A pedestrian flow model with stochastic velocities: Microscopic and macroscopic approaches

Abstract

We investigate a stochastic model hierarchy for pedestrian flow. Starting from a microscopic social force model, where the pedestrians switch randomly between the two states stop-orgo, we derive an associated macroscopic model of conservation law type. Therefore we use a kinetic mean-field equation and introduce a new problem-oriented closure function. Numerical experiments are presented to compare the above models and to show their similarities.

Density dependent diffusion models for the interaction of particle ensembles with boundaries

“…Appendix B. Simulation of swarms. To compute an approximate solution to (22), we discretize with step sizes ∆x (1) , ∆x (2) , ∆t and have to iteratively determine the velocity v s , s = 1, . .…”

“…Pedestrian crowds. The dynamical behaviour of pedestrian crowds has been modeled by the deterministic microscopic social force model [22,24]. Our microscopic model ( 2) can be written in this way by considering the velocity selection mechanism G(v) as the destination force…”

“…Since the hyperbolic part of the material flow limit ( 14) and the swarming limit (22) reduces to linear transport, we solve the advective part with flux f (l) (ρ, x) = v (l) (x)ρ, l = 1, 2 using the Upwind scheme combined with dimensional splitting…”

“…Specifically, reflection laws that show the outgoing angle of the flock as a function of the incoming angle θ 0 for different scalings of the interaction potential λ have been determined. In the following, we study the boundary behaviour for swarms and for varying diffusion coefficients C to validate the macroscopic model (22).…”

“…We study the transition from microscopic models of interacting particles to the macroscopic limit describing their motion as coherent ensembles. Such problems naturally arise in the description of biological swarms such as flocks of birds [4,15,19], schools of fish [3], ant [7] or bacterial colonies [26], the movement of pedestrian crowds [22,24] or transport of material [20,32]. In some production facilities, e.g.…”