Appendix 2.1.A. Derivation of a PDE describing traffic flow

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Appendix A. Derivation of a PDE describing traffic flow


The purpose of this discussion is to derive a toy-model PDE that describes a congested one-dimensional highway (in one direction). Let

It is directly implied by definition of $\rho(x,t)$ is \begin{equation} N(t,a,b)=\int_{a}^{b}\rho(t,x) dx. \label{eq-2.1A.1} \end{equation}

By definition of $q$ and conservation of cars we have: \begin{multline} \frac{\partial N}{\partial t}(t,a,b)=\lim_{h \rightarrow 0} \frac{N(t+h,a,b)-N(t,a,b)}{h} \\ =\lim_{h \rightarrow 0} \frac{h(q(t,a)-q(t,b))}{h} =q(t,a)-q(t,b) \label{eq-2.1A.2} \end{multline}

Differentiating (\ref{eq-2.1A.1}) with respect to $t$ \begin{equation*} \frac{\partial N}{\partial t}=\int_{a}^{b}\rho_t(t,x) dx \end{equation*} and making it equal to (\ref{eq-2.1A.2}) we get the integral form of "conservation of cars": \begin{equation*} \int_{a}^{b}\rho_t(t,x) dx=q(t,a)-q(t,b). \end{equation*}

Since $a$ and $b$ are arbitrary, it implies that $\rho_t=-q_x$. The PDE \begin{equation} \rho_t+q_x=0 \label{eq-2.1A.3} \end{equation} is conservation of cars equation.

After equation (\ref{eq-2.1A.3}) or more general equation \begin{equation} \rho_t+ \rho_x=f(x,t) \label{eq-2.1A.4} \end{equation} (where $f=f_{in}-f_{out}$, $f_{in}dx dt$ and $f_{out}dx dt$ are numbers of cars entering/exiting highway for time $dt$ at the segment of the length $dx$) has been derived we need to connect $\rho$ and $q$.

The simplest is $q=c \rho$ with a constant $c$: all cars are moving with the same speed $c$. Then (\ref{eq-2.1A.3}) becomes \begin{equation} \rho_t+ c \rho_x=0. \label{eq-2.1A.5} \end{equation} However more realistic would be $c=c(\rho)$ being monotone decreasing function of $\rho$ with $c(0)=c_0$ (speed on empty highway) and $c(\bar{\rho})=0$ where $\bar{\rho}$ is a density where movement is impossible. Assume that $q(\rho) = c(\rho)\rho$ has a single maximum at $\rho^*$. Then \begin{equation} \rho_t+ v \rho_x=0. \label{eq-2.1A.6} \end{equation} with \begin{equation} v=v(\rho)= q'(\rho) = c(\rho)+ c' (\rho)\rho \label{eq-2.1A.7} \end{equation} where $'$ is a derivative with respect to $\rho$. Therefore $\rho$ remains constant along integral line $x - v(\rho) t=\const$.

$v=v(\rho)$ is the phase velocity namely the velocity with which point where density equals given density $\rho$ is moving. Here $v(\rho)< c(\rho)$ (because $c'(\rho)<0$), so phase velocity is less than the group velocity $c(\rho)$, which means the velocity of the cars (simply cars may join the congestion from behind and leave it from its front). Further, $v(\rho)>0$ as $\rho<\rho^*$ and $v(\rho)<0$ as $\rho>\rho^*$; in the latter case the congestion moves backward: the jam grows faster than it moves.

Also the integral lines may intersect (loose and faster moving congestion catches up with dense and slower congestion). When it happens $\rho$ becomes discontinuous, (\ref{eq-2.1A.3}) still holds but (\ref{eq-2.1A.6}) fails (it is no more equivalent to (\ref{eq-2.1A.3})) and the theory becomes really complicated.

traffic

Remark 1.

  1. The notions of phase velocity and group velocity are important in study of waves, either nonlinear waves as in this case, but also multidimensional linear waves and dispersive waves.
  2. In the toy-model of gas dynamics $c(\rho)=\rho$, or more general $c'(\rho)>0$ and $v(\rho)>c(\rho)$.

$\Leftarrow$  $\Uparrow$  $\Rightarrow$