Mathematical review of second-order macroscopic traffic flow models

Abstract

As urbanization intensifies and traffic demand continues to grow, understanding and modelling vehicular dynamics in complex transportation systems has become increasingly important. Second-order macroscopic traffic flow models provide a powerful framework for capturing the evolution of both traffic density and velocity, offering significant advantages over first-order formulations. Despite extensive developments in this field, the literature lacks a unified mathematical synthesis that systematically organizes and compares the wide range of second-order models. This paper addresses this gap by presenting a comprehensive review of second-order macroscopic traffic flow models, tracing their evolution from foundational formulations in the 1970s to recent advancements up to 2024. The review adopts a structured methodology, drawing on major academic databases to identify models based on their mathematical formulation and practical relevance. The models are classified into key families, including relaxation-based, kinetic, viscous, anisotropic, nonlocal, and multi-class formulations, providing a coherent taxonomy of the field. In addition to cataloguing model equations, this study synthesizes their fundamental mathematical properties, including hyperbolicity, stability, well-posedness, and parameter identifiability. The review further examines how successive models address limitations of earlier approaches, such as non-physical wave propagation and insufficient representation of driver behaviour. Finally, emerging trends are discussed, including the integration of connected and autonomous vehicle technologies, nonlocal interactions, and data-driven modelling approaches.