LWR model
ARZ model
A PT model
Analytic theory for conservation laws on networks
III — The Riemann problem at a junction
Mauro Garavello
Department of Mathematics and Applications
University of Milano Bicocca
[email protected]
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Conservation laws at a network
Network: finite collection of directed arcs connected by nodes
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Conservation laws at a network
Network: finite collection of directed arcs connected by nodes
Ii = [ai , bi ]
I5
I6
I1
I4
I2
I3
I7
Transport Modeling and Management, March 6-10, 2017
I8
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Conservation laws at a network
Network: finite collection of directed arcs connected by nodes
Ii = [ai , bi ]
PDEs: for each i, ∂t ui + ∂x fi (ui ) = 0
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Conservation laws at a network
Network: finite collection of directed arcs connected by nodes
Ii = [ai , bi ]
PDEs: for each i, ∂t ui + ∂x fi (ui ) = 0

∂t u1 + ∂x f1 (u1 ) = 0




···



 ∂t uN + ∂x fN (uN ) = 0


u1 (0, x) = u1,0 (x)





 ···
uN (0, x) = uN,0 (x)
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Conservation laws at a network
Network: finite collection of directed arcs connected by nodes
Ii = [ai , bi ]
PDEs: for each i, ∂t ui + ∂x fi (ui ) = 0

∂t u1 + ∂x f1 (u1 ) = 0




···



 ∂t uN + ∂x fN (uN ) = 0


u1 (0, x) = u1,0 (x)





 ···
uN (0, x) = uN,0 (x)
PDEs
Initial conditions
Cauchy problem on the network
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Conservation laws at a junction
Junction: m incoming arcs and n outgoing arcs
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Conservation laws at a junction
Junction: m incoming arcs and n outgoing arcs
Ii = (−∞, 0]
Ij = [0, +∞)
I6
I1
I2
I3
I7
I4
I5
Transport Modeling and Management, March 6-10, 2017
I8
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Conservation laws at a junction
Junction: m incoming arcs and n outgoing arcs
Ii = (−∞, 0]
Ij = [0, +∞)
PDEs:
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Conservation laws at a junction
Junction: m incoming arcs and n outgoing arcs
Ii = (−∞, 0]
Ij = [0, +∞)
PDEs:

∂t u1 + ∂x f1 (u1 ) = 0




···



 ∂t um+n + ∂x fm+n (um+n ) = 0


u1 (0, x) = u1,0 (x)





 ···
um+n (0, x) = um+n,0 (x)
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Conservation laws at a junction
Junction: m incoming arcs and n outgoing arcs
Ii = (−∞, 0]
Ij = [0, +∞)
PDEs:

∂t u1 + ∂x f1 (u1 ) = 0




···



 ∂t um+n + ∂x fm+n (um+n ) = 0


u1 (0, x) = u1,0 (x)





 ···
um+n (0, x) = um+n,0 (x)
Cauchy problem at a junction
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Conservation laws at a junction
Junction: m incoming arcs and n outgoing arcs
Ii = (−∞, 0]
Ij = [0, +∞)
PDEs:

∂t u1 + ∂x f1 (u1 ) = 0




···



 ∂t um+n + ∂x fm+n (um+n ) = 0


u1 (0, x) = u1,0 (x)





 ···
um+n (0, x) = um+n,0 (x)
Constant I.C.
Riemann problem at a junction
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Riemann problem for the LWR at a junction
J: junction with m incoming arcs and n outgoing arcs
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Riemann problem for the LWR at a junction
J: junction with m incoming arcs and n outgoing arcs
PDEs:

∂t ρ1 + ∂x f (ρ1 ) = 0




···



 ∂t ρm+n + ∂x f (ρm+n ) = 0


ρ1 (0, x) = ρ1,0





 ···
ρm+n (0, x) = ρm+n,0
Transport Modeling and Management, March 6-10, 2017
f (ρ) = ρv (ρ)
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Admissible boundaries for incoming roads

 ∂t ρ + ∂x f (ρ) = 0
ρ(0, x) = ρ0

ρ(t, 0) = ρb
f
ρ
t
ρ0
ρb
x
x=0
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Admissible boundaries for incoming roads

 ∂t ρ + ∂x f (ρ) = 0
ρ(0, x) = ρ0

ρ(t, 0) = ρb
f
CASE 1
ρb < ρ 0 < σ
ρ
ρb ρ0
BOUNDARY DATUM:
t
not satisfied
ρ0
ρ0
ρb
x
x=0
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Admissible boundaries for incoming roads

 ∂t ρ + ∂x f (ρ) = 0
ρ(0, x) = ρ0

ρ(t, 0) = ρb
f
CASE 2
ρ0 < ρ b ≤ σ
ρ
ρ0 ρb
BOUNDARY DATUM:
t
not satisfied
ρ0
ρ0
ρb
x
x=0
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Admissible boundaries for incoming roads

 ∂t ρ + ∂x f (ρ) = 0
ρ(0, x) = ρ0

ρ(t, 0) = ρb
f
CASE 3
ρ0 ≤ σ ≤ ρb , f (ρ0 ) ≤ f (ρb )
ρ
ρ0
ρb
BOUNDARY DATUM:
t
not satisfied
ρ0
ρ0
ρb
x
x=0
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Admissible boundaries for incoming roads

 ∂t ρ + ∂x f (ρ) = 0
ρ(0, x) = ρ0

ρ(t, 0) = ρb
f
CASE 4
ρ0 ≤ σ ≤ ρb , f (ρ0 ) > f (ρb )
ρ
ρ0
ρb
BOUNDARY DATUM:
t
satisfied
ρb
ρ0
ρb
x
x=0
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Admissible boundaries for incoming roads

 ∂t ρ + ∂x f (ρ) = 0
ρ(0, x) = ρ0

ρ(t, 0) = ρb
f
CASE 5
ρb ≤ σ ≤ ρ0
ρ
ρb
ρ0
BOUNDARY DATUM:
t
not satisfied
σ
ρ0
ρb
x
x=0
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Admissible boundaries for incoming roads

 ∂t ρ + ∂x f (ρ) = 0
ρ(0, x) = ρ0

ρ(t, 0) = ρb
f
CASE 6
σ ≤ ρb < ρ 0
ρ
ρb ρ0
BOUNDARY DATUM:
t
satisfied
ρb
ρ0
ρb
x
x=0
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Admissible boundaries for incoming roads

 ∂t ρ + ∂x f (ρ) = 0
ρ(0, x) = ρ0

ρ(t, 0) = ρb
f
CASE 7
σ ≤ ρ0 < ρ b
ρ
ρ0 ρb
BOUNDARY DATUM:
t
satisfied
ρb
ρ0
ρb
x
x=0
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Admissible boundaries for incoming roads

 ∂t ρ + ∂x f (ρ) = 0
ρ(0, x) = ρ0

ρ(t, 0) = ρb
f
ρ
ρ0
Position of ρ0
Admissible ρb
ρ0 < σ
ρb = ρ0
ρb > σ, f (ρb ) < f (ρ0 )
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Admissible boundaries for incoming roads

 ∂t ρ + ∂x f (ρ) = 0
ρ(0, x) = ρ0

ρ(t, 0) = ρb
f
ρ
ρ0
Position of ρ0
Admissible ρb
ρ0 < σ
ρb = ρ0
ρb > σ, f (ρb ) < f (ρ0 )
ρ0 ≥ σ
ρb ≥ σ
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Admissible boundaries for incoming roads

 ∂t ρ + ∂x f (ρ) = 0
ρ(0, x) = ρ0

ρ(t, 0) = ρb
f
f (ρ0 )
ρ
ρ0
Position of ρ0
Admissible ρb
Admissible fluxes
ρ0 < σ
ρb = ρ0
ρb > σ, f (ρb ) < f (ρ0 )
[0, f (ρ0 )]
ρ0 ≥ σ
ρb ≥ σ
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Admissible boundaries for incoming roads

 ∂t ρ + ∂x f (ρ) = 0
ρ(0, x) = ρ0

ρ(t, 0) = ρb
f (σ) f
ρ
ρ0
Position of ρ0
Admissible ρb
Admissible fluxes
ρ0 < σ
ρb = ρ0
ρb > σ, f (ρb ) < f (ρ0 )
[0, f (ρ0 )]
ρ0 ≥ σ
ρb ≥ σ
[0, f (σ)]
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Admissible boundaries for outgoing roads

 ∂t ρ + ∂x f (ρ) = 0
ρ(0, x) = ρ0

ρ(t, 0) = ρb
Position of ρ0
Admissible ρb
Admissible fluxes
ρ0 < σ
ρb ≤ σ
[0, f (σ)]
ρ0 ≥ σ
ρb = ρ0
ρb < σ, f (ρb ) < f (ρ0 )
[0, f (ρ0 )]
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Algorithm

∂t ρ1 + ∂x f (ρ1 ) = 0




·
··



 ∂t ρm+n + ∂x f (ρm+n ) = 0
find (ρ1,b , · · · , ρm+n,b ) s.t.


ρ1 (0, x) = ρ1,0




···


ρm+n (0, x) = ρm+n,0
1
2
ρℓ,b is admissible for the ℓ-th road
n+m
n
∑
∑
f (ρi,b ) =
f (ρj,b ) (conservation condition)
i=1
3
4
j=n+1
drivers’ preferences
n
∑
f (ρi,b ) is the maximum possible
i=1
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Drivers’ preferences


αm+1,1 · · · αm+1,m


..
..
..
A=

.
.
.
αm+n,1 · · · αm+n,m
n rows, m columns
αj,i : ratio of drivers in the i-th incoming road which go to the
j-th outgoing one
Constraint:



f (ρm+1,b )
f (ρ1,b )




..
..

 = A

.
.
f (ρm+n,b )
f (ρm,b )

Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Example
Data of the problem(
A=
J: 2 × 2
1
3
2
3
1
4
3
4
)
Transport Modeling and Management, March 6-10, 2017
0 < ρ1,0 < σ, σ < ρ2,0 < 1
0 < ρ3,0 < σ, σ < ρ4,0 < 1
f (ρ1,0 ) = f (ρ4,0 ) = 12 , f (σ) = 1
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Example
Data of the problem(
A=
J: 2 × 2
1
3
2
3
1
4
3
4
)
0 < ρ1,0 < σ, σ < ρ2,0 < 1
0 < ρ3,0 < σ, σ < ρ4,0 < 1
f (ρ1,0 ) = f (ρ4,0 ) = 12 , f (σ) = 1
[
]
• Incoming road 1. Admissible fluxes: Ω1 = [0, f (ρ1,0 )] = 0, 12
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Example
Data of the problem(
A=
J: 2 × 2
1
3
2
3
1
4
3
4
)
0 < ρ1,0 < σ, σ < ρ2,0 < 1
0 < ρ3,0 < σ, σ < ρ4,0 < 1
f (ρ1,0 ) = f (ρ4,0 ) = 12 , f (σ) = 1
[
]
• Incoming road 1. Admissible fluxes: Ω1 = [0, f (ρ1,0 )] = 0, 12
• Incoming road 2. Admissible fluxes: Ω2 = [0, f (σ)] = [0, 1]
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Example
Data of the problem(
A=
J: 2 × 2
1
3
2
3
1
4
3
4
)
0 < ρ1,0 < σ, σ < ρ2,0 < 1
0 < ρ3,0 < σ, σ < ρ4,0 < 1
f (ρ1,0 ) = f (ρ4,0 ) = 12 , f (σ) = 1
[
]
• Incoming road 1. Admissible fluxes: Ω1 = [0, f (ρ1,0 )] = 0, 12
• Incoming road 2. Admissible fluxes: Ω2 = [0, f (σ)] = [0, 1]
• Outgoing road 3. Admissible fluxes: Ω3 = [0, f (σ)] = [0, 1]
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Example
Data of the problem(
A=
J: 2 × 2
1
3
2
3
1
4
3
4
)
0 < ρ1,0 < σ, σ < ρ2,0 < 1
0 < ρ3,0 < σ, σ < ρ4,0 < 1
f (ρ1,0 ) = f (ρ4,0 ) = 12 , f (σ) = 1
[
]
• Incoming road 1. Admissible fluxes: Ω1 = [0, f (ρ1,0 )] = 0, 12
• Incoming road 2. Admissible fluxes: Ω2 = [0, f (σ)] = [0, 1]
• Outgoing road 3. Admissible fluxes: Ω3 = [0, f (σ)] = [0, 1]
[
]
• Outgoing road 4. Admissible fluxes: Ω4 = [0, f (ρ4,0 )] = 0, 12
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Example
Data of the problem(
A=
J: 2 × 2
1
3
2
3
1
4
3
4
)
0 < ρ1,0 < σ, σ < ρ2,0 < 1
0 < ρ3,0 < σ, σ < ρ4,0 < 1
f (ρ1,0 ) = f (ρ4,0 ) = 12 , f (σ) = 1
γ2
1
[
]
Ω1 = 0, 12
Ω2 = [0, 1]
Ω3 = [0, 1]
[
]
Ω4 = 0, 12
γ1
1
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Example
Data of the problem(
A=
J: 2 × 2
1
3
2
3
1
4
3
4
)
0 < ρ1,0 < σ, σ < ρ2,0 < 1
0 < ρ3,0 < σ, σ < ρ4,0 < 1
f (ρ1,0 ) = f (ρ4,0 ) = 12 , f (σ) = 1
γ2
1
[
]
Ω1 = 0, 12
Ω2 = [0, 1]
Ω3 = [0, 1]
[
]
Ω4 = 0, 12
γ1
1/2
1
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Example
Data of the problem(
A=
J: 2 × 2
1
3
2
3
1
4
3
4
)
0 < ρ1,0 < σ, σ < ρ2,0 < 1
0 < ρ3,0 < σ, σ < ρ4,0 < 1
f (ρ1,0 ) = f (ρ4,0 ) = 12 , f (σ) = 1
γ2
1
[
]
Ω1 = 0, 12
Ω2 = [0, 1]
Ω3 = [0, 1]
[
]
Ω4 = 0, 12
γ3 = 13 γ1 + 14 γ2
γ4 = 23 γ1 + 34 γ2
γ1
1/2
1
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Example
Data of the problem(
A=
J: 2 × 2
1
3
2
3
1
4
3
4
)
0 < ρ1,0 < σ, σ < ρ2,0 < 1
0 < ρ3,0 < σ, σ < ρ4,0 < 1
f (ρ1,0 ) = f (ρ4,0 ) = 12 , f (σ) = 1
3
γ2
1
[
]
Ω1 = 0, 12
Ω2 = [0, 1]
Ω3 = [0, 1]
[
]
Ω4 = 0, 12
γ3 = 13 γ1 + 14 γ2
γ4 = 23 γ1 + 34 γ2
4
1/2
γ1
1
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Example
Data of the problem(
A=
J: 2 × 2
1
3
2
3
1
4
3
4
)
0 < ρ1,0 < σ, σ < ρ2,0 < 1
0 < ρ3,0 < σ, σ < ρ4,0 < 1
f (ρ1,0 ) = f (ρ4,0 ) = 12 , f (σ) = 1
3
γ2
1
[
]
Ω1 = 0, 12
Ω2 = [0, 1]
Ω3 = [0, 1]
[
]
Ω4 = 0, 12
γ3 = 13 γ1 + 14 γ2
γ4 = 23 γ1 + 34 γ2
4
1/2
γ1
1
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Example
Data of the problem(
A=
J: 2 × 2
1
3
2
3
1
4
3
4
)
0 < ρ1,0 < σ, σ < ρ2,0 < 1
0 < ρ3,0 < σ, σ < ρ4,0 < 1
f (ρ1,0 ) = f (ρ4,0 ) = 12 , f (σ) = 1
γ2
1
γ1
1/2
1
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Example
Data of the problem(
A=
J: 2 × 2
1
3
2
3
1
4
3
4
)
0 < ρ1,0 < σ, σ < ρ2,0 < 1
0 < ρ3,0 < σ, σ < ρ4,0 < 1
f (ρ1,0 ) = f (ρ4,0 ) = 12 , f (σ) = 1
γ2
1
f (ρ1,b ) =
1
2
f (ρ2,b ) =
2
9
f (ρ3,b ) =
1
3
·
1
2
+
1
4
·
2
9
=
2
9
f (ρ4,b ) =
2
3
·
1
2
+
3
4
·
2
9
=
1
2
2
9
γ1
1/2
1
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Example
Data of the problem(
A=
J: 2 × 2
1
3
2
3
1
4
3
4
)
0 < ρ1,0 < σ, σ < ρ2,0 < 1
0 < ρ3,0 < σ, σ < ρ4,0 < 1
f (ρ1,0 ) = f (ρ4,0 ) = 12 , f (σ) = 1
γ2
1
f (ρ1,b ) =
1
2
f (ρ2,b ) =
2
9
f (ρ3,b ) =
1
3
·
1
2
+
1
4
·
2
9
=
2
9
f (ρ4,b ) =
2
3
·
1
2
+
3
4
·
2
9
=
1
2
2
9
γ1
1/2
ρ1,b , ρ2,b , ρ3,b , ρ4,b
1
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Essential bibliography
Y. Chitour, B. Piccoli. Traffic circles and timing of traffic lights for cars
flow. Discrete Contin. Dyn. Syst. Ser. B 2005.
G.M. Coclite, M. Garavello, B. Piccoli. Traffic flow on a road network.
SIAM J. Math. Anal. 2005.
C. D’apice, R. Manzo, B. Piccoli. Packet flow on telecommunication
networks. SIAM J. Math. Anal. 2006.
M.L. Delle Monache, P. Goatin, B. Piccoli. Priority-based Riemann solver
for traffic flow on networks, submitted.
P. Goatin, S. Göttlich, O. Kolb. Speed limit and ramp meter control for
traffic flow networks, Eng. Optim. 2015.
S. Göttlich, M. Herty, A. Klar. Network models for supply chains.
Commun. Math. Sci. 2005.
M. Herty, A. Klar, B. Piccoli. Existence of solutions for supply chain
models based on partial differential equations. SIMA 2007.
H. Holden, N. Risebro. A mathematical model of traffic flow on a
network of unidirectional roads. SIAM J. Math. Anal. 1995.
A. Marigo, B. Piccoli. A fluid dynamic model for T -junctions. SIMA
2008.
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Riemann problem for the ARZ at a junction
J: junction with m incoming arcs and n outgoing arcs
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Riemann problem for the ARZ at a junction
J: junction with m incoming arcs and n outgoing arcs
PDEs:


















)
(
 ∂t ρ1 + ∂x y1 − ργ+1 = 0
1
)
( 2
 ∂t y1 + ∂x y1 − y1 ργ = 0
1
ρ1
·· ·
(
)
 ∂t ρn+m + ∂x yn+m − ργ+1
n+m = 0
( 2
)
 ∂t yn+m + ∂x yn+m − yn+m ργ
n+m = 0
ρn+m










ρ1 (0, x) = ρ1,0 y1 (0, x) = y1,0




·
··


ρm+n (0, x) = ρm+n,0 yn+m (0, x) = yn+m,0
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Algorithm I
the waves produced must have negative speed in incoming
roads and positive speed in outgoing roads
the first component of the flux (i.e., the flux of the density)
must be conserved
m (
m+n
)
)
∑
∑ (
γ+1
y i − ρi
=
yj − ργ+1
j
i=1
drivers’ preferences: matrix

ym+1 − ργ+1
m+1

..

.
ym+n − ργ+1
m+n
m
(
)
∑
yi − ργ+1
→ max
i
j=m+1
A



y1 − ργ+1
1



..

 = A
.
γ+1
y m − ρm
i=1
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Algorithm II
Previous rules are sufficient to isolate a unique solution in incoming
roads, but not in outgoing roads.
Additional rules
maximize the velocity v of cars in outgoing roads
maximize the density ρ of cars in outgoing roads
minimize the total variation of ρ along the solution of the
Riemann problem in outgoing roads
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Essential bibliography
M. Garavello, B. Piccoli. Traffic flow on a road network using
the Aw-Rascle model. Comm. Partial Differential Equations
2006.
M. Herty, S. Moutari, M. Rascle. Optimization criteria for
modelling intersections of vehicular traffic flow. Netw.
Heterog. Media 2006.
M. Herty, M. Rascle. Coupling conditions for a class of
second-order models for traffic flow. SIAM J. Math. Anal.
2006.
O. Kolb, S. Göttlich, P. Goatin. Capacity drop and traffic
control for a second order traffic model, submitted.
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Riemann problem for a PT model at a junction
J: junction with m incoming arcs and n outgoing arcs
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Riemann problem for a PT model at a junction
J: junction with m incoming arcs and n outgoing arcs
PDEs:
 {
∂t ρ1 + ∂x (ρ1 v(ρ1 , η1 )) = 0






∂t η1 + ∂x (η1 v(ρ1 , η1 )) = 0




·{· ·




∂t ρm+n + ∂x (ρm+n v(ρm+n , ηm+n )) = 0

∂t ηm+n + ∂x (ηm+n v(ρm+n , ηm+n )) = 0







ρ1 (0, x) = ρ1,0 η1 (0, x) = η1,0




···


ρm+n (0, x) = ρm+n,0 ηn+m (0, x) = ηn+m,0
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Algorithm
Admissible boundary conditions
Mass conservation:
n
∑
ρi v(ρi , ηi ) =
i=1
n+m
∑
ρj v(ρj , ηj )
j=n+1
Drivers’preferences: matrix A




ρn+1 v(ρn+1 , ηn+1 )
ρ1 v(ρ1 , η1 )




..
..

 = A

.
.
ρn+m v(ρn+m , ηn+m )
ρn v(ρn , ηn )
Rule for w:
wj =
α1,j γ1 w1 + . . . + αm,j γm wm
∑n
i=1 αi,j γi
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks
LWR model
ARZ model
A PT model
Essential bibliography
R.M. Colombo, P. Goatin, B. Piccoli. Road networks with
phase transitions. J. Hyperbolic Differ. Equ. 2010.
M. Garavello, F. Marcellini. The Riemann problem at a
junction for a phase transition traffic model. Preprint.
Transport Modeling and Management, March 6-10, 2017
Analytic theory for conservation laws on networks