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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.
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
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
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
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
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
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
u(t,x)≐(ρ(t,x),v(t,x)), |
where
Ω≐Ωf∪Ωc, |
where
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∈[R−f,∞), | (H2) |
V′(ρ)+p′(ρ)>0, V(ρ)<ρp′(ρ),ρ∈[R−f,R+f], | (H3) |
where the prime stands for the derivative with respect to the density
A typical choice for
V(ρ)≐V+f[1−ρR],p(ρ)≐{vrefγ[ρρmax]γ,γ>0,vreflog[ρρmax],γ=0, |
where
We then introduce also the following constants:
V+f≐V(0),W+c≐p(R+f)+V(R+f),R+c≐p−1(W+c),V−f≐V(R+f),W−c≐p(R−f)+V(R−f),R−c≐p−1(W−c). |
By (H2) the map
R+c>R+f>0,R−c>R−f>0,W+c>W−c. |
The above constants have the following physical meaning:
Finally, denoted by
Ωf≐{u∈[0,R+f]×[V−f,V+f]:v=V(ρ)},Ωc≐{R+fu∈[R−f,R+c]×[0,Vc]:W−c≤v+p(ρ)≤W+c}, |
respectively. Observe that
Q−c≐p−1(W−cVc)Vc,Q+c≐p−1(W+c−Vc)Vc,Qf≐R+fV−f. |
Clearly,
The traffic is governed by the PT model [9,32]
Free−flow{u∈Ωf,ρt+Q(u)x=0,v=V(ρ),Congested−flow{u∈Ωc,ρt+Q(u)x=0,[ρW(u)]t+[Q(u)W(u)]x=0, | (1) |
where the flux map
Q(u)≐ρv,W(u)≐{v+p(ρ)if u∈Ωc∪Ω+f,W−cif u∈Ω−f, | (2) |
with
Ω−f≐{u∈Ωf:ρ∈[0,R−f)},Ω+f≐{u∈Ωf:ρ∈[R−f,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)≐v−ρp′(ρ),λ2(u)≐v,∇λ1⋅r1(u)=−ρ(2p′(ρ)+ρp″(ρ)),∇λ2⋅r2(u)=0,L1(ρ;u0)≐W(u0)−p(ρ),L2(ρ;u0)≐v0. |
Above
In the subsequent definitions of the Riemann solvers we make use of the functions
uc:[W−c,W+c]→Ωc,uf:[W−c,W+c]→Ω+f, |
defined as follows
uc(w)≐(ρc(w),Vc),withρc(w)≐p−1(w−Vc),uf(w)≐(ρf(w),vf(w)),withvf(w)=V(ρf(w))=w−p(ρf(w)). |
These maps have a clear geometrical interpretation; indeed, roughly speaking,
In this section we propose two Riemann solvers
u(0,x)={uℓif x<0,urif x>0, | (3) |
where
Q(u(t,0±))≤Q0, | (4) |
where
For notational simplicity we let
qℓ≐Q(uℓ),wℓ≐W(uℓ),qr≐Q(ur),wr≐W(ur). |
Furthermore, for any
σ(u−,u+)≐Q(u+)−Q(u−)ρ+−ρ− | (5) |
to be the speed of propagation of any discontinuity between
In the following we denote by
In this subsection we first recall the Riemann solver for (1), (3) introduced in [9], here denoted by
Definition 3.1. The Riemann solver
(R1.a) If
(R1.b) If
(R1.c) If
R1[uℓ,ur](ν)≐{uℓif ν<σ(uℓ,um),RARZ[um,ur](ν)if ν>σ(uℓ,um). |
(R1.d) If
R1[uℓ,ur](ν)≐{RARZ[uℓ,uc(wℓ)](ν)if ν<σ(uc(wℓ),uf(wℓ)),RLWR[uf(wℓ),ur](ν)if ν>σ(uc(wℓ),uf(wℓ)). |
In general,
C1≐{(uℓ,ur)∈Ω2:Q(R1[uℓ,ur](0±))≤Q0},N1≐{(uℓ,ur)∈Ω2:Q(R1[uℓ,ur](0±))>Q0}, |
and for any
C1=Cf,f∪Cc,c∪Cc,f∪Cf,c1,N1=Nf,f∪Nc,c∪Nc,f∪Nf,c1, |
where
Cf,f≐{(uℓ,ur)∈Ω2f:qℓ≤Q0},Cc,c≐{(uℓ,ur)∈Ω2c:p−1(wℓ−vr)vr≤Q0},Cc,f≐{(uℓ,ur)∈Ωc×Ωf:Q(uf(wℓ))≤Q0},Cf,c1≐{(uℓ,ur)∈Ωf×Ωc:min{qℓ,p−1(wℓ−vr)vr}≤Q0}, |
and
Nf,f≐Ω2f∖Cf,f,Nc,f≐(Ωc×Ωf)∖Cc,f,Nc,c≐Ω2c∖Cc,c,Nf,c1≐(Ωf×Ωc)∖Cf,c1. |
Definition 3.2. The constrained Riemann solver
(R1ca) If
(R1cb) If
Rc1[uℓ,ur](ν)≐{R1[uℓ,ˆu1](ν)if ν<0,R1[ˇu1,ur](ν)if ν>0, | (6) |
where
ˆu1∈ˆΩ≐{u∈Ωc:Q(u)≤Q0, W(u)=wℓ},Q(ˆu1)=max{Q(u):u∈ˆΩ}, | (7) |
Q(ˇu1)=Q(ˆu1),ˇv1={vrif ur∈Ωc and Q0≥p−1(W−c−vr)vr,V(ˇρ1)otherwise. | (8) |
Observe that according to the second condition in (8) we have that
In the following proposition we show that
ˆq1≐Q(ˆu1),ˇq1≐Q(ˇu1),ˆw1≐W(ˆu1). |
Proposition 1. For any
(T11) If
(T11a) If
(T11b) If
(T12) If
(T12a) If
(T12b) If
In particular,
The proof of the above proposition is straightforward and is therefore omitted, see Figure 2 and Figure 3. Let us just underline that if
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(ν)≐{R1[uℓ,ˆu](ν)if ν<0,R1[ˇu,ur](ν)if ν>0, |
with
R1[uℓ,ˆu](0−)=ˆu1,R1[ˇu,ur](0+)=ˇu1,Q(ˆu)=Q(ˇu)≤Q0, |
Q(u(0±))≤Q(Rc1[uℓ,ur](0±)), |
with the equality holding if and only if
Definition 3.3. The Riemann solver
R2[uℓ,ur](ν)≐{uℓif ν<σ(uℓ,ur),urif ν>σ(uℓ,ur), |
for any
In analogy to the previous subsection we introduce the sets
C2≐{(uℓ,ur)∈Ω2:Q(R2[uℓ,ur](0±))≤Q0},N2≐{(uℓ,ur)∈Ω2:Q(R2[uℓ,ur](0±))>Q0}, |
and for any
C2=Cf,f∪Cc,c∪Cc,f∪Cf,c2,N2=Nf,f∪Nc,c∪Nc,f∪Nf,c2, |
where
Cf,c2≐{(uℓ,ur)∈Ωf×Ωc:wℓ≤wr and min{qℓ,qr}≤Q0, orwℓ>wr and p−1(wℓ−vr)vr≤Q0},Nf,c2≐(Ωf×Ωc)∖Cf,c2. |
Definition 3.4. The Riemann solver
(R2a) If
(R2b) If
Rc2[uℓ,ur](ν)≐{R2[uℓ,uf(wℓ)](ν)if ν<σ(uf(wℓ),ˆu2),ˆu2if σ(uf(wℓ),ˆu2)<ν<0,R2[ˇu2,ur](ν)if ν>0. |
(R2c) If
Rc2[uℓ,ur](ν)≐{R2[uℓ,ˆu2](ν)if ν<0,R2[ˇu2,ur](ν)if ν>0. |
In both cases (R2b) and (R2c),
{ˆu2∈ˆΩ≐{u∈Ωc:Q(u)=min{Q0,Q+c}, W(u)≥wℓ, v≤vr},W(ˆu2)=min{W(u):u∈ˆΩ}, | (9) |
Q(ˇu2)=min{Q0,Q+c},ˇv2={vrifur∈ΩcandQ0≥p−1(W−c−vr)vr,V(ˇρ2)otherwise. | (10) |
Observe that according to the second condition in (10) we have that
In the following proposition we show that
ˆq2≐Q(ˆu2),ˇq2≐Q(ˇu2),ˆw2≐W(ˆu2). |
Proposition 2. For any
(T21) If
(T21a) If
(T21b) If
(T22) If
(T22a) If
(T22b) If
In particular,
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
Example 1. Fix
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
Definition 4.1. Let
●
●
limn→∞∫ν2ν1|S[unℓ,unr](ν)−S[uℓ,ur](ν)|dν=0. |
●
S[uℓ,ur](ν_)=um⇒{S[uℓ,um](ν)={S[uℓ,ur](ν)if ν<ν_,umif ν≥ν_,S[um,ur](ν)={umif ν<ν_,S[uℓ,ur](ν)if ν≥ν_. | (Ⅰ) |
S[uℓ,um](ν_)=umS[um,ur](ν_)=um}⇒S[uℓ,ur](ν)={S[uℓ,um](ν)if ν<ν_,S[um,ur](ν)if ν≥ν_. | (Ⅱ) |
We recall that the consistency is a necessary condition for the well-posedness in
In the following proposition we show that a constrained Riemann solver cannot be consistent in
Proposition 3. Let
Proof. By assumption there exist
In the following propositions we collect the main properties of
Proposition 4 (Invariant domains). For any
{u∈Ωf:ρmin≤ρ≤ρmax},{u∈Ωc:wmin≤W(u)≤wmax, vmin≤v≤vmax},{u∈Ω+f:ρf(wmin)≤ρ≤ρf(wmax)}∪{u∈Ωc:wmin≤W(u)≤wmax, v≥vmin}, |
are invariant domains for both
{u∈Ωf:ρmin≤ρ≤ρf(wmax)}∪{u∈Ωc:W(u)≤wmax, v≥vmin} |
is a further invariant domain for both
The proof is straightforward and is therefore omitted.
Proposition 5.
Proof. In [9,Proposition 4.2] we already proved that
In the following propositions we collect the main properties of
Proposition 6 (Invariant domains of
(I1ca) If
(I1cb) If
(I1cc) If
(I1cd) If
Proof. (I1ca) In order to prove that if
I⊇Rc1[Ωf,Rc1[Nf,f](R)](R)⊇I0, |
where the last inclusion holds because
Rc1[Nf,f](R)⊇{u∈Ωc:Q(u)=Q0},Rc1[Ω+f,{u∈Ωc:Q(u)=Q0}](R)⊇{u∈Ωc:Q(u)≤Q0≤p−1(W+c−v)v}. |
(I1cb) In order to prove that if
I⊇Rc1[Ωf,Rc1[Nf,f](R)](R)⊇I0, |
where the last inclusion holds because
Rc1[Nf,f](R)⊇{u∈Ωc:v=Vc, Q(uf(W(u)))>Q0},Rc1[Ω+f,{u∈Ωc:v=Vc, Q(uf(W(u)))>Q0}](R)⊇{u∈Ωc:v=Vc}. |
(I1cc) In order to prove that if
(I1cd) In order to prove that if
Proposition 7 (Consistency of
(C1ca)
(C1cb)
Proof. (C1ca) Since
● Let
● Let
● Let
● Let
● Let
For each of the above cases it is easy to conclude.
(C1cb) By (C1ca) it is sufficient to prove that
Rc1[uℓ,um](ν)=R1[uℓ,um](ν)={R1[uℓ,ˆu1(wℓ,Q0)](ν)if ν<ν_umif ν≥ν_={Rc1[uℓ,ur](ν)if ν<ν_,umif ν≥ν_, |
and
Rc1[um,ur](ν)={R1[um,ˆu1(wℓ,Q0)](ν)if ν<0R1[ˇu1(wℓ,vr,Q0),ur](ν)if ν≥0={umif ν<ν_R1[uℓ,ˆu1(wℓ,Q0)](ν)if ν_≤ν<0R1[ˇu1(wℓ,vr,Q0),ur](ν)if ν≥0={umif ν<ν_,Rc1[uℓ,ur](ν)if ν≥ν_. |
We conclude the proof by observing that the maximality of
Proposition 8 (Continuity of
(L1ca)
(L1cb) If
Proof. Assume that
{uℓif x<σ(uℓ,uc(wℓ)),uc(wℓ)if σ(uℓ,uc(wℓ))<x<0. |
It remains to prove that if
The following proposition deals with the minimal invariant domains for
Proposition 9 (Invariant domains for
(I2ca) If
(I2cb) If
(I2cc) If
(I2cd) If
Concerning
Proposition 10 (Consistency of
(C2ca)
(C2cb)
Proof. (C2ca) Clearly, (Ⅱ) is not satisfied by
(C2cb) It is easy to see that
Proposition 11 (Continuity of
Proof. ● If
● If
● If
● If
● If
In this subsection we consider the total variation of the two constrained Riemann solvers in the Riemann invariant coordinates
Example 2. With reference to Figure 7, let
V(ˆu1)=V(ˆu2)=Vc,Q(ˇu2)=Q(ˆu2)=Q0,W(ˆu1)=W(u0),Q(ˇu1)=Q(ˆu1)<Q0. |
Then
W(u0)−W(ˇu1)>W(ˆu2)−W(ˇu2), |
then
Example 3. If there exist
For any fixed
{Q(u+)−Q(u−)=s(ρ+−ρ−),Q(u+)W(u+)−Q(u−)W(u−)=s(ρ+W(u+)−ρ−W(u−)). | (RH) |
By the first condition in (RH) we immediately have that
By the assumption (H3), the 1-Lax curves defined in
The extension to
In conclusion, we have that both
Let us finally underline that, even if we generalize our model to the case
In this section we apply the Riemann solvers introduced in Section 3 to simulate the traffic across a toll gate placed in
u(0,x)≐{u1if x∈(x1,x2),u2if x∈(x2,0),u0otherwise, | (11) |
where
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
W+c≐3,W−c≐2,V(ρ)≐1−ρ10,p(ρ)≐ρ2,Vc≐12. |
We use in this section the following notation
u−f≐uf(W−c),ˆu−≐uc(W−c),ˇu−≐ˇu1(W−c,V(0),Q0),ˆu+≐uc(W+c),ˇu+≐ˇu1(W+c,V(0),Q0)=ˇu2(V(0),Q0). |
Observe that by definition we have
In this subsection we apply the Riemann solver
● The Riemann problem at
● The Riemann problem at
● The Riemann problem at
To prolong then the solution we have to consider the Riemann problems arising at each interaction
● First,
● The result of the interaction between
● Each point of
● Finally, the result of the interaction between
The constructed solution is qualitatively represented in Figure 9, left, see also Figure 10 for a quantitative representation.
In this subsection we apply the Riemann solver
● the result of the interaction at
● the result of the interaction at
● the result of the interaction at
The constructed solution is qualitatively represented in Figure 9, right, see also Figure 10 and Figure 11 for a quantitative representation.
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".
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1. | Boris Andreianov, Carlotta Donadello, Ulrich Razafison, Massimiliano D. Rosini, Analysis and approximation of one-dimensional scalar conservation laws with general point constraints on the flux, 2018, 116, 00217824, 309, 10.1016/j.matpur.2018.01.005 | |
2. | Mohamed Benyahia, Carlotta Donadello, Nikodem Dymski, Massimiliano D. Rosini, An existence result for a constrained two-phase transition model with metastable phase for vehicular traffic, 2018, 25, 1021-9722, 10.1007/s00030-018-0539-1 | |
3. | Boris Andreianov, Massimiliano D. Rosini, 2020, Chapter 7, 978-3-030-46078-5, 113, 10.1007/978-3-030-46079-2_7 | |
4. | Boris Andreianov, Carlotta Donadello, Massimiliano D. Rosini, Entropy solutions for a two-phase transition model for vehicular traffic with metastable phase and time depending point constraint on the density flow, 2021, 28, 1021-9722, 10.1007/s00030-021-00689-5 | |
5. | Boris Andreianov, Carlotta Donadello, Ulrich Razafison, Massimiliano Daniele Rosini, 2018, Chapter 5, 978-3-030-05128-0, 103, 10.1007/978-3-030-05129-7_5 | |
6. | Edda Dal Santo, Massimiliano D. Rosini, Nikodem Dymski, 2018, Chapter 34, 978-3-319-91544-9, 445, 10.1007/978-3-319-91545-6_34 | |
7. | Mohamed Benyahia, Massimiliano D. Rosini, Lack of BV bounds for approximate solutions to a two‐phase transition model arising from vehicular traffic, 2020, 43, 0170-4214, 10381, 10.1002/mma.6304 | |
8. | P. Cardaliaguet, N. Forcadel, Microscopic Derivation of a Traffic Flow Model with a Bifurcation, 2024, 248, 0003-9527, 10.1007/s00205-023-01948-8 |