a piecewise linear function. While these different choices of functions $Q(\rho)$ lead to well-posed first-order models, the second-order models derived below call for further properties that the function $Q(\rho)$ must satisfy. In particular, the velocity function $U(\rho) = Q(\rho)/\rho$ must nowhere be constant, because otherwise hyperbolicity would be lost (see the analysis in §\[subsubsec:characteristics\]). This rules out the Newell-Daganzo flux; and as a consequence, in this paper we consider flow rate functions that resemble the shape of the Newell-Daganzo flux, but that are strictly concave.

The Aw-Rascle-Zhang Model
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The strict functional relationship between $\rho$ and $u$ is loosened in *second order models*, which augment by an evolution equation for the velocity field. Payne and Whitham proposed a model [@Payne1971; @Whitham1974] in which the vehicle velocity relaxes towards a velocity function, while also being affected by a “traffic pressure” that is analogous to a pressure in fluid dynamics. Because this pressure can lead to unrealistic solutions (vehicles going backwards on the road, shocks that overtake vehicles, etc., see [@Daganzo1995]), Aw and Rascle [@AwRascle2000], and independently Zhang [@Zhang2002], proposed a different form of “pressure” that remedies the shortcomings of the Payne-Whitham model. The homogeneous Aw-Rascle-Zhang (ARZ) model reads as $$\label{eq:aw_rascle_zhang_model_homogeneous}
\begin{split}
