as the plots in Fig. \[fig:fd\_lwr\_arz\] indicate, variations in $w$ can lead to significant variations in the density at which $u_w(\rho) = 0$. However, since in reality the maximum density is largely a property of the road, it should not depend (at least not strongly) on the velocity that drivers assume when alone on the road. In order to remedy this shortcoming, the relationship between $u$, $w$, and $\rho$ must be generalized.

To that end, we consider a generalized Aw-Rascle-Zhang (GARZ) model, which is a representative of the class of generic second order models (GSOM), proposed by Lebacque, Mammar, and Haj-Salem [@LebacqueMammarHajSalem2007]. Specifically, the homogeneous ARZ model generalizes to $$\label{eq:GARZ_model}
\begin{split}
\rho_t+(\rho u)_x &= 0\;, \\
w_t+uw_x &= 0\;, \\
\text{where~}u &= V(\rho,w)\;,
\end{split}$$ where we impose the following requirements on the velocity function $V(\rho,w)$, and the associated generalized flow rate function $Q(\rho,w) = \rho V(\rho,w)$:

-   $V(\rho,w)\ge 0$, i.e., vehicles never go backwards on the road.

-   $V(0,w) = w$, i.e., we gauge the convected quantity $w$ to play the role of the empty road velocity, as in the ARZ model .

-   $\frac{\partial^2 Q}{\partial \rho^2}(\rho,w) < 0$ for $w > 0$, i.e., each flow-rate curve $Q_w(\rho) = Q(\rho,w)$ 