Microscopic and Macroscopic Models for Vehicular and Pedestrian Flows

Abstract

In this chapter we develop and study microscopic and macroscopic models for car and pedestrian flows. After a general introduction on the basic features of the flows under consideration, we develop our models. We first introduce at the microscopic level few basic and well established interaction rules. We then encode them in analytical models and obtain at the microscopic level a first order follow-the-leader model and at the macroscopic level a scalar conservation law (an hyperbolic partial differential equation). These models are then generalized to the cases of heterogeneous flows and flows across point-wise inhomogeneities characterized by reduced capacity, such as doors or toll gates. At last we also consider a crowd in panic situations. Some examples are proposed and serve as benchmarks to point out the ability of the models to reproduce realistic scenarios. The solutions to these examples are constructed by applying a wave-front tracking algorithm based on Riemann solvers. We also show how macroscopic models can be obtained from a microscopic model as many-particle limit.