to all macroscopic traffic models is the continuity equation $$\label{eq:continuity_equation}
\rho_t+(\rho u)_x = 0\;,$$ which gives the conservation of vehicles. In , the vehicle density is $\rho(x,t)$, and the vehicle velocity field is $u(x,t)$, where $x$ is the position along the road, and $t$ is time. If the road has multiple lanes (in a given direction), we consider these aggregated into the scalar field quantities $\rho$ and $u$.

The Lighthill-Whitham-Richards Model and Flow Rate Functions
------------------------------------------------------------

The simplest macroscopic traffic model, the Lighthill-Whitham-Richards (LWR) model [@LighthillWhitham1955; @Richards1956], is obtained by assuming a functional relationship between $\rho$ and $u$, i.e., $u = U(\rho)$. This turns equation into a scalar hyperbolic conservation law $$\label{eq:lighthill_whitham_richards_model}
\rho_t+(Q(\rho))_x = 0\;,$$ where the flux $Q$ is given by the flow rate function $Q(\rho) = \rho U(\rho)$. Because the LWR model is a closed model consisting of a single equation, it is denoted a *first order model*. The velocity function $U(\rho)$ is commonly assumed to be decreasing in $\rho$ with $U(\rho_\text{max}) = 0$ for some maximal vehicle density $\rho_\text{max}>0$.

Popular examples of flow rate functions are the Greenshields flux [@Greenshields1935], in which $Q(\rho)$ is a quadratic function, and the Newell-Daganzo flux [@Newell1993; @Daganzo1994], in which $Q(\rho)$ is 