A multi class homogenized hyperbolic model of traffic flow

We introduce a new homogenized hyperbolic (multiclass) traffic flow model, which allows us to take into account the behaviors of different type of vehicles (cars, trucks, buses, etc.) and drivers. We discretize the starting Lagrangian system introduced below with a Godunov scheme, and we let the mesh size h in (x, t) go to 0: the typical length (of a vehicle) and time vanish. Therefore, the variables - here (w, a) - which describe the heterogeneity of the reactions of the different car-driver pairs in the traffic, develop large oscillations when h --> 0. These (known) oscillations in (w, a) persist in time, and we describe the homogenized relations between velocity and density. We show that the velocity is the unique solution "`a la Kruzkov" of a scalar conservation law, with variable coefficients, discontinuous in x. Finally, we prove that the same macroscopic homogenized model is also the hydrodynamic limit of the corresponding multiclass Follow-the-Leader model.