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

The LWR traffic model at a junction with multibuffers

Mauro Garavello 1,

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.
Keywords: junction, Lighthill-Whitham-Richards model, Conservation laws, multibuffer, traffic problems..
Mathematics Subject Classification: Primary: 35L65; Secondary: 35R0.
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), 916-938 (electronic). 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), 41-56 (electronic). 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), 107-127. 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, 20, Oxford University Press, Oxford, 2000.  Google Scholar

[5]

G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886 (electronic). doi: 10.1137/S0036141004402683.  Google Scholar

[6]

R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721 (electronic). 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), 85-106. 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), 2652-2666. 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), 717-740 (electronic). 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), 1915-1938. 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), 243-275. doi: 10.1080/03605300500358053.  Google Scholar

[12]

M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models, AIMS Series on Applied Mathematics, 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.  Google Scholar

[13]

M. Garavello and B. Piccoli, Conservation laws on complex networks, Ann. H. Poincaré, 26 (2009), 1925-1951. 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, (eds., S. V. Ukkusuri and K. Ozbay), Complex Networks and Dynamic Systems, 2, Springer New York, 2013, 143-161. 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), 287-303. 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), 315-330. 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), 160-173. 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), 813-826 (electronic). 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), 275-294 (electronic). 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), 595-616 (electronic). doi: 10.1137/05062617X.  Google Scholar

[21]

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Vonservation Laws, Applied Mathematical Sciences, 152, Springer-Verlag, New York, 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), 317-345. 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), 2016-2032. doi: 10.1137/060673060.  Google Scholar

[24]

P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. 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), 569-595 (electronic). 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), 275-290. doi: 10.1016/S0191-2615(00)00050-3.  Google Scholar

show all references

References:
[1]

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.  Google Scholar

[2]

M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56 (electronic). 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), 107-127. 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, 20, Oxford University Press, Oxford, 2000.  Google Scholar

[5]

G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886 (electronic). doi: 10.1137/S0036141004402683.  Google Scholar

[6]

R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721 (electronic). 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), 85-106. 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), 2652-2666. 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), 717-740 (electronic). 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), 1915-1938. 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), 243-275. doi: 10.1080/03605300500358053.  Google Scholar

[12]

M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models, AIMS Series on Applied Mathematics, 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.  Google Scholar

[13]

M. Garavello and B. Piccoli, Conservation laws on complex networks, Ann. H. Poincaré, 26 (2009), 1925-1951. 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, (eds., S. V. Ukkusuri and K. Ozbay), Complex Networks and Dynamic Systems, 2, Springer New York, 2013, 143-161. 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), 287-303. 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), 315-330. 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), 160-173. 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), 813-826 (electronic). 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), 275-294 (electronic). 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), 595-616 (electronic). doi: 10.1137/05062617X.  Google Scholar

[21]

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Vonservation Laws, Applied Mathematical Sciences, 152, Springer-Verlag, New York, 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), 317-345. 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), 2016-2032. doi: 10.1137/060673060.  Google Scholar

[24]

P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. 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), 569-595 (electronic). 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), 275-290. doi: 10.1016/S0191-2615(00)00050-3.  Google Scholar

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