Diving deep into area occupancy based continuum models for multi-class traffic: Stability of speed and density gradient formulations

Abstract

The paper analyses and compare the stability conditions of speed and density gradient formulations of multi-class traffic flow when using area occupancy ($AO$) as the traffic concentration measure. In a second-order continuum model of traffic flow, driver’s anticipation to traffic ahead is expressed in terms of speed and density gradient, and stability conditions are analysed based on the eigenvalues from the system of equations in both the formulations. Eigenvalues, which are characteristics speeds of traffic flow, are checked against the Banach contraction principle and the Hyers–Ulam Stability concept to check model’s oscillatory nature and to understand how well a perturbed system solution remains close to that of an unperturbed system. Anticipation terms in both formulations are overviewed to interpret stability in possible practical scenarios in traffic. Linear stability conditions are checked case by case defined based on interactions among vehicle classes. Non-linear stability conditions for the formulations are derived through wavefront expansion techniques. Further, results from numerical simulations on a hypothetical road section in various congestion scenarios show that the vehicle speeds are non-negative always in both the formulations, and characteristics speeds are real, indicating hyperbolicity of the model’s equation systems. Anisotropic behaviour of traffic is well captured in speed gradient formulation; however, characteristics speeds higher than the vehicle speeds are frequently observed in density gradient formulation. This study addresses a fundamental question: whether the classic isotropic (density gradient) or anisotropic (speed gradient) modelling approach is more physically realistic for complex, non-lane-based traffic where traditional density measure under performs. The paper is an in-depth analysis of both formulations, not only from a stability perspective, but also shares insights on the choice of $AO$ as traffic concentration measure while multi-class traffic modelling.