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

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]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301

[2]

Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185

[3]

Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097

[4]

Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020170

[5]

Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto Tesei. Signed Radon measure-valued solutions of flux saturated scalar conservation laws. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3143-3169. doi: 10.3934/dcds.2020041

[6]

Caterina Balzotti, Simone Göttlich. A two-dimensional multi-class traffic flow model. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020034

[7]

Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129

[8]

Shin-Ichiro Ei, Shyuh-Yaur Tzeng. Spike solutions for a mass conservation reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3357-3374. doi: 10.3934/dcds.2020049

[9]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[10]

Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020385

[11]

Vieri Benci, Sunra Mosconi, Marco Squassina. Preface: Applications of mathematical analysis to problems in theoretical physics. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020446

[12]

François Ledrappier. Three problems solved by Sébastien Gouëzel. Journal of Modern Dynamics, 2020, 16: 373-387. doi: 10.3934/jmd.2020015

[13]

Marek Macák, Róbert Čunderlík, Karol Mikula, Zuzana Minarechová. Computational optimization in solving the geodetic boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 987-999. doi: 10.3934/dcdss.2020381

[14]

Simone Göttlich, Elisa Iacomini, Thomas Jung. Properties of the LWR model with time delay. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020032

[15]

Ténan Yeo. Stochastic and deterministic SIS patch model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021012

[16]

M. Dambrine, B. Puig, G. Vallet. A mathematical model for marine dinoflagellates blooms. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 615-633. doi: 10.3934/dcdss.2020424

[17]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[18]

Shiqiu Fu, Kanishka Perera. On a class of semipositone problems with singular Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020452

[19]

Zhiyan Ding, Qin Li, Jianfeng Lu. Ensemble Kalman Inversion for nonlinear problems: Weights, consistency, and variance bounds. Foundations of Data Science, 2020  doi: 10.3934/fods.2020018

[20]

Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang. Duality between range and no-response tests and its application for inverse problems. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020072

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (29)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]