for the ARZ model, the forms and are equivalent, for the GARZ model the relaxations and are not. Specifically, if ${\frac{\partialV}{\partialw}}(\rho,w_\text{eq})>0$, then both forms relax to the same limit, but at different rates. A simple Taylor expansion yields that for $w$ nearby $w_\text{eq}$, the relaxation in happens ${\frac{\partialV}{\partialw}}(\rho,w_\text{eq})$ times as rapidly as the relaxation in .

Finally, in line with , the GARZ model is meant to be interpreted in the conservative form $$\label{eq:GARZ_conservative}
\begin{split}
\rho_t+{\left(V(\rho,q/\rho)\rho\right)}_x &= 0\;, \\
q_t+{\left(V(\rho,q/\rho)q\right)}_x &= \tfrac{1}{\tau}{\left(Q_\text{eq}(\rho)-Q(\rho,q/\rho)\right)}\;,
\end{split}$$ where the two conserved variables are $\rho$ and $q = \rho w$. Moreover, $Q_\text{eq}(\rho) = \rho U_\text{eq}(\rho)$ and $Q(\rho,w) = \rho V(\rho,w)$.

Properties of the GARZ Model
----------------------------

Most properties of the classical ARZ model transfer over to its generalization, the GARZ model. Here we only collect relevant results, many of which have been presented in [@LebacqueMammarHajSalem2007], or that are relatively straightforward generalizations of the results given in [@AwRascle2000; @Greenberg2001; @Rascle2002]. The theoretical results presented below are in particular important for the data-fitting methodologies conducted in §\[sec:data-fitted\_models\], and for the interpretation of the results obtained in §\[sec:inhomogeneous\_models\].

### Regions of GARZ Variables and Inverse Velocity Functions {#subsubsec:regions}

Because of their relations and because of their physical meaning, 