from the solution of the Riemann problems of the GARZ model (see below), it follows that the dynamics of the model never generate values $w\notin [w_\text{min},w_\text{max}]$ or $\rho\notin [0,\rho_\text{max})$. Hence, analogous to the ARZ model (cf. [@AwRascle2000]), the domain $\mathcal{D}_V$ is an invariant region.

### Characteristics and Associated Fields {#subsubsec:characteristics}

The homogeneous part of the GARZ model is a conservation law of the form $${\mathbf{U}}_t + {\mathbf{F}}({\mathbf{U}})_x = 0\;,$$ where $${\mathbf{U}} = \begin{pmatrix} \rho \\ q \end{pmatrix}
\quad\text{and}\quad
{\mathbf{F}}({\mathbf{U}}) = \begin{pmatrix} u\rho \\ uq \end{pmatrix}\;,
\quad\text{where}\quad
u = V(\rho,q/\rho)\;.$$ The Jacobian of the flux function ${\mathbf{F}}({\mathbf{U}})$ is $$\nabla{\mathbf{F}}({\mathbf{U}})
= \begin{pmatrix} u+\rho{\frac{\partialu}{\partial\rho}} & \rho{\frac{\partialu}{\partialq}} \\
q{\frac{\partialu}{\partial\rho}} & u+q{\frac{\partialu}{\partialq}} \end{pmatrix}\;,$$ and its eigenvalues and associated eigenvectors are $$\begin{aligned}
\lambda^{(1)} &= u+\rho{\frac{\partialu}{\partial\rho}}+q{\frac{\partialu}{\partialq}} = u+\rho{\frac{\partialV}{\partial\rho}}
\quad\text{with}\quad
{\mathbf{\gamma}}^{(1)} = \begin{pmatrix} \rho \\ q \end{pmatrix}
\quad\text{and thus}\quad
\nabla\lambda^{(1)}\cdot {\mathbf{\gamma}}^{(1)} \neq 0\;,
\intertext{and}
\lambda^{(2)} &= u \hspace{10em}
\quad\text{with}\quad
{\mathbf{\gamma}}^{(2)}
= \begin{pmatrix} -{\frac{\partialu}{\partialq}} \\ \phantom{-}{\frac{\partialu}{\partial\rho}} \end{pmatrix}
\quad\text{and thus}\quad
\nabla\lambda^{(2)}\cdot {\mathbf{\gamma}}^{(2)} = 0\;.\end{aligned}$$ Hence, like the ARZ model , the GARZ model is strictly hyperbolic for $\rho>0$. One of its characteristic velocities, $\lambda^{(1)}$, is slower than the vehicles (i.e., $\lambda^{(1)}<u$) and its associated field is genuinely nonlinear (i.e., it corresponds to shocks and rarefaction waves, see below). Its other 