Continuous-time link-based kinematic wave model: formulation, solution existence, and well-posedness

Han, Ke; Piccoli, Benedetto; Szeto, W.Y. · 2015 · OpenAlex-citations

DOI: 10.1080/21680566.2015.1064793

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Summary

This paper addresses the need for a rigorous continuous-time formulation of dynamic traffic network models, specifically focusing on the link-based kinematic wave model (LKWM). While first-order scalar conservation laws like the Lighthill-Whitham-Richards (LWR) model are widely used to capture phenomena such as shock waves and queue spillback, existing network implementations are predominantly discrete-time. The authors identify a gap in the literature regarding the qualitative properties—specifically solution existence, uniqueness, and well-posedness—of these models in the continuous-time domain. The research aims to bridge this gap by deriving a continuous-time LKWM and establishing its mathematical foundations, thereby providing a benchmark for evaluating discrete approximations like the cell transmission model. The methodology involves deriving the LKWM using variational principles for the Hamilton-Jacobi equation, assuming a triangular fundamental diagram. The model utilizes vehicle flows and binary variables to distinguish between free-flow and congested states, rather than density or speed variables. This approach eliminates partial derivatives with respect to space, allowing the network dynamics to be formulated as a system of differential-algebraic equations (DAEs). The authors define junction models for merges and diverges using Riemann Solvers based on demand and supply notions, incorporating vehicle turning ratios and right-of-way parameters. To address the mathematical rigor of the model, the paper employs the wave front tracking method to prove the global existence of weak solutions for networks containing these specific junction types. Furthermore, the authors investigate well-posedness by analyzing the continuous dependence of solutions on initial data using generalized tangent vectors. The study finds that the proposed DAE system effectively captures shock formation, propagation, and queue spillback, serving as the continuous-time counterpart to the discrete link transmission model. The authors successfully establish the existence of solutions for networks with merge and diverge junctions, filling a gap left by previous works that were limited to specific junction configurations. They also prove that the model is well-posed, meaning solutions change continuously with respect to initial conditions, which implies solution uniqueness. Numerical tests on small and large networks demonstrate the computational efficiency of the DAE system. Additionally, the paper shows that time-discretizing the LKWM yields the standard link transmission model, validating the consistency of the continuous formulation. The significance of this work lies in providing a robust theoretical foundation for continuous-time traffic network modeling. By proving existence and well-posedness, the paper ensures that the model is mathematically sound and suitable for analytical applications, such as dynamic traffic assignment and optimization problems involving traffic control. The continuous-time formulation serves as a critical benchmark for assessing the accuracy and convergence of discrete-time models, independent of numerical discretization choices. This contributes to the broader field of transport dynamics by enhancing the understanding of fluid-based traffic models and their qualitative behaviors, facilitating more reliable simulations and control strategies for complex traffic networks.

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• mental model of traffic