A Characteristic Particle Method for Traffic Flow Simulations on Highway Networks

Farjoun, Yossi; Seibold, Benjamin · 2013 · Crossref

DOI: 10.1007/978-3-642-32979-1_13

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Summary

This paper introduces a characteristic particle method for simulating first-order macroscopic traffic flow on highway networks, addressing the need for efficient numerical schemes capable of handling large-scale road graphs. The authors focus on the Lighthill-Whitham-Richards (LWR) model, a scalar hyperbolic conservation law, which describes traffic density evolution but lacks the ability to model non-equilibrium features like phantom jams. The motivation stems from the computational challenges of simulating networks with thousands of edges, where traditional fixed-grid methods suffer from numerical viscosity, limited adaptivity, and complex stencil requirements at network nodes. The proposed method generalizes the `particleclaw` approach, which solves one-dimensional hyperbolic conservation laws exactly except near shocks. In this framework, the solution on each road segment is represented by a finite set of characteristic particles that move independently according to characteristic equations. Shocks are handled by merging colliding particles while preserving the area under the similarity interpolant, introducing only small approximation errors. To extend this to networks, the authors develop a modular synchronization strategy. Edges evolve independently during a time step $\Delta t$, allowing for parallelization and larger time steps compared to grid-based methods. At the end of each step, edges are coupled at network nodes by solving generalized Riemann problems. This process involves interpolating solutions back onto edge boundaries, determining fluxes based on conservation laws and driver destination matrices, and redistributing "excess" or "virtual" area to ensure exact conservation of vehicles across the network. Numerical examples demonstrate that the method accurately approximates traffic jams and complex flow patterns while utilizing very few degrees of freedom per edge. The approach maintains high accuracy without numerical viscosity and remains second-order accurate even in the presence of shocks. By avoiding fixed grids, the method eliminates spurious discontinuities and adapts naturally to solution features. The synchronization step ensures that information propagates correctly through nodes, handling bottlenecks, bifurcations, and confluences efficiently. The significance of this work lies in providing a computationally efficient and structurally advantageous tool for traffic simulation, forecasting, and optimization on large networks. The method’s inherent adaptivity and parallelizability make it superior to traditional finite volume or finite difference schemes for network applications. It offers a robust framework for modeling large-scale, nonlinear equilibrium traffic behavior, facilitating the analysis of complex highway systems with minimal computational overhead.

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• traffic density