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

This monograph presents a systematic treatment of the theory for hyperbolic conservation laws and their applications to vehicular traffics and crowd dynamics. In the first part of the book, the author presents very basic considerations and gradually introduces the mathematical tools necessary to describe and understand the mathematical models developed in the following parts focusing on vehicular and pedestrian traffic. The book is a self-contained valuable resource for advanced courses in mathematical modeling, physics and civil engineering. A number of examples and figures facilitate a better understanding of the underlying concepts and motivations for the students. Important new techniques are presented, in particular the wave front tracking algorithm, the operator splitting approach, the non-classical theory of conservation laws and the constrained problems. This book is the first to present a comprehensive account of these fundamental new mathematical advances. Table of Contents Cover Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications - Classical and Non-classical Advanced Mathematics for Real Life Applications ISBN 9783319001548 ISBN 9783319001555 Foreword Acknowledgements Contents Part I Mathematical Theory Chapter 1 Introduction 1.1 Motivations and Applications 1.2 Mathematical Framework 1.3 Book Chapters Chapter 2 Mathematical Preliminaries 2.1 Introduction 2.2 Preliminary Lemmas 2.3 Implicit Function Theorems 2.4 Linear Algebra 2.5 Functions with Bounded Variation Chapter 3 One-Dimensional Scalar Conservation Laws 3.1 Introduction 3.2 Method of Characteristics 3.3 Loss of Regularity 3.4 Weak Solutions 3.5 Entropy Weak Solutions 3.6 Lax Inequality Chapter 4 The Riemann Problem 4.1 Introduction 4.2 Shock Waves 4.3 Non-entropy Shock Waves 4.4 Rarefaction Waves 4.5 Contact Waves 4.6 The General Case 4.7 Riemann Solver Chapter 5 The Cauchy Problem 5.1 Introduction 5.2 The Basic Case 5.3 The General Case 5.3.1 Approximation of the Initial Data 5.3.2 Approximation of the Flux 5.4 Global Existence of BV Solutions 5.5 Uniqueness Chapter 6 The Initial-Boundary Value Problem and the Constraint 6.1 Introduction 6.2 The Initial-Boundary Value Problem 6.3 The Constrained Riemann Problem 6.4 The Constrained Cauchy Problem 6.5 The Constrained Initial-Boundary Value Problem Chapter 7 One-Dimensional Systems of Conservation Laws 7.1 Introduction 7.2 Strictly Hyperbolic Linear Systems with Constant Coefficients 7.3 Riemann Problems 7.3.1 Rarefaction Waves 7.3.2 Shock Waves and Contact Discontinuities 7.3.3 General Solutions Chapter 8 One-Dimensional Systems of Balance Laws (Weakly Coupled) 8.1 Introduction 8.2 The Convective Part 8.3 The Non-local Source Term 8.4 Operator Splitting 8.5 Well Posedness of the Cauchy Problem Part II Models for Vehicular Traffic Chapter 9 Vehicular Traffic 9.1 Introduction 9.2 Mathematical Models 9.3 Computational Models 9.4 The Fundamental Macroscopic Traffic Variables 9.5 Relations between the Fundamental Traffic Variables Chapter 10 Equilibrium Traffic Models 10.1 Introduction 10.2 Riemann Problems 10.3 The Drawbacks of the Equilibrium Traffic Models Chapter 11 Generalizations of Equilibrium Traffic Models 11.1 Introduction 11.2 Highway with an Entrance and Constraints 11.3 Merging Roads 11.4 Traffic Circle 11.5 Multi-population 11.6 Multi-lane Traffic Flow Chapter 12 Cost Functionals 12.1 Introduction 12.2 Queue Length 12.3 Stop and Go Waves 12.4 Travel Times 12.5 Density Dependent Functionals Chapter 13 Numerical Applications 13.1 Introduction 13.2 Passing through a Toll Gate 13.3 Lax-Friedrichs vs. Wave Front Tracking 13.4 Synchronizing Traffic Lights Chapter 14 Non-equilibrium Traffic Models 14.1 Introduction 14.2 Generalized PW Models 14.3 AR Model Part III Models for Pedestrian Traffic Chapter 15 General Concepts 15.1 Introduction 15.2 The Need of a Non-classical Theory Chapter 16 The CR Model 16.1 Introduction 16.2 Study of the Interactions 16.3 A Weighted Total Variation 16.4 Numerical Example 16.5 The Cauchy Problem Chapter 17 Applications 17.1 Introduction 17.2 Evacuation without Obstacles 17.3 Evacuation with an Obstacle 17.4 Evacuation Time Index