A first order model in traffic flow

First order traffic flow models have been studied for long time, since they represent the simplest way to begin with the understanding of such a complex system. To our knowledge the Lighthill-Whitham-Richards (LWR) model was the first model to describe unidirectional, one-dimensional traffic flow in a highway [1], [2]. This model considers the continuity equation for the density of vehicles (number of vehicles per unit length) along the highway in terms of the flux. Such a flux must be given according to a constitutive equation which relates the flux with the density. The introduction of a constitutive relation for the flux allows us to obtain a closed equation for the density. Also, it particularizes the model and plays an important role in the behavior of the density. In the LWR model the constitutive equation taken for the flux is the Greenshields [3] expression, which tells us how the number of vehicles per unit time behave along the road. It has its maximum value at low densities and vanishes when the density grows, a situation corresponding to a completely crowded highway. It is well known that this procedure drives us to the inviscid Burgers equation, it presents the formation of shock waves and it has