Source-Destination Flow on a Road Network

Garavello, Mauro; Piccoli, Benedetto · 2005 · OpenAlex-citations

DOI: 10.4310/cms.2005.v3.n3.a1

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Summary

This paper develops a mathematical model for traffic flow on road networks that explicitly accounts for sources (origins) and destinations. The authors address the limitation of previous fluidodynamic models, which typically determine junction behavior based solely on car density and fixed distribution parameters. By incorporating "traffic-type functions," the model tracks the specific paths cars take from fixed sources to fixed destinations, allowing for a more accurate representation of driver route choices. The model is built upon two primary equations. First, car density $\rho$ evolves according to the Lighthill-Whitham-Richards scalar conservation law, $\rho_t + f(\rho)_x = 0$, where flux $f(\rho)$ depends on density-dependent average speed. Second, traffic-type functions $\pi$, representing the percentage of cars traveling between specific source-destination pairs, evolve according to semilinear equations $\pi_t + v(\rho)\pi_x = 0$. At network junctions, the authors define a solution using a distribution matrix derived from $\pi$ and a novel maximization procedure. Unlike prior approaches that simply maximize total flux, this method maximizes a functional that balances total flux with the adherence to "right of way" priorities among incoming roads. This quadratic maximization ensures that solutions depend continuously on the traffic-type coefficients, a property not guaranteed by earlier methods. The authors prove the existence of unique admissible weak solutions for the Riemann problem at junctions and for the entire network. They utilize a wave-front tracking method to construct these solutions, deriving necessary BV (bounded variation) estimates to ensure convergence. While large variations in traffic-type functions can occur at junctions, the authors establish the existence of entropic solutions for perturbations of constant initial data. The analysis demonstrates that their Riemann solver is more robust than previous solvers (such as those by Coclite, Garavello, and Piccoli), as it does not require restrictive technical conditions on the distribution matrix and handles cases where previous models failed or produced discontinuous results. The significance of this work lies in providing a rigorous mathematical framework for traffic flow that integrates microscopic route choices into macroscopic density models. By ensuring continuity with respect to traffic-type functions and handling complex junction dynamics through priority-based maximization, the model offers a more realistic and stable tool for analyzing traffic on networks with defined origins and destinations. This approach bridges the gap between simple density-based models and complex network dynamics, offering a foundation for further research in transportation science and control theory.

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• mental model of traffic