# [Traffic Flow] Concrete example of conservation equation?

1. Feb 11, 2016

### cmkluza

This is it; most likely the last time I bother the people of this website with my questions on traffic flow.

I'm trying to figure out some concrete examples to demonstrate utilization of the conservation equation in traffic flow:
$$\frac{\partial \rho }{\partial t} + \frac{\partial q(\rho )}{\partial x} = \frac{\partial \rho }{\partial t} + \frac{\partial \rho v(\rho )}{\partial x} = 0$$
where $\rho$ is density in $\frac{num. vehicles}{distance}$, $q$ is flow in $\frac{num. vehicles}{time}$, $v$ is speed/velocity in $\frac{distance}{time}$, $t$ is time, and $x$ is distance of a segment of road. $v(\rho )$ can be expressed as follows:
$$v(\rho ) = v_{max}(1 - \frac{\rho }{\rho_{max}})$$
where $v_{max}$ is maximum velocity and $\rho_{max}$ is maximum density.

Is there anyone here who knows about traffic modelling well enough to suggest some concrete examples for utilization of this equation? Alternatively, could anyone tell me what variables I would need to know to substitute into the equation in order to get something out?

I guess my ultimate question here is just, how do I use this equation for modelling traffic now that I have it?

2. Feb 14, 2016

### Ssnow

The model describe you how the density $\rho$ of cars change in the time $t$. The fixed data are $\rho_{max},v_{max}$. After fixing these values the description is given by the equation $\partial_{t}\rho+\partial_{x}q(\rho)=0$. In order to have a numerical examples you can use the discretization of the equation and approximating derivatives. To find an approximate solution there are a lot of numerical algorithms as Euler, ... (''Numerical Analysis'', Burden & Faires). Interesting can be an analysis of shock waves in this description... I remember you that the model is $1$ dimensional, you consider only one direction in the street that is the $x$-axis ...

3. Feb 14, 2016