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A Fluid Dynamic Model for T -Junctions

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A FLUID DYNAMIC MODEL FOR
T
-JUNCTIONS
ALESSIA MARIGO
AND
BENEDETTO PICCOLI
∗
Abstract.
Motivated by real road junctions, we consider a new ﬂuid dynamic model for traﬃcﬂow on networks. In particular at T-junctions, beside some ﬂows distribution and/or merging, therehappen some interactions of cars coming from diﬀerent roads and going to diﬀerent destinations.After determining some rules to uniquely solve Riemann problems, we prove existence of solutionson complete networks for initial data with bounded variation (and their limits in
L
1
loc
).
Key words.
traﬃc ﬂow on networks, conservation laws, T-junctions
AMS subject classiﬁcations.
90B20, 35L65
1. Introduction.
In recent years, many authors contributed to the developmentof ﬂuid dynamic theory for ﬂows on networks, see [4, 6, 7, 10, 11, 12, 13, 14, 15, 18].The most important applications are in urban car traﬃc [5], telecommunication datanetworks [9, 19, 21], gas pipelines [2], supply chains [1] and others.The main approach used for car traﬃc is based on the idea of a junction (cor-responding to the nodes of the network), with no particular relationships betweenincoming and outgoing roads. Then, real crossings are modeled splitting them intomany junctions of lower complexity, see e.g. [6, 12]. This means that the traﬃc ﬂowfrom incoming roads distribute to outgoing roads according to certain preferences,which in [7] were modeled by a traﬃc distribution matrix. Then, in general, to solveuniquely Riemann problems, i.e. Cauchy problems with constant initial data on eachroad, one also needs to impose the maximization of a functional, e.g. the total ﬂux.In this paper we consider a diﬀerent point of view for junctions, inspired bymodeling need. To illustrate our approach, let us focus on a simple example of
T
- junction, represented in Figure 1.1. Here we have three roads with both directionsof traﬃc. Then we can individuate the incoming ﬂows, denoted by 1, 2 and 3, andthe outgoing ones, denoted by
A
,
B
and
C
. Each incoming ﬂux at the junctionsplits into two parts depending on the ﬁnal destination. Thus ﬂux 1 is split in ﬂuxes1
B
and 1
C
(assuming that
U
-turns are not possible). As Figure 1.1 shows, thereare many interactions among the various ﬂuxes at the junction. However, not allsuch interactions can be considered in the same way. In fact, for instance, ﬂuxes 1
B
and 3
B
must ﬂow to the same ﬁnal direction, thus clearly their sum can not exceedthe possible outgoing ﬂow towards
B
. On the contrary, ﬂuxes 1
B
and 3
A
shareconﬂicting trajectories, but they do not share the same ﬁnal destination, thus theirsum is bounded only by the junction capacity.To capture this situation, we model the
T
-junction as in Figure 1.2. More precisely,to encompass the whole dynamic happening at the
T
-junction, we use nine virtual junctions denoted by letters
G
,
H
, and
K
. The three junctions
G
are formed by anincoming road and two outgoing roads, thus the described phenomenon is simply aﬂux split. Such junctions were already modeled in [7].Instead, the three junctions
H
, are formed by two incoming and one outgoing roads.In this case, clearly some right of way or yielding rule should show up to describe thetraﬃc distribution. This is in fact the case and the theory was ﬁrst developed in [6].The junctions
K
have a quite diﬀerent meaning. In this case there are two incoming
∗
Istituto per le Applicazioni del Calcolo “Mauro Picone” - C.N.R., Viale del Policlinico 137, 00161Roma, Italy(
marigo,piccoli@iac.cnr.it
)1
2
A. MARIGO and B. PICCOLI
AC312B1C1B2A2C3B3A
Fig. 1.1
.
An example of
T
-junction.
AC312BHHH KGGGKK
Fig. 1.2
.
The model for a
T
-junction.
and two outgoing roads. However, the traﬃc from each incoming road goes to aprecise outgoing road, while sharing the junction space. Our main aim is then tomodel these new types of junctions. Let us illustrate the mathematical counterpartof this example.We use the Lighthill-Whitham-Richards model ([19, 22]), which consists of a singleconservation law:
ρ
t
+
f
(
ρ
)
x
= 0
,
(1.1)where
ρ
∈
[0
,ρ
max
] is the car density,
f
(
ρ
) =
ρv
(
ρ
) is the ﬂux with
v
(
ρ
) the averagevelocity. A junction
J
is called a crossing junction, if it has the same characteristic of junctions
K
above. Thus
J
has
n
incoming roads, denoted for simplicity by
I
1
,...,I
n
,and
n
outgoing roads, denoted by
I
n
+1
,...,I
2
n
. Also, we denote with
ρ
i
(
t,x
)
, i
=1
,...,n
, and
ρ
j
(
t,x
)
, j
=
n
+ 1
,...,
2
n
, the traﬃc densities, respectively, on theincoming roads and on the outgoing ones. To describe the dynamics at
J
, we assumethe following:
1)
The ﬂux from road
I
i
is the same of the corresponding exiting road
I
n
+
i
.
2)
The total ﬂux through
J
does not exceed its maximum capacity Γ
J
.
3)
The total ﬂux through
J
is maximal respecting rules 1) and 2).In case of high traﬃc, rules 1), 2) and 3) are not enough to isolate a unique solutionto Riemann problems at
J
. This need of a new rule is not at all surprising, since such
T-JUNCTIONS MODEL
3rules were used both in car traﬃc for junctions with more incoming than outgoingroads, see [6], and also for telecommunication networks, see [9].Therefore, we introduce a
ﬂux proportion rule
, which is active only when the maximalincoming ﬂuxes overcome the maximal junction capacity Γ
J
. More precisely, weassume that there exist ideal equilibrium ﬂux proportions among incoming roads.Thus, there exist coeﬃcients ¯
r
i
so that the following holds.FPR) The ﬂux from road
I
i
+1
is ¯
r
i
times the ﬂux from road
I
i
, for
i
= 1
,...,n
−
1.The rule FPR) well captures the situation in the example of the
T
-junction above.Indeed, for instance, the ﬂux 3
A
must yield to the ﬂux 2
C
, thus the correspondingﬂux proportion coeﬃcient will be less than 1. While, usually the ﬂux 1
B
must giveprecedence to the ﬂux 3
A
, unless yielding signs are deciding the contrary.Let us further illustrate the role of rule FPR), for simplicity restricting to the case of two incoming and two outgoing roads. First, FPR) is used only if the sum of incomingmaximal ﬂuxes exceeds Γ
J
. Then, to respect FPR), we should set the incoming ﬂuxes
γ
1
and
γ
2
so that
γ
2
= ¯
r
1
γ
1
. However, this may be in contrast with 3) (if for example¯
r
1
= 1, the maximal ﬂux
γ
max
1
from road 1 exceeds Γ
J
and the maximal ﬂux fromroad 2 is less than Γ
J
/
2.) In the latter case, we set the proportion between incomingﬂuxes so to respect rule 3) (i.e. summing up to Γ
J
) and be as close as possible to thevalue prescribed by FPR).We ﬁrst show how to deﬁne the solution to Riemann problems for the new typeof junctions: the
crossing junctions
. The procedure to deﬁne the solution is based onrules 1), 2), 3) and FPR). The obtained solution eﬀectively deﬁnes a Riemann solverwith consistency properties, see Proposition 2.4.Then we provide estimates on the total variation of the ﬂux for a wave interacting witha crossing junction, having two incoming and two outgoing roads. Such estimates arethe key point to prove BV estimates on the ﬂux along wave front tracking approximatesolutions.More precisely, a wave front tracking algorithm is deﬁned as in [7], i.e. approximatinginitial data with a piecewise constant function and solving Riemann problems forinteractions between waves and of waves with junctions. To provide a well deﬁnedconstruction, estimates on the number of waves and interactions are in order. Thelatter are obtained with a careful analysis based on the special properties of theintroduced Riemann solver. Then, to pass to the limit, BV estimates on the ﬂuxare used, together with standard weak compactness arguments. The ﬁnal result isexistence of solutions to Cauchy problems on networks.The paper is organized as follows. Section 2 provides the basic deﬁnitions andresults from previous papers, while in section 2.1 we describe the Riemann solverfor crossing junctions. The following section 3 contains ﬂux variation estimates forwaves interacting with crossing junctions. Finally, in section 4, we prove existence of solutions on the whole network for
L
1
loc
initial data, which can be approximated byBV functions with uniformly bounded variation.
2. Basic deﬁnitions.
We use the same approach as in [15, 7, 11]. For readerconvenience, we recall the main notation and results.We consider a network formed by a collection
I
of unidirectional roads
I
i
, mod-eled by real (possibly unbounded) intervals [
a
i
,b
i
], whose natural order respects thedirection of the road. Roads meet at junctions: each junction
J
is given by a col-lection of incoming roads and outgoing roads, and we indicate by
J
the collection of junctions. Thus the network can be identiﬁed with a directed graph. On each roadthe evolution is given by equation (1.1) and we assume:
4
A. MARIGO and B. PICCOLI
(F) The ﬂux
f
is a smooth, strictly concave function (with
f
(0) =
f
(
ρ
max
) = 0),thus there exists a unique
σ
∈
[0
,ρ
max
] such that
f
′
(
σ
) = 0 and it is themaximum of
f
over [0
,ρ
max
].For notational simplicity, we assume that
ρ
max
= 1.
Definition 2.1.
We let
τ
: [0
,
1]
→
[0
,
1]
be the map such that
f
(
ρ
) =
f
(
τ
(
ρ
))
and
τ
(
ρ
)
=
ρ
if
ρ
=
σ
. Thus
τ
sends
ρ
to the other density value with the same ﬂux (and
τ
(
σ
) =
σ
.)
We restrict to crossing junctions as explained in the Introduction.Then each junction
J
has
n
=
n
(
J
) incoming roads and
n
outgoing roads.Let us ﬁx now a junction
J
and for simplicity assume that the incoming roads are
I
1
,...,I
n
and the outgoing roads are
I
n
+1
,...,I
2
n
. A Riemann problem for a system(1.1), on the real line, is a Cauchy problem with Heaviside type initial data. We deﬁnea Riemann problem at
J
to be a Cauchy problem with initial datum constant on eachroad. Thus let us ﬁx an initial condition
ρ
0
= (
ρ
1
,
0
,...,ρ
2
n,
0
). We look for centeredself-similar solutions (as it is natural for conservation laws see [3]), thus we want todetermine a (2
n
)-tuple ˆ
ρ
= (ˆ
ρ
1
,...,
ˆ
ρ
2
n
)
∈
[0
,
1]
2
n
, so that the following holds. Oneach incoming road
I
i
,
i
= 1
,...,n
, the solution consists of the single wave solution tothe Riemann Problem (
ρ
i,
0
,
ˆ
ρ
i
), while on each outgoing road
I
j
,
j
=
n
+1
,...,
2
n
, thesolution consists of the single wave (ˆ
ρ
j
,ρ
j,
0
).We consider waves with negative speed on incoming roads and positive on outgoingones, thus:ˆ
ρ
i
∈
{
ρ
i,
0
}∪
]
τ
(
ρ
i,
0
)
,
1]
,
if 0
≤
ρ
i,
0
≤
σ,
[
σ,
1]
,
if
σ
≤
ρ
i,
0
≤
1
,
(2.1)
i
= 1
,...,n,
andˆ
ρ
j
∈
[0
,σ
]
,
if 0
≤
ρ
j,
0
≤
σ,
{
ρ
j,
0
}∪
[0
,τ
(
ρ
j,
0
)[
,
if
σ
≤
ρ
j,
0
≤
1
,
(2.2)
j
=
n
+ 1
,...,
2
n,
. As a consequence, not every ﬂux can be obtained on each road.More precisely, we deﬁne:
γ
max
i
=
f
(
ρ
i,
0
)
,
if
ρ
i,
0
∈
[0
,σ
]
,f
(
σ
)
,
if
ρ
i,
0
∈
]
σ,
1]
, i
= 1
,...,n,
(2.3)and
γ
max
j
=
f
(
σ
)
,
if
ρ
j,
0
∈
[0
,σ
]
,f
(
ρ
j,
0
)
,
if
ρ
j,
0
∈
]
σ,
1]
, j
=
n
+ 1
,...,
2
n.
(2.4)The quantities
γ
max
i
and
γ
max
j
represent the maximum ﬂux that can be obtained bya single wave solution on each road.
Remark 1.
We may consider waves with zero speed on incoming or outgoing roads. However, this would generate shocks which stay at the intersection, without entering roads. Since solutions are usually considered in
L
1
, such shocks are not part of the solution on roads (because they aﬀect the value only at one point.)
Remark 2.
The maximum ﬂuxes on incoming and outgoing roads can be inter-preted as maximal demands and supplies, according to the theory introduced by J.P.Lebacque, see [16, 17, 11].
Notice that for each road
I
i
and possible ﬂux ˆ
γ
i
, thereexists a unique ˆ
ρ
i
, satisfying (2.1) or (2.2), such that
f
(ˆ
ρ
i
) = ˆ
γ
i
. Moreover, by rule1), outgoing ﬂuxes are obtained once incoming ﬂuxes are ﬁxed. Then we obtain thefollowing:
T-JUNCTIONS MODEL
5
Proposition 2.2.
To solve a Riemann problem at a crossing junction
J
, it is enough to determine the ﬂuxes on incoming roads.
We aim at ﬁnding a systematicway of solving RP at junctions as described by next deﬁnition.
Definition 2.3.
Given a crossing junction
J
, we call Riemann solver for
J
a map
RS
: [0
,
1]
n
×
[0
,
1]
n
→
[0
,
1]
n
×
[0
,
1]
n
that associates to Riemann data
ρ
0
=(
ρ
1
,
0
,...,ρ
2
n,
0
)
at
J
a vector
ˆ
ρ
= (ˆ
ρ
1
,...,
ˆ
ρ
2
n
)
so that the solution on an incoming road
I
i
,
i
= 1
,...,n
, is given by the wave
(
ρ
i,
0
,
ˆ
ρ
i
)
and on an outgoing one
I
j
,
j
=
n
+1
,...,
2
n
is given by the wave
(
ρ
j,
0
,
ˆ
ρ
j
)
.
We require the consistency condition:(CC)
RS
(
RS
(
ρ
0
)) =
RS
(
ρ
0
)
.
Once a Riemann solver is introduced for every junction
J
, we can deﬁne theconcept of solution on the network as in [11].
2.1. Riemann solver at crossing junctions.
The aim of this section is todescribe the solution to a Riemann problem at a crossing junction
J
, using rules 1),2), 3) and FPR).Fix a crossing junction
J
with
n
incoming roads,
I
1
,...,I
n
and
n
outgoing roads
I
n
+1
,...,I
2
n
. We denote with
ρ
i
(
t,x
)
, i
= 1
,...,n
and
ρ
j
(
t,x
)
, j
=
n
+ 1
,...,
2
n
the traﬃc densities, respectively, on the incoming roads and on the outgoing ones andby (
ρ
i,
0
,ρ
j,
0
) the initial data of a Riemann problem. The rules 1), 2) and 3) can berewritten as:
1)
f
(ˆ
ρ
i
) =
f
(ˆ
ρ
n
+
i
) for each
i
= 1
,...,n
,
2)
i
f
(ˆ
ρ
i
)
≤
Γ
J
,
3)
i
f
(ˆ
ρ
i
) is maximal respecting rules 1) and 2).The rules
1)
and
2)
alone do not give a unique solution (ˆ
ρ
1
,...,
ˆ
ρ
2
n
). Moreover,since the solution (ˆ
ρ
1
,...,
ˆ
ρ
2
n
) must satisfy the conditions
1)
and
2
), we denote byΩ the admissible region for the ﬂuxes
γ
i
=
f
(
ρ
i
)Ω =
{
(
γ
1
,...,γ
n
) :
γ
i
≤
Γ
J
,
0
≤
γ
i
≤
γ
∧
i
,i
= 1
,...,n
}
,
where
γ
∧
i
=
min
{
γ
maxi
,γ
maxn
+
i
}
.Let us now quantify rule FPR). We can rewrite the rule as:FPR) ¯
r
i
is the ratio among the ﬂuxes on two successive roads
f
(ˆ
ρ
i
) and
f
(ˆ
ρ
i
+1
).Now, we want to determine a unique solution to the Riemann problem usingrules 1), 2), 3) and FPR). More precisely, we try to ﬁt rule FPR) as much as possiblerespecting rules 1), 2) and 3).Recall that, by Proposition 2.2, to solve the Riemann problem, it is enough todetermine the ﬂuxes ˆ
γ
i
=
f
(ˆ
ρ
i
)
,i
= 1
,...,n
. (Then ˆ
γ
n
+
i
=
f
(ˆ
ρ
i
)
,i
= 1
,...,n
.)Then, let us determine ˆ
γ
i
,i
= 1
,...,n
. We denote by Γ =
i
γ
∧
i
. We have todistinguish two cases:
I
Γ
≤
Γ
J
,
II
Γ
>
Γ
J
.In the ﬁrst case we set ˆ
γ
i
=
γ
∧
i
, i
= 1
,...,n
. Let us analyse the second case in whichwe use the ﬂux proportion coeﬃcients ¯
r
1
,...,
¯
r
n
−
1
.Consider the space (
γ
1
,...,γ
n
) and denote by
γ
r
the point that satisfy the fol-lowing system of equations:
i
γ
i
= Γ
J
γ
i
+1
= ¯
r
i
γ
i
.
(2.5)Recall that the ﬁnal ﬂuxes should belong to Ω. We distinguish two cases:a)
γ
r
belongs to Ω,

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