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

• * Corresponding author: Massimiliano D. Rosini
• 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.

Mathematics Subject Classification: Primary: 35L65, 90B20; Secondary: 82C26.

 Citation:

• Figure 1.  Geometrical meaning of the notations used through the paper. In particular, $\Omega_{\rm f} = \Omega_{\rm f}^- \cup \Omega_{\rm f}^+$ and $\Omega_{\rm c}$ are the free-flow and congested domains, respectively; $V_{\rm f}^+$ and $V_{\rm f}^-$ are the maximal and minimal speeds in the free-flow phase, respectively, and $V_{\rm c}$ is the maximal speed in the congested phase

Figure 2.  Geometrical meaning of the cases (T11a) and (T11b). Above $u_\ell'$ and $u_\ell''$ are $u_\ell$ in two different cases

Figure 3.  Geometrical meaning of the cases (T12a) and (T12b). Above $u_\ell'$ and $u_\ell''$ are $u_\ell$ in two different cases

Figure 4.  Geometrical meaning of the cases (T21a), (T21b) and (T22a). Above $u_\ell'$ and $u_\ell''$ are $u_\ell$ in two different cases

Figure 5.  $(\rho_1, v_1) \doteq \mathcal{R}_1^{\rm c}[u_\ell,u_r]$ and $(\rho_2, v_2) \doteq \mathcal{R}_2^{\rm c}[u_\ell,u_r]$ in the case considered in Example 1.

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

Figure 7.  $u_1 \doteq \mathcal{R}_1^{\rm c}[u_0,u_0]$ and $u_2 \doteq \mathcal{R}_2^{\rm c}[u_0,u_0]$ in the case considered in Example 2. Above $\hat{u}_1,\check{u}_1$ are given by (T11b) and $\hat{u}_2,\check{u}_2$ by (T21b); we let $w_0 = W(u_0)$, $\check{v}_i = V(\check{u}_i)$, $\hat{w}_2 = W(\hat{u}_2)$, $\check{w}_i = W(\check{u}_i)$

Figure 8.  Notations used in Section 5

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

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 $i_5$

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

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