# American Institute of Mathematical Sciences

June 2017, 12(2): 297-317. doi: 10.3934/nhm.2017013

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

 1 Gran Sasso Science Institute, Viale F. Crispi 7,67100 L'Aquila, Italy 2 Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, Plac Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland

* Corresponding author: Massimiliano D. Rosini

Received  November 2016 Revised  January 2017 Published  May 2017

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. Networks & Heterogeneous Media, 2017, 12 (2) : 297-317. doi: 10.3934/nhm.2017013
##### References:
 [1] B. Andreianov, C. Donadello, U. Razafison and M. D. Rosini, Riemann problems with non-local point constraints and capacity drop, Math. Biosci. Eng., 12 (2015), 259-278. [2] B. Andreianov, C. Donadello, U. Razafison and M. D. Rosini, 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, 50 (2016), 1269-1287. doi: 10.1051/m2an/2015078. [3] B. Andreianov, C. Donadello and M. D. Rosini, Crowd dynamics and conservation laws with nonlocal constraints and capacity drop, Math. Models Methods Appl. Sci., 24 (2014), 2685-2722. doi: 10.1142/S0218202514500341. [4] B. Andreianov, C. Donadello and M. D. Rosini, A second-order model for vehicular traffics with local point constraints on the flow, Math. Models Methods Appl. Sci., 26 (2016), 751-802. doi: 10.1142/S0218202516500172. [5] B. Andreianov, P. Goatin and N. Seguin, Finite volume schemes for locally constrained conservation laws, Numer. Math., 115 (2010), 609-645, With supplementary material available online. doi: 10.1007/s00211-009-0286-7. [6] B. P. Andreianov, C. Donadello, U. Razafison, J. Y. Rolland and M. D. Rosini, Solutions of the Aw-Rascle-Zhang system with point constraints, Netw. Heterog. Media, 11 (2016), 29-47. doi: 10.3934/nhm.2016.11.29. [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] N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463. doi: 10.1137/090746677. [9] M. Benyahia and M. D. Rosini, Entropy solutions for a traffic model with phase transitions, Nonlinear Anal., 141 (2016), 167-190. doi: 10.1016/j.na.2016.04.011. [10] S. Blandin, D. Work, P. Goatin, B. Piccoli and A. Bayen, A general phase transition model for vehicular traffic, SIAM J. Appl. Math., 71 (2011), 107-127. doi: 10.1137/090754467. [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] C. Cancés and N. Seguin, Error estimate for Godunov approximation of locally constrained conservation laws, SIAM J. Numer. Anal., 50 (2012), 3036-3060. doi: 10.1137/110836912. [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] C. Chalons and P. Goatin, Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling, Interfaces Free Bound., 10 (2008), 197-221. doi: 10.4171/IFB/186. [15] C. Chalons, P. Goatin and N. Seguin, General constrained conservation laws. Application to pedestrian flow modeling, Netw. Heterog. Media, 8 (2013), 433-463. doi: 10.3934/nhm.2013.8.433. [16] R. M. Colombo and M. Garavello, Phase transition model for traffic at a junction, J. Math. Sci. (N. Y.), 196 (2014), 30-36. doi: 10.1007/s10958-013-1631-z. [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] R. M. Colombo and P. Goatin, A well posed conservation law with a variable unilateral constraint, J. Differential Equations, 234 (2007), 654-675. doi: 10.1016/j.jde.2006.10.014. [20] R. M. Colombo, P. Goatin and B. Piccoli, Road networks with phase transitions, J. Hyperbolic Differ. Equ., 7 (2010), 85-106. doi: 10.1142/S0219891610002025. [21] R. M. Colombo, P. Goatin and F. S. Priuli, Global well posedness of traffic flow models with phase transitions, Nonlinear Anal., 66 (2007), 2413-2426. doi: 10.1016/j.na.2006.03.029. [22] R. M. Colombo, P. Goatin and M. D. Rosini, On the modelling and management of traffic, ESAIM Math. Model. Numer. Anal., 45 (2011), 853-872. doi: 10.1051/m2an/2010105. [23] R. M. Colombo, F. Marcellini and M. Rascle, A 2-phase traffic model based on a speed bound, SIAM J. Appl. Math., 70 (2010), 2652-2666. doi: 10.1137/090752468. [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] M. Garavello and P. Goatin, The Aw-Rascle traffic model with locally constrained flow, J. Math. Anal. Appl., 378 (2011), 634-648. doi: 10.1016/j.jmaa.2011.01.033. [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] M. Garavello and B. Piccoli, Traffic flow on a road network using the Aw-Rascle model, Comm. Partial Differential Equations, 31 (2006), 243-275. doi: 10.1080/03605300500358053. [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] M. Garavello and B. Piccoli, Coupling of Lighthill-Whitham-Richards and phase transition models, J. Hyperbolic Differ. Equ., 10 (2013), 577-636. doi: 10.1142/S0219891613500215. [31] M. Garavello and B. Piccoli, Coupling of microscopic and phase transition models at boundary, Netw. Heterog. Media, 8 (2013), 649-661. doi: 10.3934/nhm.2013.8.649. [32] P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions, Math. Comput. Modelling, 44 (2006), 287-303. doi: 10.1016/j.mcm.2006.01.016. [33] P. Goatin, Traffic flow models with phase transitions on road networks, Netw. Heterog. Media, 4 (2009), 287-301. doi: 10.3934/nhm.2009.4.287. [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] R. Mohan and G. Ramadurai, State-of-the art of macroscopic traffic flow modelling, Int. J. Adv. Eng. Sci. Appl. Math., 5 (2013), 158-176. doi: 10.1007/s12572-013-0087-1. [37] L. Pan and X. Han, The generalized Riemann problem for the Aw-Rascle model with phase transitions, J. Math. Anal. Appl., 389 (2012), 685-693. doi: 10.1016/j.jmaa.2011.11.081. [38] L. Pan and X. Han, The global solution of the interaction problem for the Aw-Rascle model with phase transitions, Math. Methods Appl. Sci., 35 (2012), 1700-1711. doi: 10.1002/mma.2552. [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] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. [41] M. D. Rosini, The initial-boundary value problem and the constraint, Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, (2013), 63-91. doi: 10.1007/978-3-319-00155-5_6. [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] H. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B: Methodological, 36 (2002), 275-290. doi: 10.1016/S0191-2615(00)00050-3.

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##### References:
 [1] B. Andreianov, C. Donadello, U. Razafison and M. D. Rosini, Riemann problems with non-local point constraints and capacity drop, Math. Biosci. Eng., 12 (2015), 259-278. [2] B. Andreianov, C. Donadello, U. Razafison and M. D. Rosini, 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, 50 (2016), 1269-1287. doi: 10.1051/m2an/2015078. [3] B. Andreianov, C. Donadello and M. D. Rosini, Crowd dynamics and conservation laws with nonlocal constraints and capacity drop, Math. Models Methods Appl. Sci., 24 (2014), 2685-2722. doi: 10.1142/S0218202514500341. [4] B. Andreianov, C. Donadello and M. D. Rosini, A second-order model for vehicular traffics with local point constraints on the flow, Math. Models Methods Appl. Sci., 26 (2016), 751-802. doi: 10.1142/S0218202516500172. [5] B. Andreianov, P. Goatin and N. Seguin, Finite volume schemes for locally constrained conservation laws, Numer. Math., 115 (2010), 609-645, With supplementary material available online. doi: 10.1007/s00211-009-0286-7. [6] B. P. Andreianov, C. Donadello, U. Razafison, J. Y. Rolland and M. D. Rosini, Solutions of the Aw-Rascle-Zhang system with point constraints, Netw. Heterog. Media, 11 (2016), 29-47. doi: 10.3934/nhm.2016.11.29. [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] N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463. doi: 10.1137/090746677. [9] M. Benyahia and M. D. Rosini, Entropy solutions for a traffic model with phase transitions, Nonlinear Anal., 141 (2016), 167-190. doi: 10.1016/j.na.2016.04.011. [10] S. Blandin, D. Work, P. Goatin, B. Piccoli and A. Bayen, A general phase transition model for vehicular traffic, SIAM J. Appl. Math., 71 (2011), 107-127. doi: 10.1137/090754467. [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] C. Cancés and N. Seguin, Error estimate for Godunov approximation of locally constrained conservation laws, SIAM J. Numer. Anal., 50 (2012), 3036-3060. doi: 10.1137/110836912. [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] C. Chalons and P. Goatin, Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling, Interfaces Free Bound., 10 (2008), 197-221. doi: 10.4171/IFB/186. [15] C. Chalons, P. Goatin and N. Seguin, General constrained conservation laws. Application to pedestrian flow modeling, Netw. Heterog. Media, 8 (2013), 433-463. doi: 10.3934/nhm.2013.8.433. [16] R. M. Colombo and M. Garavello, Phase transition model for traffic at a junction, J. Math. Sci. (N. Y.), 196 (2014), 30-36. doi: 10.1007/s10958-013-1631-z. [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] R. M. Colombo and P. Goatin, A well posed conservation law with a variable unilateral constraint, J. Differential Equations, 234 (2007), 654-675. doi: 10.1016/j.jde.2006.10.014. [20] R. M. Colombo, P. Goatin and B. Piccoli, Road networks with phase transitions, J. Hyperbolic Differ. Equ., 7 (2010), 85-106. doi: 10.1142/S0219891610002025. [21] R. M. Colombo, P. Goatin and F. S. Priuli, Global well posedness of traffic flow models with phase transitions, Nonlinear Anal., 66 (2007), 2413-2426. doi: 10.1016/j.na.2006.03.029. [22] R. M. Colombo, P. Goatin and M. D. Rosini, On the modelling and management of traffic, ESAIM Math. Model. Numer. Anal., 45 (2011), 853-872. doi: 10.1051/m2an/2010105. [23] R. M. Colombo, F. Marcellini and M. Rascle, A 2-phase traffic model based on a speed bound, SIAM J. Appl. Math., 70 (2010), 2652-2666. doi: 10.1137/090752468. [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] M. Garavello and P. Goatin, The Aw-Rascle traffic model with locally constrained flow, J. Math. Anal. Appl., 378 (2011), 634-648. doi: 10.1016/j.jmaa.2011.01.033. [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] M. Garavello and B. Piccoli, Traffic flow on a road network using the Aw-Rascle model, Comm. Partial Differential Equations, 31 (2006), 243-275. doi: 10.1080/03605300500358053. [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] M. Garavello and B. Piccoli, Coupling of Lighthill-Whitham-Richards and phase transition models, J. Hyperbolic Differ. Equ., 10 (2013), 577-636. doi: 10.1142/S0219891613500215. [31] M. Garavello and B. Piccoli, Coupling of microscopic and phase transition models at boundary, Netw. Heterog. Media, 8 (2013), 649-661. doi: 10.3934/nhm.2013.8.649. [32] P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions, Math. Comput. Modelling, 44 (2006), 287-303. doi: 10.1016/j.mcm.2006.01.016. [33] P. Goatin, Traffic flow models with phase transitions on road networks, Netw. Heterog. Media, 4 (2009), 287-301. doi: 10.3934/nhm.2009.4.287. [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] R. Mohan and G. Ramadurai, State-of-the art of macroscopic traffic flow modelling, Int. J. Adv. Eng. Sci. Appl. Math., 5 (2013), 158-176. doi: 10.1007/s12572-013-0087-1. [37] L. Pan and X. Han, The generalized Riemann problem for the Aw-Rascle model with phase transitions, J. Math. Anal. Appl., 389 (2012), 685-693. doi: 10.1016/j.jmaa.2011.11.081. [38] L. Pan and X. Han, The global solution of the interaction problem for the Aw-Rascle model with phase transitions, Math. Methods Appl. Sci., 35 (2012), 1700-1711. doi: 10.1002/mma.2552. [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] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. [41] M. D. Rosini, The initial-boundary value problem and the constraint, Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, (2013), 63-91. doi: 10.1007/978-3-319-00155-5_6. [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] H. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B: Methodological, 36 (2002), 275-290. doi: 10.1016/S0191-2615(00)00050-3.
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
Geometrical meaning of the cases (T11a) and (T11b). Above $u_\ell'$ and $u_\ell''$ are $u_\ell$ in two different cases
Geometrical meaning of the cases (T12a) and (T12b). Above $u_\ell'$ and $u_\ell''$ are $u_\ell$ in two different cases
Geometrical meaning of the cases (T21a), (T21b) and (T22a). Above $u_\ell'$ and $u_\ell''$ are $u_\ell$ in two different cases
$(\rho_1, v_1) \doteq \mathcal{R}_1^{\rm c}[u_\ell,u_r]$ and $(\rho_2, v_2) \doteq \mathcal{R}_2^{\rm c}[u_\ell,u_r]$ in the case considered in Example 1.
The invariant domains described in Proposition 6 and Proposition 9
$u_1 \doteq \mathcal{R}_1^{\rm c}[u_0,u_0]$ 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 $w_0 = W(u_0)$, $\check{v}_i = V(\check{u}_i)$, $\hat{w}_2 = W(\hat{u}_2)$, $\check{w}_i = W(\check{u}_i)$
Notations used in Section 5
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
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 $i_5$
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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