On the Fundamental Diagram of Traffic Flow

We present a new fluid-dynamical model of traffic flow. This model generalizes the model of Aw and Rascle SIAM J. Appl. Math. , 60 (2000), pp. 916-938 and Greenberg SIAM J. Appl. Math. , 62 (2001), pp. 729-745 by prescribing a more general source term to the velocity equation. This source term can be physically motivated by experimental data, when taking into account relaxation and reaction time. In particular, the new model has a (linearly) unstable regime as observed in traffic dynamics. We develop a numerical code that solves the corresponding system of balance laws. Applying our code to a wide variety of initial data, we find the observed inverse- shape of the fundamental diagram of traffic flow.