Collective Behavior: Macroscopic versus Kinetic Descriptions

Pedestrian Flow Models with Slowndown Interactions

Alexander Kurganov

Tulane University
[SLIDES]

In this talk, I will present one-dimensional models for the behavior of pedestrians in a narrow street or corridor. I will first introduce a microscopic system by formulating a stochastic cellular automata model with explicit rules for pedestrians moving in two opposite directions. I will then derive coarse-grained mesoscopic and macroscopic analogs that lead to the coupled system of PDEs for the density of the pedestrian traffic. The obtained PDE system is of a mixed hyperbolic-elliptic type and therefore, I will rigorously derive higher-order nonlinear diffusive corrections for the macroscopic PDE model. Finally, I will present a series of numerical examples, in which we compare and contrast the behavior of the microscopic stochastic model and the resulting coarse-grained PDEs for various parameter settings and initial conditions in a series of numerical experiments. I will also demonstrate that the nonlinear diffusion is essential for reproducing the behavior of the stochastic system in the nonhyperbolic regime.