Pedestrian Flow Models with Slowdown Interactions
  In this paper, we introduce and study one-dimensional models for the behavior of pedestrians in a narrow street or corridor. We begin at the microscopic level by formulating a stochastic cellular automata model with explicit rules for pedestrians moving in two opposite directions. Coarse-grained mesoscopic and macroscopic analogs are derived leading 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, we rigorously derive higher-order nonlinear diffusive corrections for the macroscopic PDE model. We perform numerical experiments, which compare and contrast the behavior of the microscopic stochastic model and the resulting coarse-grained PDEs for various parameter settings and initial conditions. We also demonstrate that the nonlinear diffusion is essential for reproducing the behavior of the stochastic system in the nonhyperbolic regime. 
Alina Chertock and Alexander Kurganov and Anthony Polizzi and Ilya   Timofeyev
math.PR nlin.CG