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2 Constitutive Equations

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.