Particle Method for Macroscopic Model of Coupled Pedestrian and Vehicular Traffic Flow

In this work, we develop a model to study the interaction between pedestrians and vehicles in a 'shared sapce'. A non-local macroscopic pedestrian dynamics model is coupled to a microscopic vehicular traffic model in 1D, through eikonal equations. We highlight the use of a mesh-free particle method to solve the hydro-dynamic equations in Lagrangian form, which can then be easily combined with the kinematic equations. The eikonal equations, which provide the minimal path for pedestrians and vehicles, are solved on a fixed structured grid using a fast marching method. The variable values from the particle cloud are interpolated onto the eikonal grid and vice-versa for computation. We demonstrate the model by solving it numerically in three traffic scenarios; one, at an uncontrolled crossing in a single-lane road, second, at a zebra crossing in a single-lane road and finally, at a crossing with a traffic island in a two-lane road.