Pedestrian Flow Models with Slowdown Interactions

TL;DR

This paper develops and analyzes multi-scale models for pedestrian flow in narrow corridors, highlighting the importance of nonlinear diffusion in accurately capturing stochastic behavior.

Contribution

It introduces a new stochastic cellular automata model and derives coupled PDEs, including higher-order nonlinear diffusive corrections, for pedestrian dynamics.

Findings

Nonlinear diffusion is crucial for modeling nonhyperbolic regimes.

The PDE models effectively replicate stochastic system behavior.

Numerical experiments validate the importance of diffusive corrections.

Abstract

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.