A Multi-Class Homogenized Hyperbolic Model of Tra c Flow

We introduce a new homogenized hyperbolic (multi-class) tra c ow model which allows to take into account the behaviors of di erent 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 di erent cardriver pairs in the tra c, 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 à la Kruokov of a scalar conservation law, with variable coe cients, discontinuous in x. Finally, we prove that the same macroscopic homogenized model is also the hydrodynamic limit of the corresponding multi-class Follow-theLeader model.