# 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 & Heterogeneous Media, 2017, 12 (2) : 173-189. doi: 10.3934/nhm.2017007
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##### References:
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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