Dynamic states of a continuum traffic equation with on-ramp

Lee, H. Y.; Lee, Hyun‐Woo; Kim, D. · 1999 · OpenAlex-citations

DOI: 10.1103/physreve.59.5101

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Summary

This paper investigates the dynamic phase diagram of continuum traffic flow on a highway with an on-ramp, addressing the complex behaviors and metastabilities observed in empirical data. Motivated by the need to explain phenomena such as hysteresis, synchronized flow, and traffic jams that arise near ramps, the authors analyze how on-ramp flux influences traffic states differently than in homogeneous highways. The study aims to clarify the distinction between traffic jams and synchronized flow and to determine the stability conditions for various dynamic phases. The researchers employ a macroscopic continuum model based on hydrodynamic equations, incorporating an external source term to represent the on-ramp flux. Using open boundary conditions, they perform numerical simulations via the two-step Lax-Wendroff scheme for various upstream fluxes ($f_{up}$) and the full range of on-ramp fluxes ($f_{rmp}$). To identify multiple steady states and metastabilities, the authors utilize two search methods: applying triggering pulses to induce transitions and adiabatically sweeping parameters to determine stability ranges. They also derive nontrivial analytic solutions to validate numerical findings. The study identifies several distinct traffic states: free flow, the standing localized cluster (SLC) state, the recurring hump (RH) state, and the oscillating congested traffic (OCT) state. A key finding is that the on-ramp allows for novel traffic jams to form even when the upstream density is below the critical threshold for jam formation in homogeneous systems. The SLC state, characterized by a stationary density cluster near the ramp, transitions into the RH state—a limit cycle with localized oscillations—via a Hopf bifurcation as $f_{rmp}$ increases. The OCT state involves a train of self-generated clusters moving upstream. The authors find that previously distinct phases, such as triggered stop-and-go and homogeneous congested traffic, represent smooth crossovers within a single dynamic phase rather than separate phases with sharp boundaries. Furthermore, a metastable region exists where free flow, RH, and OCT states can coexist, with transitions dependent on finite perturbations. The significance of this work lies in its demonstration that on-ramp flux is a more critical factor for jam formation than total flux. The results provide a theoretical basis for understanding synchronized flow as originating from the RH state and explain the absence of well-defined density-flow relations in stationary synchronized traffic through the SLC state. By showing that certain phases are continuous variations of a single dynamic phase, the paper challenges previous classifications and offers a unified view of traffic dynamics influenced by boundary inhomogeneities.

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• traffic density