Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications: Classical and Non--Classical Advanced Mathematics for Real Life Applications

This book deals with modeling and analysis of the dynamics of traffic flows. The study is focused on addressing various types of congestion effects. Congestion is a phenomenon that results from a non-equilibrium development in a mobile population, hence it often may contribute to a destabilization of an involved system and result in singular dynamic behavior. This book focuses on macroscopic situations. Those models are governed by nonlinear hyperbolic balance laws. Two classes of dynamical processes are studied: vehicular traffic and pedestrian flows. In recent years, the number of different elements and aspects involved in the management of urban traffic has increased enormously. Many models for traffic flows have been developed that resort to different approaches, ranging from microscopic ones, taking into account each single individual, to kinetic and continuum ones, dealing with averaged quantities. However, recently the use of sensors imbedded in the road and cameras focused from above is becoming more common and the consequent availability of on-line data allows one to implement real-time strategies to avoid or mitigate congested traffics. Opposed to direct numerical simulations of a large number of individual interacting subjects, as is typical when dealing with microscopic models, researchers advise the use of continuum models for traffic. The main advantages of this approach as opposed to the microscopic one are the following: ∙ the model is completely evolutive and is able to rapidly describe any traffic situation at every instant of time; ∙ the resulting description of the evolution of queues and of traveling times is accurate as the position of shock waves can be exactly computed and corresponds to the tails of the queues; ∙ the theory enables the development of efficient numerical schemes also, to describe a very large number of individuals; ∙ the model can be easily calibrated, validated and implemented as the number of parameters is low; ∙ the theory allows one to state and possibly solve optimal management problems. The book is organized into three parts: I. Mathematical Theory. II. Models for Vehicular Traffic. III. Models for Pedestrian Traffic. In the first part the theory for conservation laws is carefully developed starting from the basic concepts. Besides the classical theory of entropy weak solutions, the concepts of non-entropy weak solutions are also introduced to describe phenomena typical of traffic, such as those related to the presence of constraints along the paths and, in the case of crowds, to the rise of panic. The method of characteristics, operator splitting, and the wave-front-tracking algorithm are detailed and applied to various typical traffic situations. In the second part, the author introduces the main theoretical definitions and models encountered in the study of vehicular traffic. After an overview and classification of the models for traffic, he describes the fundamental traffic variables and their relations. He continues with equilibrium models for traffic flows. The first equilibrium traffic model is the LWR model, which represents the starting point for the modeling of vehicular flows. Its basic assumptions are: ∙ there is only one class of vehicles moving along a unique homogeneous lane and overtaking is not allowed; ∙ vehicles do not enter or exit the road; ∙ the (average) speed of the cars depends on the (average) density alone. The model reads as ρt+(ρv(ρ))x=0, where ρ is the (mean) density and v=v(ρ) is the corresponding preferred (mean) velocity. The choice of a function v(ρ) depends on the behavior the model is trying to mimic. It may be either taken as a phenomenological relation extracted from empirical data or derived from more microscopic considerations. The author describes how to model a road with an entrance and a time-dependent constraint, two merging roads, a traffic circle, multi-population traffic, and a multi-lane road. He also presents some functionals related to traffic management and proves that they can be optimized. Moreover, he numerically integrates several of the previous models. Finally, he discusses two non-equilibrium models for vehicular traffic: the PW and the AR models. The third part deals with crowd dynamics characterized by a large number of individuals. The author studies so-called panic and highlights its dynamic effects, such as Braess' paradox for pedestrian flows. He explains from the modeling point of view the reasons for the failure of the classical theory of conservation laws to describe the rise of panic and, consequently, to justify the introduction of a non-classical theory. The book is self-contained and can be used for undergraduate courses in mathematical modeling, physics and civil engineering