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A macroscopic traffic model with phase transitions and local point constraints on the flow

  • Received: 01 November 2016 Revised: 01 January 2017
  • Primary: 35L65, 90B20; Secondary: 82C26

  • In this paper we present a macroscopic phase transition model with a local point constraint on the flow. Its motivation is, for instance, the modelling of the evolution of vehicular traffic along a road with pointlike inhomogeneities characterized by limited capacity, such as speed bumps, traffic lights, construction sites, toll booths, etc. The model accounts for two different phases, according to whether the traffic is low or heavy. Away from the inhomogeneities of the road the traffic is described by a first order model in the free-flow phase and by a second order model in the congested phase. To model the effects of the inhomogeneities we propose two Riemann solvers satisfying the point constraints on the flow.

    Citation: Mohamed Benyahia, Massimiliano D. Rosini. A macroscopic traffic model with phase transitions and local point constraints on the flow[J]. Networks and Heterogeneous Media, 2017, 12(2): 297-317. doi: 10.3934/nhm.2017013

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  • In this paper we present a macroscopic phase transition model with a local point constraint on the flow. Its motivation is, for instance, the modelling of the evolution of vehicular traffic along a road with pointlike inhomogeneities characterized by limited capacity, such as speed bumps, traffic lights, construction sites, toll booths, etc. The model accounts for two different phases, according to whether the traffic is low or heavy. Away from the inhomogeneities of the road the traffic is described by a first order model in the free-flow phase and by a second order model in the congested phase. To model the effects of the inhomogeneities we propose two Riemann solvers satisfying the point constraints on the flow.



    The paper deals with a phase transition model (PT model for short) that takes into account the presence along a unidirectional road of obstacles that hinder the flow of vehicles, such as speed bumps, traffic lights, construction sites, toll booths, etc. More precisely, the traffic away from these inhomogeneities of the road is described by the PT model introduced in [9], whereas the effects of these inhomogeneities are described by considering one of the two constrained Riemann solvers introduced in Section 3.

    Traffic models based on differential equations can mainly be divided in three classes: microscopic, mesoscopic and macroscopic. The present PT model belongs to the class of macroscopic traffic models. We defer to the surveys [8,36,39] and to the books [27,29,42] as general references on macroscopic models for vehicular traffic. Among these models, two of most noticeable importance are the LWR model by Lighthill, Whitham [35] and Richards [40]

    ρt+(vρ)x=0,v=V(ρ),

    and the ARZ model by Aw, Rascle [7] and Zhang [43]

    ρt+(vρ)x=0,[ρ(v+p(ρ)]t+[vρ(v+p(ρ)]x=0.

    Theses two models aim to predict the evolution in time t of the density ρ and of the (average) speed v of vehicles moving along a homogeneous road with no entries or exits and parametrized by the coordinate xR.

    Both of these models have their drawbacks. In fact, the LWR model assumes that the velocity is a function of the density alone. However empirical studies show that the density-flux diagram can be approximated by a curve only at low densities, whereas at high densities it has a multivalued structure. Hence, it is more reasonable to describe the traffic in a congested phase by a second order model, such as the ARZ model. On the other hand the ARZ model is not well-posed near the vacuum: in general the solution does not depend continuously on the initial data when the density is close to zero.

    This motivated the introduction in [32] of a PT model that couples LWR and ARZ models to describe the free-flow and the congested phases, respectively. The coupling is achieved via phase transitions, namely discontinuities that separate two states belonging to different phases and satisfying the first of the Rankine-Hugoniot conditions (RH) corresponding to the conservation of the number of vehicles. The resulting model has the advantage of correcting the aforementioned drawbacks of the LWR and ARZ models taken separately.

    We recall that the macroscopic two-phase approach was first introduced by Colombo in [17,18], where the free-flow phase is governed by the LWR model and the congested phase by a 2×2 system of conservation laws expressing the conservation of both the number of vehicles and of the linearized momentum. From the analytical point of view, in [21] this model is proved to be globally well posed for any initial datum with bounded total variation; in [13,14] a new version of Godunov scheme is proposed in order to compute numerically its solutions. In [16,20,33] this model has been generalized to the case of a network.

    The macroscopic two-phase approach was then exploited and investigated by other authors in subsequent papers, see for instance [9,10,23,24,30,31,37,38] and the references therein.

    A couple of mathematical difficulties have to be highlighted. First, one difficulty is that the curves in the (t,x)-plane dividing two regimes are not given a priori. The model cannot be reduced therefore into solving two different systems in two distinct regions with prescribed boundary conditions. Another difficulty is the possibility that two phase transitions may interact with each other and cancel themselves. In fact, for instance, it is perfectly reasonable to consider a traffic characterized by a single congested region C, with vehicles emerging at the front end of C and moving into a free-flow phase region with a velocity higher than the tail of the queue at the back end of C, so that after a certain time the congested region disappears and the whole traffic is in a free-flow phase. For this reason a global approach for the study of the corresponding Cauchy problem can not be applied, as it would require a priori knowledge of the phase transition curves; it is instead preferable to apply the wave-front tracking algorithm [11,34], as it allows to track the positions of the phase transitions.

    The present article deals with the constrained version of the PT model introduced in [9], that can be regarded as a generalization of the one given in [32]. We aim to study the PT model introduced in [9] equipped with a local point constraint on the flow, so that at the interface x=0 the flow of the solution must be lower than a given constant quantity Q0. This models, for instance, the presence of a toll gate across which the flow of the vehicles cannot exceed its capacity Q0. The additional difficulty that this adds to the mathematical modelling of the problem is that this time one can start with a traffic that is initially completely in the free-flow phase, but congested phases arise in a finite time in the upstream of x=0, as it is perfectly reasonable in the case of a toll gate with a very limited capacity. We establish two constrained Riemann solvers for this model and study their properties. These two Riemann solvers may be used in a wave-front tracking scheme to study the resulting Cauchy problem.

    Before concluding this introduction, let us briefly summarize the literature on conservation laws with point constraint on the flow recalling that:

    ● the LWR model with a local point constraint is studied analytically in [19,41] and numerically in [5,12,15,22];

    ● the LWR model with a non-local point constraint is studied analytically in [1,3] and numerically in [2];

    ● the ARZ model with a local point constraint is studied analytically in [4,25,26] and numerically in [6].

    To the best of our knowledge, the present model is the first PT model with a point constraint.

    The paper is organized as follows. In Section 2 we state carefully the model and introduce the needed notations and assumptions. In Section 3 we define four Riemann solvers. More precisely, beside the Riemann solver already presented in [9] and here denoted by R1, we propose a further Riemann solver R2. We then construct the two constrained Riemann solvers Rc1 and Rc2 corresponding to R1 and R2, respectively. In Section 4 we study their basic properties. Finally, in the last section we apply these Riemann solvers to simulate an heterogeneous traffic in the upstream of a toll booth.

    In this section, we briefly recall the PT model treated in [9].

    In this subsection we collect some useful notations, see Figure 1, and the main assumptions on parameters and functions used throughout the paper. First, at any time t0 and in any position xR along the road, the traffic is described by the vector

    Figure 1.  Geometrical meaning of the notations used through the paper. In particular, Ωf=ΩfΩ+f and Ωc are the free-flow and congested domains, respectively; V+f and Vf are the maximal and minimal speeds in the free-flow phase, respectively, and Vc is the maximal speed in the congested phase.
    u(t,x)(ρ(t,x),v(t,x)),

    where ρ is the density and v is the (average) speed. The vector u belongs to

    ΩΩfΩc,

    where Ωf and Ωc are respectively the domains of free-flow and congested phases, whose rigorous definitions are given below after the introduction of necessary parameters and functions. First, we fix two threshold densities R+f>Rf>0. Let VC2([0,R+f];R+) be the speed map and pC2([Rf,);R) be the pressure map such that

    V(ρ)<0, V(ρ)+ρV(ρ)>0, 2V(ρ)+ρV(ρ)0,ρ[0,R+f], (H1)
    p(ρ)>0, 2p(ρ)+ρp(ρ)>0,ρ[Rf,), (H2)
    V(ρ)+p(ρ)>0, V(ρ)<ρp(ρ),ρ[Rf,R+f], (H3)

    where the prime stands for the derivative with respect to the density ρ.

    A typical choice for V and p is

    V(ρ)V+f[1ρR],p(ρ){vrefγ[ρρmax]γ,γ>0,vreflog[ρρmax],γ=0,

    where R, γ, vref and ρmax are strictly positive parameters, that can be chosen so that (H1), (H2) and (H3) are satisfied, see [9] for the details.

    We then introduce also the following constants:

    V+fV(0),W+cp(R+f)+V(R+f),R+cp1(W+c),VfV(R+f),Wcp(Rf)+V(Rf),Rcp1(Wc).

    By (H2) the map p1:[WcV(Rf),W+c][Rf,R+c] is increasing and

    R+c>R+f>0,Rc>Rf>0,W+c>Wc.

    The above constants have the following physical meaning: V+f and Vf are the maximal and minimal speeds in the free-flow phase, respectively, W+c and Wc are the maximal and minimal Lagrangian markers in the congested phase, respectively, so that 1/R+c and 1/Rc are the minimal and maximal length of a vehicle, respectively.

    Finally, denoted by Vc(0,Vf) the maximal velocity in the congested phase, we can define the free-flow and congested domains

    Ωf{u[0,R+f]×[Vf,V+f]:v=V(ρ)},Ωc{R+fu[Rf,R+c]×[0,Vc]:Wcv+p(ρ)W+c},

    respectively. Observe that Ωf and Ωc are invariant domains for the LWR and the ARZ models, respectively. Let us also introduce

    Qcp1(WcVc)Vc,Q+cp1(W+cVc)Vc,QfR+fVf.

    Clearly, Qc is the flow of the state in Ωc with lowest density, Q+c is the maximal flow in Ωc and Qf is the maximal flow in Ωf (hence in Ω).

    The traffic is governed by the PT model [9,32]

    Freeflow{uΩf,ρt+Q(u)x=0,v=V(ρ),Congestedflow{uΩc,ρt+Q(u)x=0,[ρW(u)]t+[Q(u)W(u)]x=0, (1)

    where the flux map Q:Ω[0,Qf] and the Lagrangian marker map W:Ω[Wc,W+c] (extended to Ωf) are defined by

    Q(u)ρv,W(u){v+p(ρ)if uΩcΩ+f,Wcif uΩf, (2)

    with

    Ωf{uΩf:ρ[0,Rf)},Ω+f{uΩf:ρ[Rf,R+f]}.

    In the following table we collect the informations on the system governing the congested phase:

    r1(u)(ρ,ρ(v+p(ρ))),r2(u)(1,v+p(ρ)+ρp(ρ)),λ1(u)vρp(ρ),λ2(u)v,λ1r1(u)=ρ(2p(ρ)+ρp(ρ)),λ2r2(u)=0,L1(ρ;u0)W(u0)p(ρ),L2(ρ;u0)v0.

    Above ri is the i-th right eigenvector, λi is the corresponding eigenvalue and the graph of the map Li(;u0) gives the i-Lax curve passing through u0. By the assumptions (H1) and (H2) the characteristic speeds are bounded by the velocity, i.e. λf(u)v and λ1(u)λ2(u)=v, λ1 is genuinely non-linear, i.e. λ1r1(u)0, and λ2 is linearly degenerate, i.e. λ2r2(u)=0.

    In the subsequent definitions of the Riemann solvers we make use of the functions

    uc:[Wc,W+c]Ωc,uf:[Wc,W+c]Ω+f,

    defined as follows

    uc(w)(ρc(w),Vc),withρc(w)p1(wVc),uf(w)(ρf(w),vf(w)),withvf(w)=V(ρf(w))=wp(ρf(w)).

    These maps have a clear geometrical interpretation; indeed, roughly speaking, uc(w) and uf(w) are the intersections of the 1-Lax curve {uΩ:W(u)=w} with the line {uΩ:v=Vc} and with Ωf, respectively. Obviously R±f=ρf(W±c).

    In this section we propose two Riemann solvers R1, R2 for the Riemann problem of the PT model (1), namely for the Cauchy problem of (1) with an initial datum of the form

    u(0,x)={uif x<0,urif x>0, (3)

    where u,urΩ are given constants. We then construct two constrained Riemann solvers Rc1, Rc2 for the Riemann problem (1), (3) coupled with a pointwise constraint on the flux

    Q(u(t,0±))Q0, (4)

    where Q0(0,Qf) is a fixed constant.

    For notational simplicity we let

    qQ(u),wW(u),qrQ(ur),wrW(ur).

    Furthermore, for any u,u+Ω with ρρ+ we let

    σ(u,u+)Q(u+)Q(u)ρ+ρ (5)

    to be the speed of propagation of any discontinuity between u and u+. Observe that the first Rankine-Hugoniot condition (RH) is satisfied with s=σ(u,u+), therefore the number of vehicles is conserved across any discontinuity.

    In the following we denote by RLWR and RARZ the Riemann solvers for LWR and ARZ models, respectively.

    In this subsection we first recall the Riemann solver for (1), (3) introduced in [9], here denoted by R1, and then construct the corresponding constrained Riemann solver Rc1 for (1), (3), (4).

    Definition 3.1. The Riemann solver R1:Ω2L(R;Ω) is defined as follows:

    (R1.a) If u,urΩf, then R1[u,ur]RLWR[u,ur].

    (R1.b) If u,urΩc, then R1[u,ur]RARZ[u,ur].

    (R1.c) If (u,ur)Ωf×Ωc, then we let um(p1(wvr),vr)Ωc and

    R1[u,ur](ν){uif ν<σ(u,um),RARZ[um,ur](ν)if ν>σ(u,um).

    (R1.d) If (u,ur)Ωc×Ωf, then

    R1[u,ur](ν){RARZ[u,uc(w)](ν)if ν<σ(uc(w),uf(w)),RLWR[uf(w),ur](ν)if ν>σ(uc(w),uf(w)).

    In general, [(t,x)R1[u,ur](x/t)] does not satisfy the point constraint (4). For this reason we introduce the sets

    C1{(u,ur)Ω2:Q(R1[u,ur](0±))Q0},N1{(u,ur)Ω2:Q(R1[u,ur](0±))>Q0},

    and for any (u,ur)N1, we replace the self-similar weak solution [(t,x)R1[u,ur](x/t)] by a self-similar map [(t,x)Rc1[u,ur](x/t)] satisfying (3), (4) and obtained by juxtaposing maps constructed by means of R1. It is easy to see that

    C1=Cf,fCc,cCc,fCf,c1,N1=Nf,fNc,cNc,fNf,c1,

    where

    Cf,f{(u,ur)Ω2f:qQ0},Cc,c{(u,ur)Ω2c:p1(wvr)vrQ0},Cc,f{(u,ur)Ωc×Ωf:Q(uf(w))Q0},Cf,c1{(u,ur)Ωf×Ωc:min{q,p1(wvr)vr}Q0},

    and

    Nf,fΩ2fCf,f,Nc,f(Ωc×Ωf)Cc,f,Nc,cΩ2cCc,c,Nf,c1(Ωf×Ωc)Cf,c1.

    Definition 3.2. The constrained Riemann solver Rc1:Ω2L(R;Ω) is defined as follows:

    (R1ca) If (u,ur)C1, then we let Rc1[u,ur]R1[u,ur].

    (R1cb) If (u,ur)N1, then we let

    Rc1[u,ur](ν){R1[u,ˆu1](ν)if ν<0,R1[ˇu1,ur](ν)if ν>0, (6)

    where ˆu1=ˆu1(w,Q0)Ωc and ˇu1=ˇu1(w,vr,Q0)Ω satisfy

    ˆu1ˆΩ{uΩc:Q(u)Q0, W(u)=w},Q(ˆu1)=max{Q(u):uˆΩ}, (7)
    Q(ˇu1)=Q(ˆu1),ˇv1={vrif urΩc and Q0p1(Wcvr)vr,V(ˇρ1)otherwise. (8)

    Observe that according to the second condition in (8) we have that ˇu1Ωc if and only if urΩc and Q0p1(Wcvr)vr, otherwise ˇu1Ωf.

    In the following proposition we show that Rc1 is well defined, namely that for any (u,ur)N1 there exists a unique couple in (ˆu1,ˇu1) in Ω2 satisfying (7), (8). For notational simplicity we let

    ˆq1Q(ˆu1),ˇq1Q(ˇu1),ˆw1W(ˆu1).

    Proposition 1. For any (u,ur)N1, we have that (ˆu1,ˇu1)Ωc×Ω is uniquely selected by (7), (8) as follows:

    (T11) If (u,ur)Nf,fNc,f, then we distinguish the following cases:

    (T11a) If Q0>Q(uc(w)), then ˆu1=uc(w), ˇq1=ˆq1 and ˇu1Ωf.

    (T11b) If Q0Q(uc(w)), then ˆw1=w, ˆq1=ˇq1=Q0 and ˇu1Ωf.

    (T12) If (u,ur)Nc,cNf,c1, then we distinguish the following cases:

    (T12a) If Q0p1(Wcvr)vr, then ˆw1=w, ˆq1=ˇq1=Q0 and ˇv=vr.

    (T12b) If Q0<p1(Wcvr)vr, then ˆw1=w, ˆq1=ˇq1=Q0 and ˇu1Ωf.

    In particular, Rc1 is well defined in Ω2.

    The proof of the above proposition is straightforward and is therefore omitted, see Figure 2 and Figure 3. Let us just underline that if (u,ur)N1, then ˆu1 and ˇu1 must be distinct otherwise, by the consistency of R1 proved in [9,Proposition 4.2], we would have that Rc1[u,ur] coincides with R1[u,ur], and this gives a contradiction. Moreover, if (u,ur)N1, then (7), (8) imply that R1[u,ˆu1] contains only waves with negative speeds and R1[ˇu1,ur] contains only waves with positive speeds; consequently Rc1[u,ur](0)=ˆu1, Rc1[u,ur](0+)=ˇu1 and [(t,x)Rc1[u,ur](x/t)] satisfies (4) because Q(ˇu1)=Q(ˆu1)Q0.

    Figure 2.  Geometrical meaning of the cases (T11a) and (T11b). Above u and u are u in two different cases.
    Figure 3.  Geometrical meaning of the cases (T12a) and (T12b). Above u and u are u in two different cases.

    Differently from any other constrained Riemann solver available in the literature, see [1,3,4,19,24,25,26], it may well happen that (u,ur)N1 but Q(Rc1[u,ur](0±))Q0, see the case (T11a) described in Proposition 1. Moreover, for any fixed (u,ur)N1, among the self-similar maps u of the form (6), namely

    u(ν){R1[u,ˆu](ν)if ν<0,R1[ˇu,ur](ν)if ν>0,

    with (ˆu,ˇu)Ω2 eventually distinct from (ˆu1,ˇu1) but satisfying the minimal requirements

    R1[u,ˆu](0)=ˆu1,R1[ˇu,ur](0+)=ˇu1,Q(ˆu)=Q(ˇu)Q0,

    Rc1[u,ur] is the only one that maximizes the flow through x=0, namely

    Q(u(0±))Q(Rc1[u,ur](0±)),

    with the equality holding if and only if u=Rc1[u,ur]. For these reasons in this subsection we introduce a further Riemann solver R2 for (1), (3), that allows to construct a second constrained Riemann solver Rc2 for (1), (3), (4) such that Q(Rc2[u,ur](0±))=Q0 for all (u,ur)N1, at least in the case Q0Q+c.

    Definition 3.3. The Riemann solver R2:Ω2L1loc(R;Ω) is defined by letting

    R2[u,ur](ν){uif ν<σ(u,ur),urif ν>σ(u,ur),

    for any (u,ur)Ωf×Ωc with ρ0 and w<wr, and by letting R2[u,ur]R1[u,ur] in all the remaining cases.

    In analogy to the previous subsection we introduce the sets

    C2{(u,ur)Ω2:Q(R2[u,ur](0±))Q0},N2{(u,ur)Ω2:Q(R2[u,ur](0±))>Q0},

    and for any (u,ur)N2, we replace [(t,x)R2[u,ur](x/t)] by a self-similar map [(t,x)Rc2[u,ur](x/t)] satisfying (3), (4) and obtained by juxtaposing maps constructed by means of R2. It is easy to see that

    C2=Cf,fCc,cCc,fCf,c2,N2=Nf,fNc,cNc,fNf,c2,

    where

    Cf,c2{(u,ur)Ωf×Ωc:wwr and min{q,qr}Q0, orw>wr and p1(wvr)vrQ0},Nf,c2(Ωf×Ωc)Cf,c2.

    Definition 3.4. The Riemann solver Rc2:Ω2L(R;Ω) is defined as follows:

    (R2a) If (u,ur)C2, then we let Rc2[u,ur]R2[u,ur].

    (R2b) If (u,ur)Nc,f and Q0>Q(uc(w)), then we let

    Rc2[u,ur](ν){R2[u,uf(w)](ν)if ν<σ(uf(w),ˆu2),ˆu2if σ(uf(w),ˆu2)<ν<0,R2[ˇu2,ur](ν)if ν>0.

    (R2c) If (u,ur)Nc,f and Q0Q(uc(w)) or (u,ur)N2Nc,f, then we let

    Rc2[u,ur](ν){R2[u,ˆu2](ν)if ν<0,R2[ˇu2,ur](ν)if ν>0.

    In both cases (R2b) and (R2c), ˆu2=ˆu2(w,vr,Q0) and ˇu2=ˇu2(vr,Q0) are implicitly defined by

    {ˆu2ˆΩ{uΩc:Q(u)=min{Q0,Q+c}, W(u)w, vvr},W(ˆu2)=min{W(u):uˆΩ}, (9)
    Q(ˇu2)=min{Q0,Q+c},ˇv2={vrifurΩcandQ0p1(Wcvr)vr,V(ˇρ2)otherwise. (10)

    Observe that according to the second condition in (10) we have that ˇu2Ωc if and only if urΩc and Q0p1(Wcvr)vr, otherwise ˇu2Ωf.

    In the following proposition we show that Rc2 is well defined. For notational simplicity we let

    ˆq2Q(ˆu2),ˇq2Q(ˇu2),ˆw2W(ˆu2).

    Proposition 2. For any (u,ur)N2, (ˆu2,ˇu2)Ωc×Ω is uniquely selected by (9), (10) as follows:

    (T21) If (u,ur)Nf,fNc,f, then we distinguish the following cases:

    (T21a) If Q0>Q+c, then ˆq2=ˇq2=Q+c and ˇu2Ωf.

    (T21b) If Q0Q+c, then ˆw2=max{w,Vc+p(Q0/Vc)}, ˆq2=ˇq2=Q0 and ˇu2Ωf.

    (T22) If (u,ur)Nc,cNf,c2, then we distinguish the following cases:

    (T22a) If Q0p1(Wcvr)vr, then ˆw2=max{w,vr+p(Q0/vr)}, ˆq2=ˇq2=Q0 and ˇv=vr.

    (T22b) If Q0<p1(Wcvr)vr, then ˆw2=w, ˆq2=ˇq2=Q0 and ˇu2Ωf.

    In particular, Rc2 is well defined in Ω2.

    The proof of the above proposition is straightforward and is therefore omitted, see Figure 4. Let us just underline that, despite (T22a) and (T22b) are apparently the same as (T12a) and (T12b), respectively, they differ because Nf,c1Nf,c2 as shown in the following Example 1. Let us also underline that ˆu1ˇu1 for all (u,ur)N1, whereas in the case (T22a) with wvr+p(Q0/vr) we have ˆu2=ˇu2, see the last picture in Figure 4; this occurs because R1 is consistent whereas R2 is not, see Proposition 5. Clearly the map [(t,x)Rc2[u,ur](x/t)] satisfies (4).

    Figure 4.  Geometrical meaning of the cases (T21a), (T21b) and (T22a). Above u and u are u in two different cases.

    Example 1. Fix (u,ur)Ω+f×Ωc with w<wr and p1(wvr)vr<Q0<qr<q, see Figure 5. In this case Q(R1[u,ur](0))=Q(um)<Q0, where um(p1(wvr),vr), and therefore (u,ur)Cf,c1. As a consequence Rc1[u,ur] coincides with R1[u,ur] and performs a phase transition from u to um, followed by a contact discontinuity from um to ur. On the other hand, Q(R2[u,ur](0±))=qr>Q0 and therefore (u,ur)Nf,c2. By (T22a) we have that ˆu2=ˇu2=(p1(Q0/vr),vr). Hence Rc2[u,ur] performs a phase transition from u to ˆu2, followed by a contact discontinuity from ˆu2 to ur.

    Figure 5.  (ρ1,v1)Rc1[u,ur] and (ρ2,v2)Rc2[u,ur] in the case considered in Example 1..

    In this section we expose the main properties of the Riemann solvers constructed in the previous sections. This study may be useful to compare the difficulty of applying one of these Riemann solvers in a wave-front tracking scheme [11,34]. In particular, we introduce their invariant domains and discuss their consistency and L1loc-continuity. In this regard, we recall the following definition.

    Definition 4.1. Let S:Ω2L(R;Ω) be a Riemann solver.

    IΩ is an invariant domain for S if S[I,I](R)I.

    S is L1loc-continuous in DΩ if for any ν1,ν2R and for any sequences un,unrD converging to u,urD:

    limnν2ν1|S[un,unr](ν)S[u,ur](ν)|dν=0.

    S is consistent in an invariant domain IΩ if for any u,um,urI and ν_R:

    S[u,ur](ν_)=um{S[u,um](ν)={S[u,ur](ν)if ν<ν_,umif νν_,S[um,ur](ν)={umif ν<ν_,S[u,ur](ν)if νν_. (Ⅰ)
    S[u,um](ν_)=umS[um,ur](ν_)=um}S[u,ur](ν)={S[u,um](ν)if ν<ν_,S[um,ur](ν)if νν_. (Ⅱ)

    We recall that the consistency is a necessary condition for the well-posedness in L1 of the Cauchy problem.

    In the following proposition we show that a constrained Riemann solver cannot be consistent in Ω because it cannot satisfy (Ⅰ) of Definition 4.1 in Ω. As a consequence, none of the constrained Riemann solvers Rc1 and Rc2 is consistent in Ω and in the forthcoming propositions we consider in Ω only (Ⅱ).

    Proposition 3. Let S:Ω2L(R;Ω) be a Riemann solver satisfying (4). If I is an invariant domain for S and Q0<maxuIQ(u), then S does not satisfy (Ⅰ) of Definition 4.1 in I.

    Proof. By assumption there exist u,urI such that qr>Q0. By the finite speed of propagation of the waves there exists ν_>0 such that S[u,ur](ν_)=ur. Let umur. Then the property S[um,ur](ν)=um for any ν<ν_ required in (Ⅰ) cannot be satisfied. Indeed, if by contradiction S[um,ur](ν)=um for any ν<ν_, then Q(S[um,ur](0±))=qr>Q0 and this gives a contradiction because by assumption (t,x)S[um,ur](x/t) satisfies (4).

    In the following propositions we collect the main properties of R1 and R2.

    Proposition 4 (Invariant domains). For any ρmin,ρmax[0,R+f], vmin,vmax[0,Vc] and wmin,wmax[Wc,W+c] such that ρmin<ρmax, vmin<vmax and wmin<wmax, we have that

    {uΩf:ρminρρmax},{uΩc:wminW(u)wmax, vminvvmax},{uΩ+f:ρf(wmin)ρρf(wmax)}{uΩc:wminW(u)wmax, vvmin},

    are invariant domains for both R1 and R2. If moreover ρmin<Rf, then

    {uΩf:ρminρρf(wmax)}{uΩc:W(u)wmax, vvmin}

    is a further invariant domain for both R1 and R2.

    The proof is straightforward and is therefore omitted.

    Proposition 5. R1 is L1loc-continuous and consistent in Ω; whereas R2 is L1loc-continuous but not consistent in Ω.

    Proof. In [9,Proposition 4.2] we already proved that R1 is L1loc-continuous and consistent. By taking u, um and ur as in the Example 1, see Figure 5, we have that R2 does not satisfy (Ⅱ) of Definition 4.1, hence it is not consistent. Finally proceeding as in [9,Proposition 4.2] it can be proved that R2 is L1loc-continuous.

    In the following propositions we collect the main properties of Rc1. We start by studying the invariant domains of Rc1, see Figure 6. Clearly, Ω is an invariant domain for both R1 and Rc1. Moreover, Ωf and Ωc are invariant domains for R1 but not for Rc1. For this reason we look for minimal (with respect to the inclusion) invariant domains for Rc1 containing Ωf or Ωc.

    Figure 6.  The invariant domains described in Proposition 6 and Proposition 9.

    Proposition 6 (Invariant domains of Rc1).

    (I1ca) If Q0<Q+c, then Ωf{uΩc:Q(u)Q0p1(W+cv)v} is the smallest invariant domain for Rc1 containing Ωf.

    (I1cb) If Q0Q+c, then Ωf{uΩc:v=Vc} is the smallest invariant domain for Rc1 containing Ωf.

    (I1cc) If Q0Qc, then Ωc is the smallest invariant domain for Rc1 containing Ωc.

    (I1cd) If Q0<Qc, then Ωc{uΩf:Q(u)=Q0} is the smallest invariant domain for Rc1 containing Ωc.

    Proof. (I1ca) In order to prove that if Q0<Q+c, then the smallest invariant domain containing Ωf is I0Ωf{uΩc:Q(u)Q0p1(W+cv)v} it suffices to observe that I0 is an invariant domain and that if I is an invariant domain containing Ωf, then

    IRc1[Ωf,Rc1[Nf,f](R)](R)I0,

    where the last inclusion holds because

    Rc1[Nf,f](R){uΩc:Q(u)=Q0},Rc1[Ω+f,{uΩc:Q(u)=Q0}](R){uΩc:Q(u)Q0p1(W+cv)v}.

    (I1cb) In order to prove that if Q0Q+c, then the smallest invariant domain containing Ωf is I0Ωf{uΩc:v=Vc} it suffices to observe that I0 is an invariant domain and that if I is an invariant domain containing Ωf, then

    IRc1[Ωf,Rc1[Nf,f](R)](R)I0,

    where the last inclusion holds because

    Rc1[Nf,f](R){uΩc:v=Vc, Q(uf(W(u)))>Q0},Rc1[Ω+f,{uΩc:v=Vc, Q(uf(W(u)))>Q0}](R){uΩc:v=Vc}.

    (I1cc) In order to prove that if Q0Qc, then the smallest invariant domain containing Ωc is Ωc it suffices to prove that Ωc is an invariant domain. Since Ωc is an invariant domain for RARZ we immediately have that Rc1[Cc,c](R)=RARZ[Cc,c](R)Ωc. Moreover, if (u,ur)Nc,c, then ˆu1,ˇu1Ωc because Q0Qcp1(Wcvr)vr, see (T12a). As a consequence Rc1[u,ur](R)=RARZ[u,ˆu](R)RARZ[ˇu,ur](R)Ωc.

    (I1cd) In order to prove that if Q0<Qc, then the smallest invariant domain containing Ωc is I0Ωc{u0}, where u0 is the unique element of {uΩf:Q(u)=Q0}, it suffices to observe that I0 is an invariant domain and that if I is an invariant domain containing Ωc, then IRc1[Nc,c](R){u0}, where the last inclusion holds because by (T12) we have that for any (u,ur)Nc,c either ˇu1Ωc or ˇu1=u0.

    Proposition 7 (Consistency of Rc1).

    (C1ca) Rc1 satisfies (Ⅱ) of Definition 4.1 in Ω.

    (C1cb) Rc1 is consistent in the invariant domain I1{uΩ:Q(u)Q0}; moreover it is not consistent in any other invariant domain containing I1.

    Proof. (C1ca) Since R1 satisfies (Ⅱ), it suffices to consider the cases where at least one among (u,um), (um,ur) and (u,ur) belongs to N1. We observe that Rc1[u,um] cannot perform any contact discontinuity, otherwise it would not be possible to juxtapose Rc1[u,um] and Rc1[um,ur]. For the same reason (u,um) cannot belong to Cf,f. Moreover, (u,um) cannot belong to Cc,f, because in this case Rc1[u,um] and Rc1[um,ur] can be juxtaposed if and only if um=uf(w)Ω+f (hence Q(um)Q0 because (u,um)Cc,f) and urΩf, but then also (u,ur) and (um,ur) belong to C1. We are then left to consider the following cases.

    ● Let (u,um)Nf,f and um=ˇu1(w,vm,Q0), namely q>Q0Q(um). We have then that either urΩc and p1(Wcvr)vr>Q0 or urΩf.

    ● Let u,umΩc. In this case, we have either (u,um)Nc,c or (u,um)Cc,c and w=W(um). In the first case, whether um=ˇu1(w,vm,Q0)Ωc and W(um)<w or ˇu1(w,vm,Q0)Ωf and Q(um)>Q0, we have that vr=vm. In the latter case, we have Q(um)=Q0, vr>vm and (um,ur)Nc,fNc,c.

    ● Let (u,um)Nc,f and um=ˇu1(w,vm,Q0). Then either urΩc satisfies p1(Wcvr)vr>Q0 or urΩf.

    ● Let (u,um)Ωf×Ωc. In this case, we have W(um)=Wc and either Q(um)=Q0<q and vr>vm or Q(um)>Q0 and vr=vm.

    ● Let (u,um)Ω+f×Ωc. In this case, we have Q(um)=Q0<q and either ˆv1(u,Q0)<vm=vr or ˆv1(u,Q0)=vmvr.

    For each of the above cases it is easy to conclude.

    (C1cb) By (C1ca) it is sufficient to prove that Rc1 satisfies (Ⅰ) in I1. Fix u,um,urI1 and ν_R such that Rc1[u,ur](ν_)=um. If (u,ur)C1I21, then also (u,um),(um,ur)C1 and (Ⅰ) comes from the consistency of R1. On the other hand, if (u,ur)N1I21Nc,cNc,f and ν_0 (the case ν_0 is analogous), then um=Rc1[u,ur](ν_)=R1[u,ˆu1(w,Q0)](ν_), W(um)=w and by exploiting the consistency of R1 we have

    Rc1[u,um](ν)=R1[u,um](ν)={R1[u,ˆu1(w,Q0)](ν)if ν<ν_umif νν_={Rc1[u,ur](ν)if ν<ν_,umif νν_,

    and

    Rc1[um,ur](ν)={R1[um,ˆu1(w,Q0)](ν)if ν<0R1[ˇu1(w,vr,Q0),ur](ν)if ν0={umif ν<ν_R1[u,ˆu1(w,Q0)](ν)if ν_ν<0R1[ˇu1(w,vr,Q0),ur](ν)if ν0={umif ν<ν_,Rc1[u,ur](ν)if νν_.

    We conclude the proof by observing that the maximality of I1 follows directly from Proposition 3.

    Proposition 8 (Continuity of Rc1).

    (L1ca) Rc1 is L1loc-continuous in Ω2 if and only if Q0Qc.

    (L1cb) If Q0>Qc, then Rc1 is L1loc-continuous in Ω2(C1¯Nf,f) and is not L1loc-continuous in any point of C1¯Nf,f.

    Proof. Assume that Q0>Qc and take (u,ur)C1¯Nf,f, namely u,urΩf with Q(u)=Q0. Let unΩf with Q(un)=Q0+1/n. Then un converges to u but Rc1[un,ur] does not converge to Rc1[u,ur] in L1loc(R;Ω). Indeed, Rc1[u,ur]u in R and by (T11a) the restriction of Rc1[un,ur] to R converges to

    {uif x<σ(u,uc(w)),uc(w)if σ(u,uc(w))<x<0.

    It remains to prove that if Q0>Qc and (u,ur)Ω2(C1¯Nf,f) or Q0Qc and (u,ur)Ω2, then Rc1[un,unr] converges to Rc1[u,ur] in L1loc(R;Ω) for all (un,unr)Ω2 converging to (u,ur). Since we already know that R1 is L1loc-continuous in Ω2, we can assume that (un,unr)N1. Thus, by Definition 3.2, completing the proof is a matter of showing that R1[un,ˆu1(wn,Q0)]Rc1[u,ur] pointwise in {x<0}, R1[ˇu1(wn,vnr,Q0),unr]Rc1[u,ur] pointwise in {x>0}, and applying the dominated convergence theorem of Lebesgue. For this, it suffices to observe that either ˆu1(wn,Q0)R1[u,ur](0) and the result follows then by the L1loc-continuity of R1, or σ(un,ˆu1(wn,Q0))0, R1[u,ur]u in {x<0} and therefore R1[un,ˆu1(wn,Q0)]R1[u,ur] pointwise in {x<0}. A similar analysis proves that R1[ˇu1(wn,vnr,Q0),unr]R1[u,ur] pointwise in {x>0}.

    The following proposition deals with the minimal invariant domains for Rc2 containing Ωf or Ωc, see Figure 6; its proof is analogous to that of Proposition 6 and is therefore omitted.

    Proposition 9 (Invariant domains for Rc2).

    (I2ca) If Q0<Q+c, then Ωf{uΩc:Q(u)Q0p1(W+cv)v} is the smallest invariant domain for Rc2 containing Ωf.

    (I2cb) If Q+cQ0, then Ωf{(Vc,W+c)} is the smallest invariant domain for Rc2 containing Ωf.

    (I2cc) If Q0Qc, then Ωc is the smallest invariant domain for Rc2 containing Ωc.

    (I2cd) If Q0<Qc, then Ωc{uΩf:Q(u)=Q0} is the smallest invariant domain for Rc2 containing Ωc.

    Concerning Rc2, in general no significant positive result for consistency can be expected because R2 is not consistent, see Proposition 5.

    Proposition 10 (Consistency of Rc2).

    (C2ca) Rc2 does not satisfy (Ⅱ) of Definition 4.1 in Ω.

    (C2cb) Rc2 does not satisfy (Ⅱ) of Definition 4.1 in any invariant domain containing I1{uΩ:Q(u)Q0}.

    Proof. (C2ca) Clearly, (Ⅱ) is not satisfied by Rc2 because we already know by Proposition 5 that it is not satisfied by R2.

    (C2cb) It is easy to see that I1 is an invariant domain if and only if Q0Qc. In any case, by taking ν_<0 sufficiently close to zero, u as the unique element of {uΩf:Q(u)=Q0} and um,urΩc such that vm=0=vr, W(um)=w and wr=W+c, we have that u,um,urI1 but Rc2 does not satisfy (Ⅱ).

    Proposition 11 (Continuity of Rc2). Rc2 is L1loc-continuous in Ω2.

    Proof. ● If u,urΩf, then the L1loc-continuity of Rc2 follows from the continuity of σ(u,ˆu), σ(ˇu,ur) with respect to (u,ur) and from the continuity of RLWR.

    ● If u,urΩc or (u,ur)Ωc×Ωf and Q(uc(w))>Q0, then Rc2[u,ur]=Rc1[u,ur] and the continuity follows from Proposition 8.

    ● If (u,ur)Ωc×Ωf and Q(uc(w))<Q0, then the continuity follows from the continuity of uc(w), uf(w), σ(uf(w),ˆu) with respect to u and Proposition 8.

    ● If (u,ur)Ωc×Ωf and Q(uc(w))=Q0, then it suffices to consider for n sufficiently large un defined by vnv and wnw1/n. Clearly unu and Q(uc(wn))<Q0. Moreover, Rc2[un,ur] has two phase transitions, one from uc(wn) to uf(wn) and one from uf(wn) to ˆu, that are not performed by Rc2[u,ur]. Since both σ(uc(wn),uf(wn)) and σ(uf(wn),ˆu) converge to σ(uf(w),ˆu), also in this case we have that Rc2[un,ur]Rc2[u,ur] in L1loc.

    ● If (u,ur)Ωf×Ωc, then the continuity comes from the continuity of σ(u,ur), σ(u,ˆu), σ(ˇu,ur) and Rc1 with respect to (u,ur).

    In this subsection we consider the total variation of the two constrained Riemann solvers in the Riemann invariant coordinates (v,w). We provide two examples showing that in general the comparison of their total variations can go in both ways. This suggests that the total variation is not a relevant selection criteria for choosing a wave-front tracking algorithm based on one or the other constrained Riemann solver.

    Example 2. With reference to Figure 7, let Q0(Qc,Q+c) and u0Ω+f with Q(u0)(Q0,Qf) be such that there exist ˇu1,ˇu2Ωf and ˆu1,ˆu2Ωc satisfying

    Figure 7.  u1Rc1[u0,u0] and u2Rc2[u0,u0] in the case considered in Example 2. Above ˆu1,ˇu1 are given by (T11b) and ˆu2,ˇu2 by (T21b); we let w0=W(u0), ˇvi=V(ˇui), ˆw2=W(ˆu2), ˇwi=W(ˇui).
    V(ˆu1)=V(ˆu2)=Vc,Q(ˇu2)=Q(ˆu2)=Q0,W(ˆu1)=W(u0),Q(ˇu1)=Q(ˆu1)<Q0.

    Then TV(VRc1[u0,u0])=2[V(ˇu1)Vc]>TV(VRc2[u0,u0])=2[V(ˇu2)Vc]. If we further assume that

    W(u0)W(ˇu1)>W(ˆu2)W(ˇu2),

    then TV(WRc1[u0,u0])=2[W(u0)W(ˇu1)]>TV(WRc2[u0,u0])=2[W(ˆu2)W(ˇu2)].

    Example 3. If there exist (u,ur)Ωc×Ωf and Q0 such that v=Vc and q=qr<Q0<Q(uf(w)), then TV(VRc1[u,ur])=vrVc<TV(VRc2[u,ur])=vr+2V(uf(w))3Vc and TV(WRc1[u,ur])=wwr<TV(WRc2[u,ur])=2ˆw2wwr, where ˆw2Vc+p(Q0/Vc).

    For any fixed u,urΩ, let us consider uiRi[u,ur], i=1,2. Since both R1 and R2 coincide with RARZ in Ω2c, all the possible discontinuities of u1 and u2 in Ωc satisfy the Rankine-Hugoniot conditions. This means that if u1 or u2 performs a discontinuity from uΩc to u+Ωc with speed of propagation sR, then ρρ+ and

    {Q(u+)Q(u)=s(ρ+ρ),Q(u+)W(u+)Q(u)W(u)=s(ρ+W(u+)ρW(u)). (RH)

    By the first condition in (RH) we immediately have that s=σ(u,u+), with σ defined in (5). We recall that the first and second conditions in (RH) express the conservation across the discontinuity of the number of vehicles and the linearized momentum, respectively. As a consequence, both the number of vehicles and the linearized momentum are conserved across discontinuities in Ωc performed by u1 or u2.

    By the assumption (H3), the 1-Lax curves defined in Ωc can be extended in a natural way up to reach Ω+f. Any point of the curve Ω+f is reached by exactly one extended 1-Lax curve. Hence, since the Lagrangian marker is constant along the 1-Lax curves, there is a natural way to define the Lagrangian marker in Ω+f, see the definition of W given in (2). It is then easy to see that also all the possible phase transitions between Ω+f and Ωc performed by u1 satisfy (RH), whereas those performed by u2 satisfy in general only the first condition in (RH). In fact, this is the case if u2 performs a phase transition from uΩ+f to u+Ωc with W(u)<W(u+).

    The extension to Ωf of the Lagrangian marker given in (2) ensures that any phase transition away from the vacuum performed by u1 satisfies (RH). Now, since the extended Lagrangian marker is defined in Ωf, we can question whether the shocks between states in Ωf satisfy (RH) or not. It is easy to see that the answer is positive if and only if Vf=V+f, however this contradicts our assumption (H1).

    In conclusion, we have that both R1 and R2 conserve the number of vehicles but not the (extended) linearized momentum; consequently also Rc1 and Rc2 do so. This is in the same spirit of the Riemann solvers introduced for traffic through locations with reduced capacity in [24,25,26] and for traffic at junctions in [28].

    Let us finally underline that, even if we generalize our model to the case Vf=V+f as done in [24] and consider only solutions away from the vacuum so that R1 conserves also the linearized momentum, the corresponding constrained Riemann solver Rc1 would not conserve it.

    In this section we apply the Riemann solvers introduced in Section 3 to simulate the traffic across a toll gate placed in x=0 and with capacity Q0(Q+c,Q(uf(Wc))), see Figure 8. We consider two types of vehicles: the 1-vehicles and the 2-vehicles. Fix x1<x2<0 and assume that initially the 1-vehicles and the 2-vehicles are stopped in (x1,x2) and (x2,0), respectively, and have Lagrangian markers Wc and W+c, respectively. Then we are led to consider the Cauchy problem for (1) with initial datum

    Figure 8.  Notations used in Section 5.
    u(0,x){u1if x(x1,x2),u2if x(x2,0),u0otherwise, (11)

    where u0(0,V(0)) belongs to Ωf and u1(Rc,0), u2(R+c,0) belong to Ωc.

    In subsections 5.1 and 5.2 we construct the solutions obtained by applying the wave-front tracking method [11,34] based on the Riemann solvers R1, Rc1 and R2, Rc2, respectively. The simulations presented in Figure 9 are obtained by the explicit analysis of the wave-fronts interactions with computer-assisted computation of the interaction times and front slopes and correspond to the following choice of the parameters

    Figure 9.  The solutions constructed in Subsection 5.1 on the left and in Subsection 5.2 on the right represented in the (x,t)-plane. The red thick curves are phase transitions. In particular, those along x=0 are stationary undercompressive phase transitions.
    W+c3,Wc2,V(ρ)1ρ10,p(ρ)ρ2,Vc12.

    We use in this section the following notation

    ufuf(Wc),ˆuuc(Wc),ˇuˇu1(Wc,V(0),Q0),ˆu+uc(W+c),ˇu+ˇu1(W+c,V(0),Q0)=ˇu2(V(0),Q0).

    Observe that by definition we have ˆu=ˆu1(Wc,Q0) and ˆu+=ˆu1(Wc,Q0)=ˆu2(Wc,V(0),Q0)=ˆu2(W+c,V(0),Q0), see Figure 8.

    In this subsection we apply the Riemann solver R1 away from x=0 and the constrained Riemann solver Rc1 at x=0 to construct the solution to the Cauchy problem (1), (11). The first step consists in solving the Riemann problems at the points (x,t)=(x1,0),(x2,0),(0,0).

    ● The Riemann problem at (x1,0) is solved by a stationary phase transition PT1 from u0 to u1.

    ● The Riemann problem at (x2,0) is solved by a stationary contact discontinuity C1 from u1 to u2.

    ● The Riemann problem at (0,0) is solved by a rarefaction R1 from u2 to ˆu+, followed by a stationary undercompressive phase transition U1 from ˆu+ to ˇu+ and then by another rarefaction R2 from ˇu+ to u0.

    To prolong then the solution we have to consider the Riemann problems arising at each interaction i[x1,0]×(0,) as follows.

    ● First, C1 starts to interact with R1 at i1. The result of this interaction is a contact discontinuity C2, which accelerates during its interaction with R1. C2 stops to interact with R1 once it reaches i2. Then, a contact discontinuity C3 from ˆu to ˆu+ starts from i2.

    ● The result of the interaction between C3 and U1 at i5 is a stationary undercompressive phase transition U2 from ˆu to ˇu followed by a shock S1 from ˇu to ˇu+.

    ● Each point of C2 is the center of a rarefaction appearing on its left. Let R3 be the juxtaposition of these rarefactions. Then PT1 starts to interact with R3 at i3. The result of this interaction is a phase transition PT2, which accelerates during its interaction with R3. PT2 stops to interact with R3 once it reaches i4. Then, a phase transition PT3 from u0 to ˆu starts from i4.

    ● Finally, the result of the interaction between PT3 and U2 at i6 is a shock S2 from u0 to ˇu.

    The constructed solution is qualitatively represented in Figure 9, left, see also Figure 10 for a quantitative representation.

    Figure 10.  Quantitative representation of density, on the left, and velocity, on the right, corresponding to the solutions constructed in Subsection 5.1 and Subsection 5.2. Recall that the two solutions coincide up to the interaction i5.

    In this subsection we apply the Riemann solver R2 away from x=0 and the constrained Riemann solver Rc2 at x=0 to construct the solution to the Cauchy problem (1), (11). The solution coincides with that constructed in Subsection 5.1 up to the interaction i5. The result of the interaction at i5 is now a phase transition PT4 from ˆu to uf, followed by another phase transition PT5 from uf to ˆu+ and then by a stationary undercompressive phase transition U3 from ˆu+ to ˇu+.To prolong then the solution it is sufficient to observe that:

    ● the result of the interaction at i7 between PT3 and PT4 is a shock S3 from u0 and uf;

    ● the result of the interaction at i8 between S3 and PT5 is a phase transition PT6 from u0 and ˆu+;

    ● the result of the interaction at i9 between PT6 and U3 is a shock S3 from u0 and ˇu+.

    The constructed solution is qualitatively represented in Figure 9, right, see also Figure 10 and Figure 11 for a quantitative representation.

    Figure 11.  Quantitative representation of density, on the left, and velocity, on the right, corresponding to the solution constructed in Subsection 5.2.

    The authors thank Boris Andreianov, Edda Dal Santo and Carlotta Donadello for very helpful discussions. The first author thanks the Faculty of Mathematics, Physics and Computer Science of Maria Curie-Sklodowska-University (UMCS), for the hospitality during the preparation of this paper. The first author also thanks the Gran Sasso Science Institute (GSSI), with a special mention to Pierangelo Marcati, for the support they brought during the time this work was accomplished. The last author was also supported by the INdAM -GNAMPA Project 2017 "Equazioni iperboliche con termini nonlocali: teoria e modelli".

    [1] Riemann problems with non-local point constraints and capacity drop. Math. Biosci. Eng. (2015) 12: 259-278.
    [2] 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. ESAIM: M2AN (2016) 50: 1269-1287.
    [3] Crowd dynamics and conservation laws with nonlocal constraints and capacity drop. Math. Models Methods Appl. Sci. (2014) 24: 2685-2722.
    [4] A second-order model for vehicular traffics with local point constraints on the flow. Math. Models Methods Appl. Sci. (2016) 26: 751-802.
    [5] Finite volume schemes for locally constrained conservation laws. Numer. Math. (2010) 115: 609-645, With supplementary material available online.
    [6] Solutions of the Aw-Rascle-Zhang system with point constraints. Netw. Heterog. Media (2016) 11: 29-47.
    [7] A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916–938 (electronic). doi: 10.1137/S0036139997332099
    [8] On the modeling of traffic and crowds: A survey of models, speculations, and perspectives. SIAM Rev. (2011) 53: 409-463.
    [9] Entropy solutions for a traffic model with phase transitions. Nonlinear Anal. (2016) 141: 167-190.
    [10] A general phase transition model for vehicular traffic. SIAM J. Appl. Math. (2011) 71: 107-127.
    [11] A. Bressan, Hyperbolic Systems of Conservation Laws, vol. 20 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2000, The one-dimensional Cauchy problem.
    [12] Error estimate for Godunov approximation of locally constrained conservation laws. SIAM J. Numer. Anal. (2012) 50: 3036-3060.
    [13] C. Chalons and P. Goatin, Computing phase transitions arising in traffic flow modeling, in Hyperbolic Problems: Theory, Numerics, Applications, Springer, Berlin, 2008,559–566. doi: 10.1007/978-3-540-75712-2_54
    [14] Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling. Interfaces Free Bound. (2008) 10: 197-221.
    [15] General constrained conservation laws. Application to pedestrian flow modeling. Netw. Heterog. Media (2013) 8: 433-463.
    [16] Phase transition model for traffic at a junction. J. Math. Sci. (N. Y.) (2014) 196: 30-36.
    [17] R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708–721 (electronic). doi: 10.1137/S0036139901393184
    [18] R. M. Colombo, Phase transitions in hyperbolic conservation laws, in Progress in Analysis, Vol. I, II (Berlin, 2001), World Sci. Publ., River Edge, NJ, 2003,1279–1287.
    [19] A well posed conservation law with a variable unilateral constraint. J. Differential Equations (2007) 234: 654-675.
    [20] Road networks with phase transitions. J. Hyperbolic Differ. Equ. (2010) 7: 85-106.
    [21] Global well posedness of traffic flow models with phase transitions. Nonlinear Anal. (2007) 66: 2413-2426.
    [22] On the modelling and management of traffic. ESAIM Math. Model. Numer. Anal. (2011) 45: 853-872.
    [23] A 2-phase traffic model based on a speed bound. SIAM J. Appl. Math. (2010) 70: 2652-2666.
    [24] E. Dal Santo, M. D. Rosini, N. Dymski and M. Benyahia, General phase transition models for vehicular traffic with point constraints on the flow, arXiv preprint, arXiv: 1608.04932.
    [25] The Aw-Rascle traffic model with locally constrained flow. J. Math. Anal. Appl. (2011) 378: 634-648.
    [26] M. Garavello and S. Villa, The Cauchy problem for the Aw-Rascle-Zhang traffic model with locally constrained flow, 2016, URL https://www.math.ntnu.no/conservation/2016/007.pdf.
    [27] M. Garavello, K. Han and B. Piccoli, Models for Vehicular Traffic on Networks, vol. 9 of AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2016, Conservation laws models.
    [28] Traffic flow on a road network using the Aw-Rascle model. Comm. Partial Differential Equations (2006) 31: 243-275.
    [29] M. Garavello and B. Piccoli, Traffic Flow on Networks, vol. 1 of AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006, Conservation laws models.
    [30] Coupling of Lighthill-Whitham-Richards and phase transition models. J. Hyperbolic Differ. Equ. (2013) 10: 577-636.
    [31] Coupling of microscopic and phase transition models at boundary. Netw. Heterog. Media (2013) 8: 649-661.
    [32] The Aw-Rascle vehicular traffic flow model with phase transitions. Math. Comput. Modelling (2006) 44: 287-303.
    [33] Traffic flow models with phase transitions on road networks. Netw. Heterog. Media (2009) 4: 287-301.
    [34] H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, vol. 152 of Applied Mathematical Sciences, 2nd edition, Springer, Heidelberg, 2015. doi: 10.1007/978-3-662-47507-2
    [35] M. J. Lighthill and G. B. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, in Proceedings of the Royal Society. London. Series A. Mathematical, Physical and Engineering Sciences, 229 (1955), 317–345. doi: 10.1098/rspa.1955.0089
    [36] State-of-the art of macroscopic traffic flow modelling. Int. J. Adv. Eng. Sci. Appl. Math. (2013) 5: 158-176.
    [37] The generalized Riemann problem for the Aw-Rascle model with phase transitions. J. Math. Anal. Appl. (2012) 389: 685-693.
    [38] The global solution of the interaction problem for the Aw-Rascle model with phase transitions. Math. Methods Appl. Sci. (2012) 35: 1700-1711.
    [39] B. Piccoli and A. Tosin, Vehicular traffic: A review of continuum mathematical models, in Mathematics of complexity and dynamical systems. Vols. 1–3, Springer, New York, 2012, 1748–1770. doi: 10.1007/978-1-4614-1806-1_112
    [40] Shock waves on the highway. Operations Res. (1956) 4: 42-51.
    [41] The initial-boundary value problem and the constraint. Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications (2013) 63-91.
    [42] M. D. Rosini, Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, Understanding Complex Systems, Springer, Heidelberg, 2013. doi: 10.1007/978-3-319-00155-5
    [43] A non-equilibrium traffic model devoid of gas-like behavior. Transportation Research Part B: Methodological (2002) 36: 275-290.
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