THE RIEMANN PROBLEM AT A JUNCTION FOR A PHASE TRANSITION TRAFFIC MODEL

. We extend the Phase Transition model for traﬃc proposed in [8], by Colombo, Marcellini, and Rascle to the network case. More precisely, we consider the Riemann problem for such a system at a general junction with n incoming and m outgoing roads. We propose a Riemann solver at the junction which conserves both the number of cars and the maximal speed of each vehicle, which is a key feature of the Phase Transition model. For special junctions, we prove that the Riemann solver is well deﬁned.


1.
Introduction. This paper deals with Riemann problems at junctions for a macroscopic phase transition traffic model. More precisely, we consider the 2-Phase Traffic Model, proposed by Colombo, Marcellini and Rascle in [8], given by the system in conservation form ∂ t ρ + ∂ x (ρ v(ρ, η)) = 0 ∂ t η + ∂ x (η v(ρ, η)) = 0 with v(ρ, η) = min V max , η ρ ψ(ρ) , where ρ denotes the car traffic density, η is a generalized momentum, v ∈ [0, V max ] is the speed of cars, and ψ is a decreasing function. This model has been derived as an extension of the famous Lighthill-Whitham-Richards (LWR) model (see [17,20]), by assuming that different typologies of drivers have different maximal speed w, where η = ρw. A key feature of this model is that there are two different traffic regimes: the free one and the congested one. Consequently, the fundamental diagram is composed by the Free phase F and the Congested phase C. In the free phase the model is the classical LWR one, while in the congested phase it consists on a system of two differential equations. The phase transitions traffic models belong to the class of macroscopic second order models, started by the Aw-Rascle-Zhang (ARZ) model, see [1] and [21]. The first phase transition model for traffic has been introduced by Colombo in 2002, 2. The Phase Transition Model. We recall at first the Phase Transition model, introduced in [8] as an extension of the LWR model, since it allows different speeds for different typology of drivers. The LWR model is given by the following scalar conservation law where ρ is the traffic density and V = V (t, x, ρ) is the speed. Assume now that V = w ψ(ρ), where ψ = ψ(ρ) is a C 2 function and w = w(t, x) is the maximal speed of a driver, located at position x at time t. Introducing a uniform bound V max > 0 on the speed of vehicles, we obtain the model With the change of variables η = ρw, the former system can be written in conservation form (1), where the conserved quantities are ρ and η.
Here, R is the maximal possible density, whilew, respectively,ŵ, is the minimum, respectively, maximum, of the maximal speeds of each vehicle. The two phases, free and congested, are described by the sets see Figure 1. Both F and C are closed sets and F ∩ C = ∅. Note also that F is one-dimensional in the (ρ, ρv) plane, while it is two-dimensional in the (ρ, η) coordinates. Figure 1, left, also contains the curves η =wρ, η =ŵρ, and the curve η = Vmax ψ(ρ) ρ that separates the two phases. Note that, in the free phase F , the system (1) reduces to ∂ t ρ + ∂ x (ρ V max ) = 0 ∂ t η + ∂ x (ηV max ) = 0 , while, in the congested phase C, it is given by By (H-1), (H-2), and (H-3), system (2) is strictly hyperbolic in C, see [8], and where λ i and r i are respectively the eigenvalues and right eigenvectors of the Jacobian matrix of the flux, and L i are the Lax curves. When ρ o = R, the 2-Lax curve through (ρ o , η o ) is given by the segment ρ = R, η ∈ [Rw, Rŵ].
Introduce also the following technical assumption: (H-4): the waves of the first family in C have negative speed.

Remark 1.
It is possible to choose the parameters such that (H-4) is satisfied.
• Linear wave: a wave connecting two states in the free phase.
• Phase transition wave: a wave connecting a left state (ρ l , η l ) ∈ F with a right state (ρ r , η r ) ∈ C satisfying η l ρ l = ηr ρr . • First family wave: a wave connecting a left state (ρ l , η l ) ∈ C with a right state (ρ r , η r ) ∈ C such that η l ρ l = ηr ρr . • Second family wave: a wave connecting a left state (ρ l , η l ) ∈ C with a right state (ρ r , η r ) ∈ C such that v (ρ l , η l ) = v (ρ r , η r ).
3. The Riemann Problem at a generic node. Consider a node J with n incoming arcs I 1 , ..., I n and m outgoing arcs I n+1 , ..., I n+m , where each incoming arc is modeled by I i = ]−∞, 0] and each outgoing arc by I j = [0, +∞[. On each arc we consider the phase transition model in (1). A Riemann problem at J is the following Cauchy problem where (ρ i ,η i ) ∈ F ∪ C are the initial data in each incoming arc I i , i = 1, . . . , n, and (ρ j ,η j ) ∈ F ∪ C are the initial data in each outgoing arc I j , j = n + 1, . . . , n + m.
Next, we analyze all the possible traces, and the corresponding flows, at x = 0 for self-similar solutions, separately in the incoming arcs and in the outgoing arcs. Incoming Arc. We define T inc (ρ,η) as the set of all the possible traces at x = 0 of a solution in the incoming arc when the initial condition is (ρ,η). More precisely, the set T inc (ρ,η) is composed by all the points (ρ * , η * ) ∈ F ∪ C such that the classical Riemann problem x > 0 is solved with waves with negative speed, i.e., by (H-4) with waves of the first family or with phase transition waves with negative speed. Moreover we define the   All the points (ρ * , η * ) ∈ T inc (ρ,η) have maximal speed w * equal tow. The following cases hold.
1. Case (ρ,η) ∈ C. The set T inc (ρ,η) consists of all the points in the congested phase C belonging to the Lax curve of the first family passing through (ρ,η) .
There exists a unique point (ρ 1 ,η 1 ) ∈ C such that the set T inc (ρ,η) consists of the point (ρ,η) itself and of all the points in the congested phase C belonging to the Lax curve of the first family passing through (ρ 1 ,η 1 ), with density strictly bigger thanρ 1 . Moreover T f inc (ρ,η) = [0,ρV max ], see Figure 3.
Proof. The waves with negative speed could be wave of the first family (see assumption (H-4)) and phase-transition waves. Thus, sinceη ρ =w, we deduce that w * =w. Case 1. Since (ρ,η) ∈ C, phase transitions waves do not appear. Therefore the set T inc (ρ,η) consists of all the points in the congested phase C of the Lax curve of the first family passing through (ρ,η), that is see Figure 2, left. Next, in the (ρ, ρv) plane, the Lax curve passing through (ρ,η) is the graph of the function ρ →η ρ ρψ(ρ). By imposing ρV max =η ρ ρψ(ρ), we obtain Figure 2, Since (ρ,η) ∈ F , one can use only phase transition waves with negative speed. By the Rankine-Hugoniot condition, a phase transition wave connecting (ρ l , η l ) ∈ F and (ρ r , η r ) ∈ C has strictly negative speed if and only if ρ l V max > η r ψ(ρ r ) and has zero speed if and only if ρ l V max = η r ψ(ρ r ). Define (ρ 1 ,η 1 ) ∈ C by the unique solution to In particular the first equation, by the Rankine-Hugoniot conditions, means that the wave between (ρ,η) and (ρ 1 ,η 1 ) has zero speed. The set T inc (ρ,η) consists of (ρ,η) and of all the points in the congested phase C of the Lax curve of the first family passing through (ρ,η), with ρ >ρ 1 ; that is see Figure 3, left. Clearly, the set of flows in the (ρ, ρv) plane is T f inc (ρ,η) = [0,ρV max ], see Figure 3, right.
Outgoing Arc. We define T out (w,ρ,η) as the set of all the possible traces at x = 0 of a solution, having w as maximal speed, in the outgoing arc when the initial condition is (ρ,η). More precisely, the set T out (w,ρ,η) is composed by all the points (ρ * , η * ) ∈ F ∪ C such that η * = wρ * and the classical Riemann problem x > 0 is solved with waves with positive speed, i.e., with waves of the second family, with phase transition waves with positive speed or with linear waves connecting two states in F . Moreover we define the corresponding set of flows  1. Case (ρ,η) ∈ F . The set T out (w,ρ,η) consists of all the points (ρ * , η * ) of the free phase F such that η * /ρ * = w.
Proof. Case 1. Since (ρ,η) ∈ F , phase transitions waves do not appear and we use only linear waves. Once fixed the maximal speed w, since w = η * /ρ * , we have see Figure 4, left.
Since we fixed w, we consider only the point (ρ + , η + ), which is the point of intersection between w = η * /ρ * and the Lax curve of the second family through (ρ,η); Moreover T out (w,ρ,η) contains also points in F which belong to the curve w = η * /ρ * and which can be connected by a phase transition wave with positive speed to the point (ρ + , η + ). By the Rankine-Hugoniot condition, a phase transition wave connecting (ρ l , η l ) ∈ F and (ρ r , η r ) ∈ C has strictly positive speed if and only if ρ l V max < η r ψ(ρ r ) and has zero speed if and only if ρ l V max = η r ψ(ρ r ). In particular, define (ρ 2 ,η 2 ) ∈ F by the unique solution to The first equation above, by the Rankine-Hugoniot conditions, means that the wave between (ρ + , η + ) and (ρ 2 ,η 2 ) has zero speed. Therefore see Figure 5, left. Finally, we obtain that the maximum for the flow is attained at the point Figure 5, right.
Admissible Solutions at J. Define Γ i = max T f inc (ρ i ,η i ), for i = 1, . . . , n in the incoming arcs and, for every w ∈ [w,ŵ], Γ w j = max T f out (w,ρ j ,η j ) for j = n + 1, . . . , n + m in the outgoing arcs. Fix a matrix A ∈ A, where ..,n, j=n+1,...,n+m indicates the percentage of traffic that passes from I i to I j . Define, for every i ∈ {1, · · · , n}, and consider the set where the maximal speeds w j , for j ∈ {n + 1, · · · , n + m}, are defined by . . .
in the case (γ 1 , · · · , γ n ) = (0, · · · , 0) or by w n+1 = · · · = w n+m =w in the other case. Note that every point in the set Ω is a tuple of admissible fluxes at the junction.
Remark 2. In equation (4) the choicew, ifρ i = 0, is arbitrary, but it does not influence the set Ω, in the sense that every other choice in the set [w,ŵ] produces the same set Ω. The same consideration holds also for the choice in the outgoing arcs in the case (γ 1 , · · · , γ n ) = (0, · · · , 0).
We define the concept of Riemann solver at a generic node.
1. The consistency condition is solved with waves with negative speed. 3. For every i ∈ {n + 1, . . . , n + m}, the classical Riemann problem x > 0 is solved with waves with positive speed.

4.1.
A different approach. In this subsection, we outline the fact that it is fundamental to impose the constraint w j =w 1 (j ∈ {2, . . . , 1 + m}) before calculating the admissible fluxes in the outgoing roads. Indeed in the point 3. of the construction of the Riemann solver, the number Γw 1 j depends explicitly on that constraint. The approach, similar to that of Garavello and Piccoli [10] or Herty and Rascle [14]  in the case of the Aw-Rascle-Zhang traffic model (see [1,21]), which consists of first calculating all the possible admissible fluxes at the junction and then imposing the constraint on the maximum speed, is not working for the phase transition model, considered in this paper. We propose the following example. Choose the constants R = 1, V max = 1, w = 2,ŵ = 3, and the function ψ(ρ) = 1 − ρ. In this way the hypothesis We can easily check that (ρ 1 ,η 1 ) ∈ C, (ρ 2 ,η 2 ) ∈ F , and (ρ 3 ,η 3 ) ∈ C.
We construct a Riemann solver RS J with the following procedure. 1. Define the maximal speeds  (5). In this situation, givenw 1 ,w 2 , it becomes 5204 MAURO GARAVELLO AND FRANCESCA MARCELLINI Figure 8. The situation in the outgoing road related to the approach of Subsection 4.1. Left, in the (ρ, η)-plane, the states (ρ * 3 , η * 3 ) and (ρ 3 ,η 3 ), connected through the middle state (ρ m , η m ). Right, in the (t, x)-plane, the waves generated by the Riemann problem. Note that the first wave has negative speed, so that it is not contained in the feasible region of the outgoing road.
Proof. Clearly Ω = ∅, since (0, 0) ∈ Ω. Assume by simplicity that w 1 ≤w 2 the other cases can be treated in a similar way.
By points 2 and 3 of the construction of RS J , Γ i = max T f inc (ρ i ,η i ), for i = 1, 2, and Γ w 3 = max T f out (w,ρ 3 ,η 3 ) for every w. In a similar way, we define Γ * i = max T f inc (ρ * i , η * i ), for every i = 1, 2, and Γ * ,w

3
= max T f out (w, ρ * 3 , η * 3 ) for every w. We consider the following two cases. The details similar to those in proof of Theorem 5.2 are omitted.