# American Institute of Mathematical Sciences

October 2017, 37(10): 5191-5209. doi: 10.3934/dcds.2017225

## The Riemann Problem at a Junction for a Phase Transition Traffic Model

 Department of Mathematics and its Applications, University of Milano Bicocca, Via R. Cozzi 55, 20125 Milano, Italy

* Corresponding author: M. Garavello

Received  October 2016 Revised  May 2017 Published  June 2017

Fund Project: The authors were partially supported by the INdAM-GNAMPA 2015 project "Balance Laws in the Modeling of Physical, Biological and Industrial Processes"

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.

Citation: Mauro Garavello, Francesca Marcellini. The Riemann Problem at a Junction for a Phase Transition Traffic Model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5191-5209. doi: 10.3934/dcds.2017225
##### References:
 [1] A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938. doi: 10.1137/S0036139997332099. [2] 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. [3] G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683. [4] R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721. doi: 10.1137/S0036139901393184. [5] R. M. Colombo, Phase transitions in hyperbolic conservation laws, In Progress in analysis, Vol. I, II (Berlin, 2001), pages 1279-1287. World Sci. Publ., River Edge, NJ, 2003. [6] 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. [7] R. M. Colombo and F. Marcellini, A mixed ODE-PDE model for vehicular traffic, Math. Methods Appl. Sci., 38 (2015), 1292-1302. doi: 10.1002/mma.3146. [8] 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. [9] M. Garavello, K. Han and B. Piccoli, Models for Vehicular Traffic on Networks Volume 9 of AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2016. [10] 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. [11] M. Garavello and B. Piccoli, Traffic Flow on Networks volume 1 of AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006. [12] 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. [13] M. Herty, S. Moutari and M. Rascle, Optimization criteria for modelling intersections of vehicular traffic flow, Netw. Heterog. Media, 1 (2006), 275-294. doi: 10.3934/nhm.2006.1.275. [14] M. Herty and M. Rascle, Coupling conditions for a class of second-order models for traffic flow, SIAM Journal on Mathematical Analysis, 38 (2006), 595-616. doi: 10.1137/05062617X. [15] H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), 999-1017. doi: 10.1137/S0036141093243289. [16] J. P. Lebacque, X. Louis, S. Mammar, B. Schnetzlera and H. Haj-Salem, Modélisation du trafic autoroutier au second ordre, Comptes Rendus Mathematique, 346 (2008), 1203-1206. doi: 10.1016/j.crma.2008.09.024. [17] M. J. Lighthill and G. B. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. [18] F. Marcellini, Free-congested and micro-macro descriptions of traffic flow, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 543-556. doi: 10.3934/dcdss.2014.7.543. [19] S. Moutari and M. Rascle, A hybrid Lagrangian model based on the Aw-Rascle traffic flow model, SIAM J. Appl. Math., 68 (2007), 413-436. doi: 10.1137/060678415. [20] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. [21] 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] A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938. doi: 10.1137/S0036139997332099. [2] 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. [3] G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683. [4] R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721. doi: 10.1137/S0036139901393184. [5] R. M. Colombo, Phase transitions in hyperbolic conservation laws, In Progress in analysis, Vol. I, II (Berlin, 2001), pages 1279-1287. World Sci. Publ., River Edge, NJ, 2003. [6] 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. [7] R. M. Colombo and F. Marcellini, A mixed ODE-PDE model for vehicular traffic, Math. Methods Appl. Sci., 38 (2015), 1292-1302. doi: 10.1002/mma.3146. [8] 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. [9] M. Garavello, K. Han and B. Piccoli, Models for Vehicular Traffic on Networks Volume 9 of AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2016. [10] 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. [11] M. Garavello and B. Piccoli, Traffic Flow on Networks volume 1 of AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006. [12] 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. [13] M. Herty, S. Moutari and M. Rascle, Optimization criteria for modelling intersections of vehicular traffic flow, Netw. Heterog. Media, 1 (2006), 275-294. doi: 10.3934/nhm.2006.1.275. [14] M. Herty and M. Rascle, Coupling conditions for a class of second-order models for traffic flow, SIAM Journal on Mathematical Analysis, 38 (2006), 595-616. doi: 10.1137/05062617X. [15] H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), 999-1017. doi: 10.1137/S0036141093243289. [16] J. P. Lebacque, X. Louis, S. Mammar, B. Schnetzlera and H. Haj-Salem, Modélisation du trafic autoroutier au second ordre, Comptes Rendus Mathematique, 346 (2008), 1203-1206. doi: 10.1016/j.crma.2008.09.024. [17] M. J. Lighthill and G. B. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. [18] F. Marcellini, Free-congested and micro-macro descriptions of traffic flow, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 543-556. doi: 10.3934/dcdss.2014.7.543. [19] S. Moutari and M. Rascle, A hybrid Lagrangian model based on the Aw-Rascle traffic flow model, SIAM J. Appl. Math., 68 (2007), 413-436. doi: 10.1137/060678415. [20] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. [21] 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.
The free phase $F$ and the congested phase $C$ resulting from (1) in the coordinates, from left to right, $(\rho,\eta)$ and $(\rho, \rho v)$. In the $(\rho,\eta)$ plane, the curves $\eta= \check w \rho$, $\eta= \hat w \rho$ and the curve $\eta= \frac{V_{\max}}{\psi(\rho)}\rho$ that divides the two phases are represented. The densities $\sigma_-$ and $\sigma_+$ are given by the intersections between the previous curves. Similarly in the $(\rho, \rho v)$ plane, the curves $\rho v= \check w \psi(\rho)\rho$, $\rho v= \hat w \psi(\rho)\rho$ and the densities $\sigma_-$ and $\sigma_+$ are represented
The case $(\bar \rho,\bar \eta)\in C$. The set $\mathcal T_{inc}\left(\bar \rho, \bar \eta\right)$ it is represented in red in the coordinates, from left to right, $(\rho,\eta)$ and $(\rho, \rho v)$. The set $\mathcal T_{inc}^f \left(\bar \rho, \bar \eta\right)$ is represented on the $\rho v$ axis in the $(\rho, \rho v)$ plane
The case $(\bar \rho,\bar \eta)\in F$. The set $\mathcal T_{inc}\left(\bar \rho, \bar \eta\right)$ it is represented in red in the coordinates, from left to right, $(\rho,\eta)$ and $(\rho, \rho v)$. The set $\mathcal T_{inc}^f \left(\bar \rho, \bar \eta\right)$ is represented on the $\rho v$ axis in the $(\rho, \rho v)$ plane
The case $(\bar \rho,\bar \eta)\in F$. The set $\mathcal T_{out}\left(w, \bar \rho, \bar \eta \right)$ it is represented in red in the coordinates, from left to right, $(\rho,\eta)$ and $(\rho, \rho v)$. The set $\mathcal T_{out}^f \left(w,\bar \rho, \bar \eta\right)$ is represented on the $\rho v$ axis in the $(\rho, \rho v)$ plane
The case $(\bar \rho,\bar \eta)\in C$. The set $\mathcal T_{out}\left(w, \bar \rho, \bar \eta \right)$ it is represented in red in the coordinates, from left to right, $(\rho,\eta)$ and $(\rho, \rho v)$. The set $\mathcal T_{out}^f \left(w,\bar \rho, \bar \eta\right)$ is represented on the $\rho v$ axis in the $(\rho, \rho v)$ plane
The case $(\bar \rho,\bar \eta)\in F$ in an outgoing road for the approach in Subsection 4.1. The set of all the possible traces it is represented in red in the coordinates, from left to right, $(\rho,\eta)$ and $(\rho, \rho v)$. The corresponding set of flows is represented on the $\rho v$ axis in the $(\rho, \rho v)$ plane
The case $(\bar \rho,\bar \eta)\in C$ in an outgoing road for the approach in Subsection 4.1. The set of all the possible traces it is represented in red in the coordinates, from left to right, $(\rho,\eta)$ and $(\rho, \rho v)$. The corresponding set of flows is represented on the $\rho v$ axis in the $(\rho, \rho v)$ plane
The situation in the outgoing road related to the approach of Subsection 4.1. Left, in the $\left(\rho, \eta\right)$-plane, the states $\left(\rho_3^\ast, \eta_3^\ast\right)$ and $\left(\bar \rho_3, \bar \eta_3\right)$, connected through the middle state $\left(\rho^m, \eta^m\right)$. 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
The case $\gamma_{1}^{*}+\gamma_{2}^{*}=\Gamma_{3}^{w_{3}}$. At left the case $\gamma_1^* < \Gamma_1$ and $\gamma_2^* < \Gamma_2$. At right the case $\gamma_1^* = \Gamma_1$
The case $\gamma_{1}^{*}+\gamma_{2}^{*}<\Gamma_{3}^{w_{3}}$
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