Derivation of Continuum Traffic Flow Models from Microscopic Follow-the-Leader Models

In this paper we establish a connection between a microscopic follow-the-leader model based on ordinary differential equations and a semidiscretization of a macroscopic continuum model based on a conservation law. Naturally, it also turns out that the natural discretization of the conservation law in Lagrangian coordinates is equivalent to a straightforward time discretization of the microscopic model. We also show rigorously that, at least in the homogeneous case, the macroscopic model can be viewed as the limit of the time discretization of the microscopic model as the number of vehicles increases, with a scaling in space and time (a zoom) for which the density and the velocity remain fixed. Moreover, a numerical investigation and comparison is presented for the different models.