Kinetic Description of Social Dynamics: From Consensus to Flocking

Macroscopic Pedestrian and Traffic Flow Models: Derivation and Numerics

Alina Chertock

North Carolina State University

In this talk, I will first present a one-dimensional PDE model for the behavior of pedestrians in a narrow street or corridor. The model is derived from a microscopic cellular automata model and governed by a 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. If time permits, I will also present a new multi-class traffic flow model with Arrhenius look-ahead dynamics. Similarly to the pedestrian flow case, we first derive a cellular automatum model and then pass to the PDE limit, which is a hyperbolic system of conservation laws with global fluxes. The solution of both pedestrian and traffic PDE systems are numerically studied using the central-upwind scheme. I will present a series of numerical examples, in which we compare and contrast the behavior of the microscopic stochastic models and the corresponding PDEs for various parameter settings and initial conditions. Joint work with A. Kurganov, A. Polizzi and I. Timofeyev.