June  2015, 10(2): 255-293. doi: 10.3934/nhm.2015.10.255

Conservation law models for traffic flow on a network of roads

1. 

Penn State University Mathematics Dept., University Park, State College, PA 16802

2. 

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

Received  June 2014 Revised  January 2015 Published  April 2015

The paper develops a model of traffic flow near an intersection, where drivers seeking to enter a congested road wait in a buffer of limited capacity. Initial data comprise the vehicle density on each road, together with the percentage of drivers approaching the intersection who wish to turn into each of the outgoing roads.
    If the queue sizes within the buffer are known, then the initial-boundary value problems become decoupled and can be independently solved along each incoming road. Three variational problems are introduced, related to different kind of boundary conditions. From the value functions, one recovers the traffic density along each incoming or outgoing road by a Lax type formula.
    Conversely, if these value functions are known, then the queue sizes can be determined by balancing the boundary fluxes of all incoming and outgoing roads. In this way one obtains a contractive transformation, whose fixed point yields the unique solution of the Cauchy problem for traffic flow in an neighborhood of the intersection.
    The present model accounts for backward propagation of queues along roads leading to a crowded intersection, it achieves well-posedness for general $L^\infty $ data, and continuity w.r.t. weak convergence of the initial densities.
Citation: 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
References:
[1]

M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,, Birkhäuser, (1997).  doi: 10.1007/978-0-8176-4755-1.  Google Scholar

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C. I. Bardos, A. Y. Leroux and J. C. Nedelec, First order quasilinear equations with boundary conditions,, Comm. P.D.E., 4 (1979), 1017.  doi: 10.1080/03605307908820117.  Google Scholar

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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

[4]

A. Bressan and K. Han, Optima and equilibria for a model of traffic flow,, SIAM J. Math. Anal., 43 (2011), 2384.  doi: 10.1137/110825145.  Google Scholar

[5]

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

[6]

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

[7]

A. Bressan and F. Yu, Continuous Riemann solvers for traffic flow at a junction,, to appear in Discr. Cont. Dyn. Syst., ().   Google Scholar

[8]

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

[9]

C. D'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization of Supply Chains. A Continuous Approach,, SIAM, (2010).  doi: 10.1137/1.9780898717600.  Google Scholar

[10]

L. C. Evans, Partial Differential Equations,, Second edition, (2010).   Google Scholar

[11]

T. Friesz, Dynamic Optimization and Differential Games,, Springer, (2010).  doi: 10.1007/978-0-387-72778-3.  Google Scholar

[12]

T. Friesz and K. Han, Dynamic Network User Equilibrium,, Springer-Verlag, ().   Google Scholar

[13]

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

[14]

M. Garavello and B. Piccoli, Source-destination flow on a road network,, Commun. Math. Sci., 3 (2005), 261.  doi: 10.4310/CMS.2005.v3.n3.a1.  Google Scholar

[15]

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

[16]

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

[17]

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), (2013), 143.  doi: 10.1007/978-1-4614-6243-9_6.  Google Scholar

[18]

M. Herty, C. Kirchner, S. Moutari and M. Rascle, Multicommodity flows on road networks,, Commun. Math. Sci., 6 (2008), 171.  doi: 10.4310/CMS.2008.v6.n1.a8.  Google Scholar

[19]

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

[20]

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

[21]

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

[22]

P. D. Lax, Hyperbolic systems of conservation laws,, Comm. Pure Appl. Math., 10 (1957), 537.  doi: 10.1002/cpa.3160100406.  Google Scholar

[23]

P. Le Floch, Explicit formula for scalar non-linear conservation laws with boundary condition,, Math. Methods Appl. Sciences, 10 (1988), 265.  doi: 10.1002/mma.1670100305.  Google Scholar

[24]

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

[25]

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

[26]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, Second edition, (1994).  doi: 10.1007/978-1-4612-0873-0.  Google Scholar

show all references

References:
[1]

M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,, Birkhäuser, (1997).  doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[2]

C. I. Bardos, A. Y. Leroux and J. C. Nedelec, First order quasilinear equations with boundary conditions,, Comm. P.D.E., 4 (1979), 1017.  doi: 10.1080/03605307908820117.  Google Scholar

[3]

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

[4]

A. Bressan and K. Han, Optima and equilibria for a model of traffic flow,, SIAM J. Math. Anal., 43 (2011), 2384.  doi: 10.1137/110825145.  Google Scholar

[5]

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

[6]

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

[7]

A. Bressan and F. Yu, Continuous Riemann solvers for traffic flow at a junction,, to appear in Discr. Cont. Dyn. Syst., ().   Google Scholar

[8]

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

[9]

C. D'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization of Supply Chains. A Continuous Approach,, SIAM, (2010).  doi: 10.1137/1.9780898717600.  Google Scholar

[10]

L. C. Evans, Partial Differential Equations,, Second edition, (2010).   Google Scholar

[11]

T. Friesz, Dynamic Optimization and Differential Games,, Springer, (2010).  doi: 10.1007/978-0-387-72778-3.  Google Scholar

[12]

T. Friesz and K. Han, Dynamic Network User Equilibrium,, Springer-Verlag, ().   Google Scholar

[13]

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

[14]

M. Garavello and B. Piccoli, Source-destination flow on a road network,, Commun. Math. Sci., 3 (2005), 261.  doi: 10.4310/CMS.2005.v3.n3.a1.  Google Scholar

[15]

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

[16]

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

[17]

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), (2013), 143.  doi: 10.1007/978-1-4614-6243-9_6.  Google Scholar

[18]

M. Herty, C. Kirchner, S. Moutari and M. Rascle, Multicommodity flows on road networks,, Commun. Math. Sci., 6 (2008), 171.  doi: 10.4310/CMS.2008.v6.n1.a8.  Google Scholar

[19]

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

[20]

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

[21]

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

[22]

P. D. Lax, Hyperbolic systems of conservation laws,, Comm. Pure Appl. Math., 10 (1957), 537.  doi: 10.1002/cpa.3160100406.  Google Scholar

[23]

P. Le Floch, Explicit formula for scalar non-linear conservation laws with boundary condition,, Math. Methods Appl. Sciences, 10 (1988), 265.  doi: 10.1002/mma.1670100305.  Google Scholar

[24]

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

[25]

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

[26]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, Second edition, (1994).  doi: 10.1007/978-1-4612-0873-0.  Google Scholar

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