Understanding Complex Traffic Dynamics with the Nondimensionalisation Technique

• Published in: Mathematics (Basel) Vol. 12; no. 4; p. 532
• Main Authors: Sánchez-Pérez, Juan Francisco, Oviedo-Casado, Santiago, García-Ros, Gonzalo, Conesa, Manuel, Castro, Enrique
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.02.2024
• Subjects: Differential equations; Extreme values; Inflow; Initial conditions; mathematical modelling; Mathematical models; Methods; nondimensionalisation; Parameter sensitivity; Partial differential equations; sensitivity; Traffic congestion; traffic dynamics; Traffic flow; Traffic jams; Traffic models; Urban planning; Variables; Vehicles; Velocity
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math12040532

Hydrodynamic traffic models are crucial to optimizing transportation efficiency and urban planning. They usually comprise a set of coupled partial differential equations featuring an arbitrary number of terms that aim to describe the different nuances of traffic flow. Consequently, traffic models quickly become complicated to solve and difficult to interpret. In this article, we present a general traffic model that includes a relaxation term and an inflow of vehicles term and utilize the mathematical technique of nondimensionalisation to obtain universal solutions to the model. Thus, we are able to show extreme sensitivity to initial conditions and parameter changes, a classical signature of deterministic chaos. Moreover, we obtain simple relations among the different variables governing traffic, thus managing to efficiently describe the onset of traffic jams. We validate our model by comparing different scenarios and highlighting the model’s applicability regimes in traffic equations. We show that extreme speed values, or heavy traffic inflow, lead to divergences in the model, showing its limitations but also demonstrating how the problem of traffic jams can be alleviated. Our results pave the way to simulating and predicting traffic accurately on a real-time basis.