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June  2017, 12(2): 173-189. doi: 10.3934/nhm.2017007

The Riemann solver for traffic flow at an intersection with buffer of vanishing size

Alberto Bressan 1, and  Anders Nordli 2,

1. 

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

Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway

Received  December 2015 Revised  March 2016 Published  May 2017

Fund Project: The first author was partially supported by NSF, with grant DMS-1411786: "Hyperbolic Conservation Laws and Applications".

The paper examines the model of traffic flow at an intersection introduced in [2], containing a buffer with limited size. As the size of the buffer approaches zero, it is proved that the solution of the Riemann problem with buffer converges to a self-similar solution described by a specific Limit Riemann Solver (LRS). Remarkably, this new Riemann Solver depends Lipschitz continuously on all parameters.

Keywords: Hyperbolic conservation laws, continuum traffic models, Riemann problem, Riemann solver.
Mathematics Subject Classification: Primary: 35L65, 90B20; Secondary: 35R02.
Citation: 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
References:
[1]

A. Bressan, S. Canic, M. Garavello, M. Herty and B. Piccoli, Flow on networks: Recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111.  doi: 10.4171/EMSS/2.  Google Scholar

[2]

A. Bressan and K. Nguyen, Conservation law models for traffic flow on a network of roads, Netw. Heter. Media, 10 (2015), 255-293.  doi: 10.3934/nhm.2015.10.255.  Google Scholar

[3]

A. Bressan and F. Yu, Continuous Riemann solvers for traffic flow at a junction, Discr. Cont. Dyn. Syst., 35 (2015), 4149-4171.  doi: 10.3934/dcds.2015.35.4149.  Google Scholar

[4]

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

[5]

M. Garavello, Conservation laws at a node, in Nonlinear Conservation Laws and Applications (eds A. Bressan, G. Q. Chen, M. Lewicka), The IMA Volumes in Mathematics and its Applications 153 (2011), 293-302. doi: 10.1007/978-1-4419-9554-4_15.  Google Scholar

[6]

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

[7]

M. Garavello and B. Piccoli, Conservation laws on complex networks, Ann. Inst. H. Poincaré, 26 (2009), 1925-1951.  doi: 10.1016/j.anihpc.2009.04.001.  Google Scholar

[8]

M. Herty, J. P. Lebacque and S. Moutari, A novel model for intersections of vehicular traffic flow, Netw. Heterog. Media, 4 (2009), 813-826.  doi: 10.3934/nhm.2009.4.813.  Google Scholar

[9]

H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), 999-1017.  doi: 10.1137/S0036141093243289.  Google Scholar

[10]

C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM-COCV, 19 (2013), 129-166.  doi: 10.1051/cocv/2012002.  Google Scholar

[11]

P. Le Floch, Explicit formula for scalar non-linear conservation laws with boundary condition, Math. Methods Appl. Sciences, 10 (1988), 265-287.  doi: 10.1002/mma.1670100305.  Google Scholar

[12]

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), 317-345.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[13]

P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.  Google Scholar

show all references

References:
[1]

A. Bressan, S. Canic, M. Garavello, M. Herty and B. Piccoli, Flow on networks: Recent results and perspectives, EMS Surv. Math. Sci., 1 (2014), 47-111.  doi: 10.4171/EMSS/2.  Google Scholar

[2]

A. Bressan and K. Nguyen, Conservation law models for traffic flow on a network of roads, Netw. Heter. Media, 10 (2015), 255-293.  doi: 10.3934/nhm.2015.10.255.  Google Scholar

[3]

A. Bressan and F. Yu, Continuous Riemann solvers for traffic flow at a junction, Discr. Cont. Dyn. Syst., 35 (2015), 4149-4171.  doi: 10.3934/dcds.2015.35.4149.  Google Scholar

[4]

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

[5]

M. Garavello, Conservation laws at a node, in Nonlinear Conservation Laws and Applications (eds A. Bressan, G. Q. Chen, M. Lewicka), The IMA Volumes in Mathematics and its Applications 153 (2011), 293-302. doi: 10.1007/978-1-4419-9554-4_15.  Google Scholar

[6]

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

[7]

M. Garavello and B. Piccoli, Conservation laws on complex networks, Ann. Inst. H. Poincaré, 26 (2009), 1925-1951.  doi: 10.1016/j.anihpc.2009.04.001.  Google Scholar

[8]

M. Herty, J. P. Lebacque and S. Moutari, A novel model for intersections of vehicular traffic flow, Netw. Heterog. Media, 4 (2009), 813-826.  doi: 10.3934/nhm.2009.4.813.  Google Scholar

[9]

H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), 999-1017.  doi: 10.1137/S0036141093243289.  Google Scholar

[10]

C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM-COCV, 19 (2013), 129-166.  doi: 10.1051/cocv/2012002.  Google Scholar

[11]

P. Le Floch, Explicit formula for scalar non-linear conservation laws with boundary condition, Math. Methods Appl. Sciences, 10 (1988), 265-287.  doi: 10.1002/mma.1670100305.  Google Scholar

[12]

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), 317-345.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[13]

P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.  Google Scholar

Figure 1.  The flux $f_k$ as a function of the density $\rho$, along the $k$-th road
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Figure 2.  Constructing the solution of the the Riemann problem, according to the limit Riemann solver (LRS), with two incoming and two outgoing roads. The vector $\mathbf{f}=(\bar f_1,\bar f_2)$ of incoming fluxes is the largest point on the curve $\gamma$ that satisfies the two constraints $\sum_{i\in\mathcal{I}} \gamma_i(s) \theta_{ij}\leq \omega_j$, $j\in\mathcal{O}$
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Figure 3.  Left: an incoming road which is initially free. For $t_1<t<t_2$ part of the road is congested (shaded area). Right: an outgoing road which is initially congested. For $0<t<t_3$ part of the road is free (shaded area). In both cases, a shock marks the boundary between the free and the congested region
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Figure 4.  The two cases in the proof of Theorem 2.3. Left: none of the outgoing roads provides a restriction on the fluxes of the incoming roads. The queues are zero. Right: one of the outgoing roads is congested and restricts the maximum flux through the node
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Figure 5.  A case with three incoming roads. For large times, the first two roads become free, while the third road remains congested
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