Suppose that is some observable function (field variable) that
determines the density of **M** through a relationship of the form

In practice, the constitutive relationship between the
field variable **u** and
the flux often takes one of the following forms:

or

A mathematical model of traffic flow affords an example of a pair of
constituitive
relationships of the form of Eq. 4
and Eq. 5.
In this case the field variable and
the traffic density are the same, with .
One example of a flux rule of the form
of Eq. 5 for traffic flow would have **q = uv** where traffic
speed **v**
is given by ; **V** is a constant representing maximum speed (at
low density); **U** represents the
traffic density at which traffic stalls.
Thus, in this case, Eq. 5 would read .
Conservation law PDEs with the flux defined by Eq. 5 are of
hyperbolic type.
A numerical treatment of hyperbolic PDEs requires a thorough understanding of
the notion of characteristic curves.
A discussion of numerical methods for hyperbolic PDEs
is beyond the scope of this brief introduction to the numerical treatment of
PDEs by the finite difference method.