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