# American Institute of Mathematical Sciences

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

 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.

Citation: Alberto Bressan, Anders Nordli. The Riemann solver for traffic flow at an intersection with buffer of vanishing size. Networks and 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. [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. [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. [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. [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. [6] M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models, AIMS Series on Applied Mathematics, Springfield, Mo., 2006. [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. [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. [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. [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. [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. [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. [13] P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.

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##### 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. [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. [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. [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. [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. [6] M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models, AIMS Series on Applied Mathematics, Springfield, Mo., 2006. [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. [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. [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. [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. [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. [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. [13] P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.
The flux $f_k$ as a function of the density $\rho$, along the $k$-th road
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 $\sum_{i\in\mathcal{I}} \gamma_i(s) \theta_{ij}\leq \omega_j$, $j\in\mathcal{O}$
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
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
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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