# American Institute of Mathematical Sciences

June  2014, 7(3): 463-482. doi: 10.3934/dcdss.2014.7.463

## The LWR traffic model at a junction with multibuffers

 1 Dipartimento di Matematica e Applicazioni, Via R. Cozzi 55, 20125 Milano, Italy

Received  May 2013 Revised  July 2013 Published  January 2014

We consider the Lighthill-Whitham-Richards traffic flow model on a network composed by a single junction $J$ with $n$ incoming roads, $m$ outgoing roads and $m$ buffers, one for each outgoing road. We introduce a concept solution at $J$, which is compared with that proposed in [14]. Finally we study the Cauchy problem and, in the special case of $n \le 2$ and $m \le 2$, we prove existence of solutions to the Cauchy problem, via the wave-front tracking method.
Citation: Mauro Garavello. The LWR traffic model at a junction with multibuffers. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 463-482. doi: 10.3934/dcdss.2014.7.463
##### References:
 [1] A. Aw and M. Rascle, Resurrection of "second order'' models of traffic flow,, SIAM J. Appl. Math., 60 (2000), 916. doi: 10.1137/S0036139997332099. Google Scholar [2] M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks,, Netw. Heterog. Media, 1 (2006), 41. doi: 10.3934/nhm.2006.1.41. Google Scholar [3] 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. doi: 10.1137/090754467. Google Scholar [4] A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem,, Oxford Lecture Series in Mathematics and its Applications, (2000). Google Scholar [5] G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network,, SIAM J. Math. Anal., 36 (2005), 1862. doi: 10.1137/S0036141004402683. Google Scholar [6] R. M. Colombo, Hyperbolic phase transitions in traffic flow,, SIAM J. Appl. Math., 63 (2002), 708. doi: 10.1137/S0036139901393184. Google Scholar [7] R. M. Colombo, P. Goatin and B. Piccoli, Road networks with phase transitions,, J. Hyperbolic Differ. Equ., 7 (2010), 85. doi: 10.1142/S0219891610002025. Google Scholar [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. doi: 10.1137/090752468. Google Scholar [9] C. D'apice, R. Manzo and B. Piccoli, Packet flow on telecommunication networks,, SIAM J. Math. Anal., 38 (2006), 717. doi: 10.1137/050631628. Google Scholar [10] M. Garavello and P. Goatin, The Cauchy problem at a node with buffer,, Discrete Contin. Dyn. Syst., 32 (2012), 1915. doi: 10.3934/dcds.2012.32.1915. Google Scholar [11] M. Garavello and B. Piccoli, Traffic flow on a road network using the Aw-Rascle model,, Comm. Partial Differential Equations, 31 (2006), 243. doi: 10.1080/03605300500358053. Google Scholar [12] M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models,, AIMS Series on Applied Mathematics, (2006). Google Scholar [13] M. Garavello and B. Piccoli, Conservation laws on complex networks,, Ann. H. Poincaré, 26 (2009), 1925. doi: 10.1016/j.anihpc.2009.04.001. Google Scholar [14] M. Garavello and B. Piccoli, A multibuffer model for lwr road networks,, in Advances in Dynamic Network Modeling in Complex Transportation Systems, (2013), 143. doi: 10.1007/978-1-4614-6243-9_6. Google Scholar [15] P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions,, Math. Comput. Modelling, 44 (2006), 287. doi: 10.1016/j.mcm.2006.01.016. Google Scholar [16] S. Göttlich, M. Herty and A. Klar, Modelling and optimization of supply chains on complex networks,, Commun. Math. Sci., 4 (2006), 315. doi: 10.4310/CMS.2006.v4.n2.a3. Google Scholar [17] M. Herty, A. Klar and B. Piccoli, Existence of solutions for supply chain models based on partial differential equations,, SIAM J. Math. Anal., 39 (2007), 160. doi: 10.1137/060659478. Google Scholar [18] M. Herty, J.-P. Lebacque and S. Moutari, A novel model for intersections of vehicular traffic flow,, Netw. Heterog. Media, 4 (2009), 813. doi: 10.3934/nhm.2009.4.813. Google Scholar [19] M. Herty, S. Moutari and M. Rascle, Optimization criteria for modelling intersections of vehicular traffic flow,, Netw. Heterog. Media, 1 (2006), 275. doi: 10.3934/nhm.2006.1.275. Google Scholar [20] M. Herty and M. Rascle, Coupling conditions for a class of second-order models for traffic flow,, SIAM J. Math. Anal., 38 (2006), 595. doi: 10.1137/05062617X. Google Scholar [21] H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Vonservation Laws,, Applied Mathematical Sciences, (2002). doi: 10.1007/978-3-642-56139-9. Google Scholar [22] 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. doi: 10.1098/rspa.1955.0089. Google Scholar [23] A. Marigo and B. Piccoli, A fluid dynamic model for $T$-junctions,, SIAM J. Math. Anal., 39 (2008), 2016. doi: 10.1137/060673060. Google Scholar [24] P. I. Richards, Shock waves on the highway,, Operations Res., 4 (1956), 42. doi: 10.1287/opre.4.1.42. Google Scholar [25] D. Sun, I. S. Strub and A. M. Bayen, Comparison of the performance of four Eulerian network flow models for strategic air traffic management,, Netw. Heterog. Media, 2 (2007), 569. doi: 10.3934/nhm.2007.2.569. Google Scholar [26] H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior,, Transportation Research Part B, 236 (2002), 275. doi: 10.1016/S0191-2615(00)00050-3. Google Scholar

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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. doi: 10.1137/S0036139997332099. Google Scholar [2] M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks,, Netw. Heterog. Media, 1 (2006), 41. doi: 10.3934/nhm.2006.1.41. Google Scholar [3] 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. doi: 10.1137/090754467. Google Scholar [4] A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem,, Oxford Lecture Series in Mathematics and its Applications, (2000). Google Scholar [5] G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network,, SIAM J. Math. Anal., 36 (2005), 1862. doi: 10.1137/S0036141004402683. Google Scholar [6] R. M. Colombo, Hyperbolic phase transitions in traffic flow,, SIAM J. Appl. Math., 63 (2002), 708. doi: 10.1137/S0036139901393184. Google Scholar [7] R. M. Colombo, P. Goatin and B. Piccoli, Road networks with phase transitions,, J. Hyperbolic Differ. Equ., 7 (2010), 85. doi: 10.1142/S0219891610002025. Google Scholar [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. doi: 10.1137/090752468. Google Scholar [9] C. D'apice, R. Manzo and B. Piccoli, Packet flow on telecommunication networks,, SIAM J. Math. Anal., 38 (2006), 717. doi: 10.1137/050631628. Google Scholar [10] M. Garavello and P. Goatin, The Cauchy problem at a node with buffer,, Discrete Contin. Dyn. Syst., 32 (2012), 1915. doi: 10.3934/dcds.2012.32.1915. Google Scholar [11] M. Garavello and B. Piccoli, Traffic flow on a road network using the Aw-Rascle model,, Comm. Partial Differential Equations, 31 (2006), 243. doi: 10.1080/03605300500358053. Google Scholar [12] M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models,, AIMS Series on Applied Mathematics, (2006). Google Scholar [13] M. Garavello and B. Piccoli, Conservation laws on complex networks,, Ann. H. Poincaré, 26 (2009), 1925. doi: 10.1016/j.anihpc.2009.04.001. Google Scholar [14] M. Garavello and B. Piccoli, A multibuffer model for lwr road networks,, in Advances in Dynamic Network Modeling in Complex Transportation Systems, (2013), 143. doi: 10.1007/978-1-4614-6243-9_6. Google Scholar [15] P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions,, Math. Comput. Modelling, 44 (2006), 287. doi: 10.1016/j.mcm.2006.01.016. Google Scholar [16] S. Göttlich, M. Herty and A. Klar, Modelling and optimization of supply chains on complex networks,, Commun. Math. Sci., 4 (2006), 315. doi: 10.4310/CMS.2006.v4.n2.a3. Google Scholar [17] M. Herty, A. Klar and B. Piccoli, Existence of solutions for supply chain models based on partial differential equations,, SIAM J. Math. Anal., 39 (2007), 160. doi: 10.1137/060659478. Google Scholar [18] M. Herty, J.-P. Lebacque and S. Moutari, A novel model for intersections of vehicular traffic flow,, Netw. Heterog. Media, 4 (2009), 813. doi: 10.3934/nhm.2009.4.813. Google Scholar [19] M. Herty, S. Moutari and M. Rascle, Optimization criteria for modelling intersections of vehicular traffic flow,, Netw. Heterog. Media, 1 (2006), 275. doi: 10.3934/nhm.2006.1.275. Google Scholar [20] M. Herty and M. Rascle, Coupling conditions for a class of second-order models for traffic flow,, SIAM J. Math. Anal., 38 (2006), 595. doi: 10.1137/05062617X. Google Scholar [21] H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Vonservation Laws,, Applied Mathematical Sciences, (2002). doi: 10.1007/978-3-642-56139-9. Google Scholar [22] 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. doi: 10.1098/rspa.1955.0089. Google Scholar [23] A. Marigo and B. Piccoli, A fluid dynamic model for $T$-junctions,, SIAM J. Math. Anal., 39 (2008), 2016. doi: 10.1137/060673060. Google Scholar [24] P. I. Richards, Shock waves on the highway,, Operations Res., 4 (1956), 42. doi: 10.1287/opre.4.1.42. Google Scholar [25] D. Sun, I. S. Strub and A. M. Bayen, Comparison of the performance of four Eulerian network flow models for strategic air traffic management,, Netw. Heterog. Media, 2 (2007), 569. doi: 10.3934/nhm.2007.2.569. Google Scholar [26] H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior,, Transportation Research Part B, 236 (2002), 275. doi: 10.1016/S0191-2615(00)00050-3. Google Scholar
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