Continuum approach to car-following models

A continuum version of the car-following Bando model is developed using a series expansion of the headway in terms of the density. This continuum model obeys the same stability criterion as its discrete counterpart. To compare both models we show that traveling wave solutions of the Bando model are very similar to those of the continuum model in the limit of small changes of headway. As the change of headway across the wave increases the solutions gradually diverge. Our transformation relating headway to density enables predictions of the global impact and characteristics of any car-following model using the analogous continuum model. In contrast, we show that the conventional continuum models which account for effects of pressure and dispersion predict behavior which is distinct from the global behavior of discrete models.