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

Figure 1.  The flux $f_k$ as a function of the density $\rho$, along the $k$-th road

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 $\sum_{i\in\mathcal{I}} \gamma_i(s) \theta_{ij}\leq \omega_j$, $j\in\mathcal{O}$

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

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

Figure 5.  A case with three incoming roads. For large times, the first two roads become free, while the third road remains congested