Moving bottlenecks for the Aw-Rascle-Zhang traffic flow model

Figure 1.  Representation of the phase plane in the fixed and in the bus reference frames

Figure 2.  Notations used in the definition of the Riemann solvers

Figure 3.  Example of an invariant domain (the shaded area) for $\mathcal{RS}^\alpha_1$ (right) and $\mathcal{RS}^\alpha_2$ (left), for $v_1>V_b$ (case (ⅱ) in Theorems 4.2 and 4.3 respectively). Geometrically, the condition (10) in Theorem 4.2 states that the first Lax curves passing through the points of intersection between the constraint line and the second Lax curves $v=v_2$ and $v=v_1$ respectively, are both above the first Lax curve $w=w_2$. A similar interpretation applies to condition (12) in Theorem 4.3

Figure 4.  Representation of the points used in Section 4.2.1. The invariant domain is the colored area. If by contradiction $h_\alpha(v_1)<w_2$ (case (a)), then the two points $(\rho^*, v^*)$ and $(\hat{\rho}, \hat{v})$ cannot both belong to $\mathcal{D}_{v_1, v_2, w_1, w_2}$. The same is for $(\rho^*, v^*)$ and $(\check{\rho}_1, \check{v}_1)$ if $h_\alpha(v_2)<w_2$ (case (b)).

Figure 5.  Representation of the points used in the proof of Lemmas 4.8 and 4.9

Figure 6.  An example of a discontinuity reconstruction for the Riemann solver $\mathcal{RS}^\alpha_1$.

Figure 7.  Comparison between the solution obtained with both conditions (16) and (17) (the dot-dashed line) and the one obtained when only (17) is enforced (the continuous line): the first has undesirable oscillation, while the latter is correct. The initial data are: $(\rho^l, v^l)=(7, 3)$, $(\rho^r, v^r)=(6, 4)$, $\alpha = 0.4$, $V_b=1.5$, $R = 15$ and $y_0=0$.

Figure 8.  Representation of the reconstruction method (v-component)

Figure 9.  Solutions obtained with the discontinuity reconstruction method for the $(\rho, v)$ coordinates. The dot-dashed line is obtained without the correction for the contact discontinuity, while the continuous line is obtained with the correction: in the first case, the velocity downstream the shock is overestimated, in the latter it is correct. The initial data are $(\rho^l, v^l)=(\rho^r, v^r)=(7, 3)$, $V_b=1$, $\alpha = 0.25$ and $y_0=0$.

Figure 10.  Spatio-temporal evolution of traffic density and bus trajectory given by $\mathcal{RS}^\alpha_1$ (top) and $\mathcal{RS}^\alpha_2$ (bottom) corresponding to the data $(\rho^l, v^l)=(9, 1)$ for $x<0$, $(\rho^r, v^r)=(2, 8)$ for $x>0$, $V_b=4$, $\alpha =0.5$, $R=15$ and $y_0=-0.1$.