On Hughes' Model for Pedestrian Traffic

Abstract

In this thesis, we explore the modeling and application of the pedestrian fl w model proposed by Roger L. Hughes in 2002, a system of a hyperbolic conservation law to describe crowd densities (ρ) and an Eikonal equation to describe the path potential (φ) of this crowd:

ρt − div(ρf 2(ρ)∇φ) = 0 ||∇φ|| =1 f(p)

Throughout this work, we explain this model in the context of a one dimensional walking facility, like a bridge or a hallway, and a two dimensional one, such as an open room with obstacles and obstructions. We revisit the motivation for the model, as well as properties and qualities of the resulting weak entropy so- lutions, some of which will aid in understanding numerical results. After this, we describe numerical methods to use in order to solve the eikonal equation for the path potential, then use this quantity to solve the conservation law for the density after a certain time step. With these methods established, we proceed to provide meaningful simulations in both the 1D and 2D cases, describing what our results mean from a mathematical perspective, then a real-life explanation, alongside a brief critique of the simulated situations with comments on how to improve the walking facility conditions from a design perspective.