Macroscopic pedestrian dynamics modelling

Abstract

The dynamics of pedestrian crowds can involve very different scales. While situations involving only a few pedestrians are better described by microscopic models, large crowds can exhibit collective behavior which can be captured by macroscopic equations. Macroscopic models describe crowds as fluids of pedestrians, where individuals cannot be distinguished anymore. This fluid is characterized by some local averages of pedestrian density and velocity. These macroscopic variables are shown to obey conservation equations, which can be solved using the method of characteristics. In contrast with classical fluid equations, the evolution of density and velocity depends on some target or preferred velocity that can be specific to different pedestrian groups. We review the advantages and drawbacks of these conservation laws adapted to the pedestrian case. We also discuss the associated numerical methods, which can be Eulerian or Lagrangian. Particular attention will be devoted to the link between models at microscopic, mesoscopic, and macroscopic scales. Using macroscopic approaches give access to a whole set of methods developed for this kind of partial differential equation, including the study of phase transition, or of travelling waves. Eventually, recent variants that have been proposed in the literature will be outlined.