A fluid-dynamic model for traffic flow on a road network
LWR Model

The basic macroscopic model is based on the conservation laws
              formulation, expressed as the conservation of cars (nonlinear equation):

$\displaystyle \partial_t \rho + \partial_x f(\rho)= 0,$ (1)

where

$\displaystyle \rho=\rho(t,x),$

and

$\displaystyle f(\rho),$

are, respectively, the density of cars and the flux function.

This model properly reveals shocks formation and propagation backwards along the road.
An example: a bottleneck



   The fundamental traffic variables are velocity, density and traffic flow:

    Velocity:
       instead of measuring the speed of a single car, consider a mean velocity;

    Density:
      the number of cars per distance unit;

    Traffic Flow:
       the average number of cars passing per time unit.






The velocity

    The velocity of cars at any point of the road is a regular decreasing function of the density:

$\displaystyle v=v(\rho),$




    at maximum density (bumper-to-bumper traffic) one has:



    at very low density (no cars) corresponds the maximum speed:






The flux function

    Flow equals density times velocity:

$\displaystyle f(x,t)=\rho(x,t) v(x,t),$

    Taking the velocity as a linear function, the flux is

$\displaystyle f(\rho)= v_{max} \left(1-\frac{\rho}{\rho_{max}}\right)\rho,$

    Then, setting , the flux function assumes the following form:

$\displaystyle f(\rho)= \rho (1-\rho),$



    The flux function is concave with a unique maximum.