Abstract
Recently different formulations of the first-order Lighthill-Whitham-Richards (LWR) model have been identified in different coordinates and state variables. However, relationships between higher-order continuum and car-following traffic flow models are still not well understood. In this study, we first categorize traffic flow models according to their coordinates, state variables, and orders in the three-dimensional representation of traffic flow and propose a unified approach to convert higher-order car-following models into continuum models and vice versa. The conversion method consists of two steps: equivalent transformations between the secondary Eulerian (E-S) formulations and the primary Lagrangian (L-P) formulations, and approximations of L-P derivatives with anisotropic (upwind) finite differences. We use the method to derive continuum models from general second- and third-order car-following models and derive car-following models from second-order continuum models. Furthermore, we demonstrate that corresponding higher-order continuum and car-following models have the same fundamental diagrams, and that the string stability conditions for vehicle-continuous car-following models are the same as the linear stability conditions for the corresponding continuum models. A numerical example verifies the analytical results. In a sense, we establish a weak equivalence between continuum and car-following models, subject to errors introduced by the finite difference approximation. Such an equivalence relation can help us to pick out anisotropic solutions of higher-order models with non-concave fundamental diagrams.
Highlights
From the studies on different formulations of the LWR model in (Daganzo, 2006a,b; Leclercq et al, 2007; Laval and Leclercq, 2013), we can see that car-following models can be considered as discrete primary formulations in the Lagrangian coordinates, and continuum models as secondary formulations in the Eulerian coordinates
In this subsection we analyze the stability of the general second-order car-following model, (19), and the corresponding continuum model, (21)
The modification may not apply to general models. Such an inconsistency could be interpreted in two ways: first, it is possible that string stability is different from linear stability; second, it is possible that the continuum and car-following models obtained with the conversion method may not be equivalent, due to the upwind difference method for discretization of the L-P models, (12)
Summary
In the three-dimensional representation of traffic flow (Makigami et al, 1971), as shown in Figure 1, the evolution of a traffic stream can be captured by a surface in a three-dimensional space of time t, location x, whose positive direction is the same as the traffic direction, and. From the studies on different formulations of the LWR model in (Daganzo, 2006a,b; Leclercq et al, 2007; Laval and Leclercq, 2013), we can see that car-following models can be considered as discrete primary formulations in the Lagrangian coordinates, and continuum models as secondary formulations in the Eulerian coordinates Based on this observation, in this study we will develop a systematic method for conversions between higher-order continuum models and car-following models. The framework is different from that for the first-order models: first, the higher-order L-P models are no longer HamiltonJacobi equations, and the variational principle or the Hopf-Lax formula cannot be applied to obtain car-following models from the L-P models; second, as higher-order derivatives of the primary variables are involved, the inverse function theorem cannot be used to transform variables between Eulerian and Lagrangian coordinates.
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