ABSTRACT
CONTENTS 3.1 Lighthill-Whitham-Richards Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 Payne-Whitham Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3 Aw-Rascle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.4 Zhang Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.5 Pedestrian and Control Models in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . 39
In this chapter we review macroscopic traffic models and how they relate to conservation equations. We consider one-dimensional and two-dimensional vehicular and pedestrian traffic models. Traffic models can be microscopic (see [14]), mesoscopic or macroscopic (see [22], [61]). Macroscopic models treat traffic as a continuum and these are the models of interest to this dissertation. Microscopic models treat each vehicle or pedestrian as an individual entity and treats acceleration as the control variable that depends on inter-vehicular or inter-pedestrian density (see [4], [14], [44]). Mesoscopic models use kinetic models for traffic using Boltzmann equation from statistical mechanics (see [74]). Some mesoscopic models model each vehicle individually but obtain behaviour of multiple vehicles based on macroscopic traffic variables.
