American Institute of Mathematical Sciences

Advanced
September  2015, 35(9): 4149-4171. doi: 10.3934/dcds.2015.35.4149

Continuous Riemann solvers for traffic flow at a junction

Alberto Bressan 1, and  Fang Yu 2,

1. 

Department of Mathematics, Penn State University, University Park, Pa.16802
2. 

Department of Mathematics, Penn State University, University Park, PA. 16802, United States

Received  May 2014 Revised  August 2014 Published  April 2015

The paper studies a class of conservation law models for traffic flow on a family of roads, near a junction. A Riemann Solver is constructed, where the incoming and outgoing fluxes depend Hölder continuously on the traffic density and on the drivers' turning preferences. However, various examples show that, if junction conditions are assigned in terms of Riemann Solvers, then the Cauchy problem on a network of roads can be ill posed, even for initial data having small total variation.
Keywords: network of roads., Riemann solvers, traffic flow.
Mathematics Subject Classification: Primary: 35Q93, 35L65; Secondary: 90B2.
Citation: 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
References:
[1]

A. Bressan, Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem,, Oxford University Press, (2000). Google Scholar

[2]

A. Bressan, S. Canic, M. Garavello, M. Herty and B. Piccoli, Flow on networks: Recent results and perspectives,, EMS Surv. Math. Sci., 1 (2014), 47. doi: 10.4171/EMSS/2. Google Scholar

[3]

A. Bressan and K. Han, Existence of optima and equilibria for traffic flow on networks,, Networks & Heter. Media, 8 (2013), 627. doi: 10.3934/nhm.2013.8.627. Google Scholar

[4]

A. Bressan and K. Nguyen, Conservation law models for traffic flow on a network of roads,, Networks & Heter. Media, (). Google Scholar

[5]

A. Bressan and K. Nguyen, Optima and equilibria for traffic flow on networks with backward propagating queues,, Networks & Heter. Media, (). Google Scholar

[6]

A. Bressan and W. Shen, Uniqueness for discontinuous O.D.E. and conservation laws,, Nonlinear Analysis, 34 (1998), 637. doi: 10.1016/S0362-546X(97)00590-7. Google Scholar

[7]

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

[8]

C. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law,, J. Math. Anal. Appl., 38 (1972), 33. doi: 10.1016/0022-247X(72)90114-X. Google Scholar

[9]

C. Daganzo, Fundamentals of Transportation and Traffic Operations,, Pergamon-Elsevier, (1997). Google Scholar

[10]

L. C. Evans, Partial Differential Equations,, $2^{nd}$ edition, (2010). Google Scholar

[11]

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

[12]

M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models,, in AIMS Series on Applied Mathematics, (2006). Google Scholar

[13]

M. Garavello and B. Piccoli, Conservation laws on complex networks,, Ann. Inst. H. Poincaré, 26 (2009), 1925. doi: 10.1016/j.anihpc.2009.04.001. Google Scholar

[14]

M. Herty, S. Moutari and M. Rascle, Optimization criteria for modeling intersections of vehicular traffic flow,, Netw. Heter. Media., 1 (2006), 275. doi: 10.3934/nhm.2006.1.275. Google Scholar

[15]

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

[16]

C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows,, ESAIM Control Optim. Calc. Var., 19 (2013), 129. doi: 10.1051/cocv/2012002. Google Scholar

[17]

M. Lighthill and G. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads,, Proceedings of the Royal Society of London: Series A, 229 (1955), 317. doi: 10.1098/rspa.1955.0089. Google Scholar

[18]

P. I. Richards, Shock waves on the highway,, Oper. Res., 4 (1956), 42. doi: 10.1287/opre.4.1.42. Google Scholar

[19]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, $2^{nd}$ edition, (1994). doi: 10.1007/978-1-4612-0873-0. Google Scholar

show all references

References:
[1]

A. Bressan, Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem,, Oxford University Press, (2000). Google Scholar

[2]

A. Bressan, S. Canic, M. Garavello, M. Herty and B. Piccoli, Flow on networks: Recent results and perspectives,, EMS Surv. Math. Sci., 1 (2014), 47. doi: 10.4171/EMSS/2. Google Scholar

[3]

A. Bressan and K. Han, Existence of optima and equilibria for traffic flow on networks,, Networks & Heter. Media, 8 (2013), 627. doi: 10.3934/nhm.2013.8.627. Google Scholar

[4]

A. Bressan and K. Nguyen, Conservation law models for traffic flow on a network of roads,, Networks & Heter. Media, (). Google Scholar

[5]

A. Bressan and K. Nguyen, Optima and equilibria for traffic flow on networks with backward propagating queues,, Networks & Heter. Media, (). Google Scholar

[6]

A. Bressan and W. Shen, Uniqueness for discontinuous O.D.E. and conservation laws,, Nonlinear Analysis, 34 (1998), 637. doi: 10.1016/S0362-546X(97)00590-7. Google Scholar

[7]

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

[8]

C. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law,, J. Math. Anal. Appl., 38 (1972), 33. doi: 10.1016/0022-247X(72)90114-X. Google Scholar

[9]

C. Daganzo, Fundamentals of Transportation and Traffic Operations,, Pergamon-Elsevier, (1997). Google Scholar

[10]

L. C. Evans, Partial Differential Equations,, $2^{nd}$ edition, (2010). Google Scholar

[11]

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

[12]

M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models,, in AIMS Series on Applied Mathematics, (2006). Google Scholar

[13]

M. Garavello and B. Piccoli, Conservation laws on complex networks,, Ann. Inst. H. Poincaré, 26 (2009), 1925. doi: 10.1016/j.anihpc.2009.04.001. Google Scholar

[14]

M. Herty, S. Moutari and M. Rascle, Optimization criteria for modeling intersections of vehicular traffic flow,, Netw. Heter. Media., 1 (2006), 275. doi: 10.3934/nhm.2006.1.275. Google Scholar

[15]

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

[16]

C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows,, ESAIM Control Optim. Calc. Var., 19 (2013), 129. doi: 10.1051/cocv/2012002. Google Scholar

[17]

M. Lighthill and G. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads,, Proceedings of the Royal Society of London: Series A, 229 (1955), 317. doi: 10.1098/rspa.1955.0089. Google Scholar

[18]

P. I. Richards, Shock waves on the highway,, Oper. Res., 4 (1956), 42. doi: 10.1287/opre.4.1.42. Google Scholar

[19]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, $2^{nd}$ edition, (1994). doi: 10.1007/978-1-4612-0873-0. Google Scholar

[1]

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
[2]

Ángela Jiménez-Casas, Aníbal Rodríguez-Bernal. Linear model of traffic flow in an isolated network. Conference Publications, 2015, 2015 (special) : 670-677. doi: 10.3934/proc.2015.0670
[3]

Alberto Bressan, Anders Nordli. The Riemann solver for traffic flow at an intersection with buffer of vanishing size. Networks & Heterogeneous Media, 2017, 12 (2) : 173-189. doi: 10.3934/nhm.2017007
[4]

David J. Silvester, Alex Bespalov, Catherine E. Powell. A framework for the development of implicit solvers for incompressible flow problems. Discrete & Continuous Dynamical Systems - S, 2012, 5 (6) : 1195-1221. doi: 10.3934/dcdss.2012.5.1195
[5]

Dengfeng Sun, Issam S. Strub, Alexandre M. Bayen. Comparison of the performance of four Eulerian network flow models for strategic air traffic management. Networks & Heterogeneous Media, 2007, 2 (4) : 569-595. doi: 10.3934/nhm.2007.2.569
[6]

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
[7]

Yacine Chitour, Benedetto Piccoli. Traffic circles and timing of traffic lights for cars flow. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 599-630. doi: 10.3934/dcdsb.2005.5.599
[8]

Yinfei Li, Shuping Chen. Optimal traffic signal control for an $M\times N$ traffic network. Journal of Industrial & Management Optimization, 2008, 4 (4) : 661-672. doi: 10.3934/jimo.2008.4.661
[9]

R.L. Sheu, M.J. Ting, I.L. Wang. Maximum flow problem in the distribution network. Journal of Industrial & Management Optimization, 2006, 2 (3) : 237-254. doi: 10.3934/jimo.2006.2.237
[10]

Mary Luz Mouronte, Rosa María Benito. Structural analysis and traffic flow in the transport networks of Madrid. Networks & Heterogeneous Media, 2015, 10 (1) : 127-148. doi: 10.3934/nhm.2015.10.127
[11]

Gabriella Bretti, Roberto Natalini, Benedetto Piccoli. Numerical approximations of a traffic flow model on networks. Networks & Heterogeneous Media, 2006, 1 (1) : 57-84. doi: 10.3934/nhm.2006.1.57
[12]

Gabriella Bretti, Roberto Natalini, Benedetto Piccoli. Fast algorithms for the approximation of a traffic flow model on networks. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 427-448. doi: 10.3934/dcdsb.2006.6.427
[13]

Tong Li. Qualitative analysis of some PDE models of traffic flow. Networks & Heterogeneous Media, 2013, 8 (3) : 773-781. doi: 10.3934/nhm.2013.8.773
[14]

Rinaldo M. Colombo, Andrea Corli. Dynamic parameters identification in traffic flow modeling. Conference Publications, 2005, 2005 (Special) : 190-199. doi: 10.3934/proc.2005.2005.190
[15]

Paola Goatin. Traffic flow models with phase transitions on road networks. Networks & Heterogeneous Media, 2009, 4 (2) : 287-301. doi: 10.3934/nhm.2009.4.287
[16]

Alberto Bressan, Ke Han. Existence of optima and equilibria for traffic flow on networks. Networks & Heterogeneous Media, 2013, 8 (3) : 627-648. doi: 10.3934/nhm.2013.8.627
[17]

Wen Shen, Karim Shikh-Khalil. Traveling waves for a microscopic model of traffic flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2571-2589. doi: 10.3934/dcds.2018108
[18]

Michael Herty, J.-P. Lebacque, S. Moutari. A novel model for intersections of vehicular traffic flow. Networks & Heterogeneous Media, 2009, 4 (4) : 813-826. doi: 10.3934/nhm.2009.4.813
[19]

Luisa Fermo, Andrea Tosin. Fundamental diagrams for kinetic equations of traffic flow. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 449-462. doi: 10.3934/dcdss.2014.7.449
[20]

Michael Herty, S. Moutari, M. Rascle. Optimization criteria for modelling intersections of vehicular traffic flow. Networks & Heterogeneous Media, 2006, 1 (2) : 275-294. doi: 10.3934/nhm.2006.1.275

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences