$\rho_\text{L}<\rho_\text{M}$ the 1-wave is a shock wave (moving with speed $s = \frac{\rho_\text{M}V(\rho_\text{M},w_\text{M}) - \rho_\text{L}V(\rho_\text{L},w_\text{L})}{\rho_\text{M}-\rho_\text{L}}$, given by the Rankine-Hugoniot conditions [@Evans1998]), while for $\rho_\text{L}>\rho_\text{M}$ it is a rarefaction wave. The condition $w_\text{L}\ge u_\text{R}$ means that drivers on the left wish to drive at least as fast as the vehicles on the right are driving. If this is not the case, i.e., if $w_\text{L} < u_\text{R}$, then there is no non–negative density at which the 1-wave and the 2-wave intersect. Here, a vacuum state will be generated, analogously to the construction for the ARZ model [@AwRascle2000; @Rascle2002]. The left state $(\rho_\text{L},w_\text{L})$ is connected by a rarefaction wave to a left vacuum state $(0,w_\text{L})$; this state is connected to a right vacuum state $(0,u_\text{R})$ via another rarefaction (which is feasible because $w_\text{L} < u_\text{R}$); and this then connects to the right state $(\rho_\text{R},w_\text{R})$ via a 2-contact discontinuity.

### Relaxation of GARZ to LWR {#subsubsec:relaxtion_GARZ_LWR}

Smooth solutions to the Cauchy problem of the *inhomogeneous* GARZ model relax in time towards solutions of the LWR model , because $w$ tends to $w_\text{eq}$ along characteristic curves. Note that for general relaxation systems, convergence to a first-order equation is only warranted if a sub-characteristic condition is 