A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow

Monache, Maria Laura Delle; Goatin, Paola · 2014 · OpenAlex-citations

DOI: 10.3934/dcdss.2014.7.435

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Summary

This paper introduces a numerical method for solving a strongly coupled partial differential equation (PDE) and ordinary differential equation (ODE) system that models traffic flow with a moving bottleneck, such as a bus. The research addresses the challenge of accurately simulating the interaction between macroscopic traffic density and a slower vehicle that reduces road capacity. While previous engineering approaches approximated moving constraints as a sequence of fixed bottlenecks, this work employs a rigorous mathematical framework where the traffic flow is described by a scalar hyperbolic conservation law (the LWR model) and the bus trajectory by an ODE. The coupling is enforced through a moving density constraint, represented as an inequality on the flux, which accounts for the reduced capacity caused by the bus. The authors develop a finite volume algorithm utilizing a locally non-uniform moving mesh to track the bus position. Standard Godunov schemes fail to capture the non-classical shocks that arise when the density constraint is active; therefore, the proposed method combines a Lagrangian front-tracking algorithm with a specific tracking procedure for the bus trajectory. The numerical scheme dynamically adjusts the mesh grid points to align with the bus position whenever the constraint is enforced. Specifically, if the flux condition indicates the constraint is active, the algorithm shifts grid interfaces to match the bus trajectory, recomputing cell averages to maintain conservation. When the constraint is not active, the bus position is updated by tracking its interaction with density waves (shocks or rarefactions) using an explicit ODE solver. This approach ensures that the non-classical discontinuities, which satisfy the Rankine-Hugoniot condition but violate the Lax entropy condition, are correctly resolved. Numerical tests demonstrate the effectiveness of the proposed scheme on a road segment of length 1, with parameters set for a bus velocity of 0.3 and a capacity reduction factor of 0.6. In the first test case, with uniform initial density, the simulation correctly reproduces a solution consisting of two classical shocks separated by a non-classical discontinuity associated with the bus. In the second test case, involving a Riemann problem with different initial densities, the method accurately captures a rarefaction wave followed by both non-classical and classical shocks. The results confirm that the moving mesh and front-tracking strategy successfully resolve the complex interactions between the bus trajectory and the surrounding traffic density waves, providing a robust tool for analyzing urban transport systems.

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• traffic density