American Institute of Mathematical Sciences

Advanced
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.  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

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.  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

[1]

Helge Holden, Nils Henrik Risebro. Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow. Networks & Heterogeneous Media, 2018, 13 (3) : 409-421. doi: 10.3934/nhm.2018018
[2]

Guillaume Costeseque, Jean-Patrick Lebacque. Discussion about traffic junction modelling: Conservation laws VS Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 411-433. doi: 10.3934/dcdss.2014.7.411
[3]

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
[4]

Nicolas Forcadel, Wilfredo Salazar, Mamdouh Zaydan. Specified homogenization of a discrete traffic model leading to an effective junction condition. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2173-2206. doi: 10.3934/cpaa.2018104
[5]

Wen Shen. Traveling waves for conservation laws with nonlocal flux for traffic flow on rough roads. Networks & Heterogeneous Media, 2019, 14 (4) : 709-732. doi: 10.3934/nhm.2019028
[6]

Mapundi K. Banda, Michael Herty. Numerical discretization of stabilization problems with boundary controls for systems of hyperbolic conservation laws. Mathematical Control & Related Fields, 2013, 3 (2) : 121-142. doi: 10.3934/mcrf.2013.3.121
[7]

Yu Zhang, Yanyan Zhang. Riemann problems for a class of coupled hyperbolic systems of conservation laws with a source term. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1523-1545. doi: 10.3934/cpaa.2019073
[8]

Alberto Bressan, Fang Yu. Continuous Riemann solvers for traffic flow at a junction. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4149-4171. doi: 10.3934/dcds.2015.35.4149
[9]

Avner Friedman. Conservation laws in mathematical biology. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3081-3097. doi: 10.3934/dcds.2012.32.3081
[10]

Mauro Garavello. A review of conservation laws on networks. Networks & Heterogeneous Media, 2010, 5 (3) : 565-581. doi: 10.3934/nhm.2010.5.565
[11]

Len G. Margolin, Roy S. Baty. Conservation laws in discrete geometry. Journal of Geometric Mechanics, 2019, 11 (2) : 187-203. doi: 10.3934/jgm.2019010
[12]

Mauro Garavello, Roberto Natalini, Benedetto Piccoli, Andrea Terracina. Conservation laws with discontinuous flux. Networks & Heterogeneous Media, 2007, 2 (1) : 159-179. doi: 10.3934/nhm.2007.2.159
[13]

Leo G. Rebholz, Dehua Wang, Zhian Wang, Camille Zerfas, Kun Zhao. Initial boundary value problems for a system of parabolic conservation laws arising from chemotaxis in multi-dimensions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3789-3838. doi: 10.3934/dcds.2019154
[14]

Wen-Xiu Ma. Conservation laws by symmetries and adjoint symmetries. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 707-721. doi: 10.3934/dcdss.2018044
[15]

Tai-Ping Liu, Shih-Hsien Yu. Hyperbolic conservation laws and dynamic systems. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 143-145. doi: 10.3934/dcds.2000.6.143
[16]

Yanbo Hu, Wancheng Sheng. The Riemann problem of conservation laws in magnetogasdynamics. Communications on Pure & Applied Analysis, 2013, 12 (2) : 755-769. doi: 10.3934/cpaa.2013.12.755
[17]

Stefano Bianchini, Elio Marconi. On the concentration of entropy for scalar conservation laws. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 73-88. doi: 10.3934/dcdss.2016.9.73
[18]

Alberto Bressan, Marta Lewicka. A uniqueness condition for hyperbolic systems of conservation laws. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 673-682. doi: 10.3934/dcds.2000.6.673
[19]

Christophe Prieur. Control of systems of conservation laws with boundary errors. Networks & Heterogeneous Media, 2009, 4 (2) : 393-407. doi: 10.3934/nhm.2009.4.393
[20]

Alberto Bressan, Khai T. Nguyen. Conservation law models for traffic flow on a network of roads. Networks & Heterogeneous Media, 2015, 10 (2) : 255-293. doi: 10.3934/nhm.2015.10.255

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (22)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences