants coordinates

~W = (v , w) ∈ W =

(v , w) ∈ R2+ : v ≤ w

= W0 ∪ Wc

0

given by

Y ( ~W ) = p−1(w − v)

(1w

), W (~Y ) =

y

ρ− p(ρ)

y

ρ

,

where v is the velocity,

w is the Lagrangian marker.

Crucial for the existence results the BV-estimate

TV( ~W (t)) ≤ TV( ~W0)

37 / 53

ARZ and the vacuum

Introduce in PC the concept of physically admissiblediscontinuities involving a vacuum state:

(~W (t, x−), ~W (t, x+)

)∈ G

where

G.=

( ~Wℓ, ~Wr ) ∈ W2 :

~Wℓ ∈ W0

~Wr ∈ W0

⇒ ~Wℓ = ~Wr ,

~Wℓ ∈ Wc0

~Wr ∈ W0

⇒ ~Wr = (wℓ, wℓ)

.

38 / 53

ARZ and the vacuum

Then define the Riemann solver RARZ

RARZ : G → C0 (]0, +∞[ ; L∞ (R; W)) ,

( ~Wℓ, ~Wr ) 7→ RARZ[ ~Wℓ, ~Wr ],

by introducing first the elementary waves: shocks, rarefactions andcontact discontinuities.

Details on elementary waves

39 / 53

The Riemann solver RARZ

Definition (roughly speaking)

The solution performs either a rarefaction or a shock, followed by acontact discontinuity.

Definition

1~Wℓ ∈ Wc

0~Wr ∈ W0

⇒

RARZ[ ~Wℓ, ~Wr ] ≡ R[ ~Wℓ, ~Wm]~Wm = (wℓ, wℓ) ∈ W0

2~Wℓ ∈ W0~Wr ∈ Wc

0

⇒ RARZ[ ~Wℓ, ~Wr ]

(xt

)=

~Wℓ if x < vr t~Wr if x > vr t

3~Wℓ, ~Wr ∈ Wc

0vr < vℓ < wℓ

⇒

RARZ[ ~Wℓ, ~Wr ] ≡ S[ ~Wℓ, ~Wm] + C[ ~Wm, ~Wr ]~Wm = (vr , wℓ) ∈ Wc

0

4~Wℓ, ~Wr ∈ Wc

0vℓ < vr < wℓ

⇒

RARZ[ ~Wℓ, ~Wr ] ≡ R[ ~Wℓ, ~Wm] + C[ ~Wm, ~Wr ]~Wm = (vr , wℓ) ∈ Wc

0

5Wℓ, ~Wr ∈ Wc

0vℓ <wℓ ≤vr <wr

⇒

RARZ[ ~Wℓ, ~Wr ](

xt

)=

R[ ~Wℓ, ~Wm]

(xt

)if x

t< vr

~Wr if xt

> vr

~Wm = (wℓ, wℓ) ∈ W0

40 / 53

Section 6

Constrained ARZ

41 / 53

ARZ as extension of LWR

ARZ can be interpreted as a generalization of LWR possessing afamily of fundamental diagram curves, rather than a single one,corresponding to different Lagrangian markers w .

ARZ can be interpreted as a multi population version of LWR:

The population corresponding to the La-grangian marker w is characterized bymaximal speed w and length 1/p−1(w).

42 / 53

ARZ as extension of LWR

Can we extend also the concept of point constraints?

Technically YES because ARZ has the Panov renormalizationproperty

F (~Y ) = v ~Y

This ensures the existence of the traces of the solutions at x = 0.f

ρ

F

ρ ρρ

f

ρ

f

F

ρρ ρ

f

43 / 53

ARZ with constraint

Assume to have at x = 0 a constraint and consider the problem

~Yt + F (~Y )x = ~0, ~Y (0, x) = ~Y0, f (~Y (t, 0±)) ≤ F . (cARZ)

The Riemann problem

If f (RARZ[ ~Wℓ, ~Wr ]) ≤ F , then CRARZ[ ~Wℓ, ~Wr ] ≡ RARZ[ ~Wℓ, ~Wr ].

If f (RARZ[ ~Wℓ, ~Wr ]) > F , then

CRARZ[ ~Wℓ, ~Wr ](x/t) =

RARZ[ ~Wℓ, W (wℓ)](x/t) if x < 0,

RARZ[W (wℓ), ~Wr ](x/t) if x > 0.

Garavello, Goatin, The Aw-Rascle traffic model with locallyconstrained flow, J. Math. Anal. Appl. 2011

44 / 53

ARZ with constraint



The Cauchy problem

The first results in this direction are in

Andreianov, Donadello, Razafison, Rolland, Rosini, Solutionsof the Aw-Rascle-Zhang system with point constraints, toappear on Networks and Heterogeneous Media

44 / 53

ARZ with constraint



The Cauchy problem

We revisit the wave-front tracking construction of physically admis-sible solutions possibly involving vacuum states.

G.=

( ~Wℓ, ~Wr ) ∈ W2 :

~Wℓ ∈ W0

~Wr ∈ W0

⇒ ~Wℓ = ~Wr ,

~Wℓ ∈ Wc0

~Wr ∈ W0

⇒ ~Wr = (wℓ, wℓ)

44 / 53

ARZ with constraint



The Cauchy problem

We introduce a Kruzhkov-like family of entropies to select the ad-missible shocks.

Ek( ~W ) =

0 if v ≤ k,

1 −p−1(w − v)

p−1(w − k)if v > k,

Qk( ~W ) =

0 if v ≤ k,

k −q( ~W )

p−1(w − k)if v > k.

This tool allows to define rigorously the appropriate notion of ad-missible solution and to approximate the solutions of (cARZ).

44 / 53

ARZ with constraint



The Cauchy problem

Definition (Constrained entropy solution)

~W ∈ L∞ (]0, +∞[ ; BV(R; W)) ∩ C0(R+; L1loc (R; W)) is a

constrained entropy solution if it is a constrained weak solution and

∫∫

R+×R

[Ek( ~W ) φt +Qk( ~W ) φx

]dx dt+

∫

R+

Nk( ~W , q0) φ(t, 0) dt ≥ 0

for any φ ∈ C∞c (]0, +∞[ × R;R), for any k ∈ ]0, V0], where

Nk( ~W , q0)=

q( ~W (t, 0))[

kq0(t) − 1

p−1([w(t,0)−k]+)

]+if q0(t) 6= 0

k otherwise.

44 / 53

ARZ with constraint



The Cauchy problem

We propose a finite volumes numerical scheme for (cARZ) and weshow that it can correctly locate contact discontinuities and takethe constraint into account.

Chalons, Goatin, “Transport-equilibrium schemes forcomputing contact discontinuities in traffic flow modeling”,Communications in Mathematical Sciences, 2007

44 / 53

ARZ with constraint



The Cauchy problem

The convergence of the wave-front tracking is proved in

B.Andreianov, C.Donadello, M.D.Rosini, A second order modelfor vehicular traffics with local point constraints on the flow,to appear on Mathematical Models and Methods in AppliedSciences

44 / 53

ARZ with constraint



The Cauchy problem

The constraint F is essentially constant.

f

ρ

F

ρ ρρ

fw

v

ww

vv v

ρ 7→ [w − p(ρ)] ρ v 7→ [v + p(F /v)]44 / 53

Section 7

Two phase model (M.Benyahia, GSSI L’Aquila)

45 / 53

Free flow (LWR)

u = (ρ, v) ∈ Ωf ,

ρt + [ρ v ]x = 0,

v = vf(ρ),

Congested flow (ARZ)

u = (ρ, v) ∈ Ωc,

ρt + [ρ v ]x = 0,

[ρ w ]t + [ρ w v ]x = 0.

(PT)

ρ

qmax

ρ v Vmax Vf

Vc

Ω′

f

Ωc

Ω′′

fΩc Ω′′

f

Ω′

f

v

w

Wc

Wmin

Wmax

VfVc

P. Goatin, “The Aw-Rascle vehicular traffic flow with phasetransitions”, Mathematical and Computer Modeling, 2006

46 / 53

Definition (Benyahia, Rosini work in progress)

Fix an initial distribution u in BV(R; Ω). Let u be a function inL∞(]0, +∞[; BV(R; Ω)) ∩ C0(R+; L1

loc(R; Ω)) satisfying the initialcondition u(0, x) = u(x) for a.e. x ∈ R. We say that u is anentropy solution to the two phase model (PT) if:

1 It is an entropy solution of LWR in u−1(Ωf).

2 It is an entropy solution of ARZ in u−1(Ωc).

3 If u performs a phase transition from u− to u+, then thespeed of propagation of the phase transition satisfies theRankine-Hugoniot conditions and (u−, u+) ∈ Ge .

47 / 53

Theorem (Benyahia, Rosini work in progress)

For any initial distribution u in BV(R; Ω), the approximatesolution for the Cauchy problem of (PT) with initial datum uconverges (up to a subsequence) in L1

loc(R+ × R; Ω) to a functionu and for all t, s ≥ 0 we have that

TV(u(t)) ≤ TV(u), ‖u(t)‖L∞(R;Ω) ≤ C ,

‖u(t) − u(s)‖L1(R;Ω) ≤ L |t − s|,

where L = TV(u) maxVmax, p−1(Wmax) p′(p−1(Wmax)) andC = max|Wmax|, |Wmin| + Vf .Moreover, if Vmax = Vf and u performs a finite number of phasetransitions, then u is an entropy solution.

48 / 53

Let R : Ω × Ω → L∞(R; Ω) be the Riemann solver associated tothe constrained (PT).

Properties of R

R is neither L1loc-continuous, nor consistent.

R introduces non-classical shocks that do not “maximize” the fluxthrough the constraint.

The study of the constrained Cauchy problem for (PT) will be veryinteresting. . .

49 / 53

THANK YOU

FOR YOUR

ATTENTION

Shocks

If ~Wℓ ∈ W, ~Wr ∈ Wc0 with wℓ = wr = w and vr < vℓ, then we let

S[ ~Wℓ, ~Wr ](x/t) =

~Wℓ x < σ t,~Wr x > σ t,

σ =fr − fℓρr − ρℓ

.

The solution is then

f

fℓfr

vℓ

vr

ρρℓ ρr

w

w

vvr vℓBack

Shocks

If ~Wℓ ∈ W, ~Wr ∈ Wc0 with wℓ = wr = w and vr < vℓ, then we let

S[ ~Wℓ, ~Wr ](x/t) =

~Wℓ x < σ t,~Wr x > σ t,

σ =fr − fℓρr − ρℓ

.


tx = σ

~Wℓ~Wr

x

v(t)vℓ

vr

x

ρ(t)

ρℓ

ρr

x

Back

Rarefactions

If ~Wℓ ∈ Wc0 , ~Wr ∈ W with wℓ = wr = w and vℓ < vr , then we let

R[ ~Wℓ, ~Wr ]( xt )=

~Wℓ x < λ1( ~Wℓ) t,

w − p

(R

(w −

x

t

))

w

λ1( ~Wℓ) < x

t< λ1( ~Wr ),

~Wr x > λ1( ~Wr ) t.

Above

λ1( ~W ) =

v − ρ( ~W ) p′(ρ( ~W )

)if ~W ∈ Wc

0 ,

w if ~W ∈ W0,

and R is the inverse function of ρ 7→ p(ρ) + ρ p′(ρ).

Back

Rarefactions


R[ ~Wℓ, ~Wr ]( xt )=

~Wℓ x < λ1( ~Wℓ) t,

w − p

(R

(w −

x

t

))

w

λ1( ~Wℓ) < x

t< λ1( ~Wr ),

~Wr x > λ1( ~Wr ) t.

The solution is thenf

frfℓ

vr

vℓ

ρρr ρℓ

w

w

vvℓ vrBack

Rarefactions


R[ ~Wℓ, ~Wr ]( xt )=

~Wℓ x < λ1( ~Wℓ) t,

w − p

(R

(w −

x

t

))

w

λ1( ~Wℓ) < x

t< λ1( ~Wr ),

~Wr x > λ1( ~Wr ) t.


tx = λ1( ~Wℓ) x = λ1( ~Wr )

~Wℓ~Wr

x

v(t)

vℓ

vr

x

ρ(t)ρℓ

ρr

xBack

Contact discontinuities

If ~Wℓ, ~Wr ∈ Wc0 with vℓ = vr = v and wℓ 6= wr , then we let

C[ ~Wℓ, ~Wr ](x/t) =

~Wℓ x < v t,~Wr x > v t.


f

fℓ

fr

v

ρρrρℓ

w

wℓ

wr

v vBack

Contact discontinuities

If ~Wℓ, ~Wr ∈ Wc0 with vℓ = vr = v and wℓ 6= wr , then we let

C[ ~Wℓ, ~Wr ](x/t) =

~Wℓ x < v t,~Wr x > v t.


t x = v

~Wℓ~Wr

x

v(t)

v v

x

ρ(t)ρℓ

ρr

x

Back

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