Abstract
The kinematic wave model of traffic flow on a road network is a system of hyperbolic conservation laws, for which the Riemann solver is of physical, analytical, and numerical importance. In this paper, we present a new Riemann solver at a general network junction in the demand-supply space. In the Riemann solutions, traffic states on a link include the initial, stationary, and interior states, and a discrete Cell Transmission Model flux function in interior states is used as an entropy condition, which is consistent with fair merging and first-in-first-out diverging rules. After deriving the feasibility conditions for both stationary and interior states, we obtain a set of algebraic equations, and prove that the Riemann solver is well-defined, in the sense that the stationary states, the out-fluxes of upstream links, the in-fluxes of downstream links, and kinematic waves on all links can be uniquely solved. In addition, we show that the resulting global flux function in initial states is the same as the local one in interior states. Hence we presents a new definition of invariant junction models, in which the global and local flux functions are the same. We also present a simplified framework for solving the Riemann problem with invariant junction flux functions.
Highlights
A better understanding of traffic dynamics on a road network is critical for improving safety, mobility, and environmental impacts of modern surface transportation systems [45]: practically, it is helpful for efficient implementations of ramp metering [41], evacuation [46], signal control, and other management and control strategies; theoretically, it can yield better network loadingIn a road network, such bottlenecks as merging, diverging, and general junctions play a critical role in initiating, propagating, and dissipating traffic congestion
Some interesting traffic dynamics can be caused by interactions among these network bottlenecks: for examples, a beltway network can be totally gridlocked [12], and periodic oscillations can occur in a diverge-merge network [26]
Since traffic dynamics inside a link can be described by the LWR model and are well understood, the most important component of network kinematic wave models is related to how merging and diverging behaviors would impact the formation of shock and rarefaction waves at a general network junction shown in Figure 2, which has m upstream links and n downstream links
Summary
A better understanding of traffic dynamics on a road network is critical for improving safety, mobility, and environmental impacts of modern surface transportation systems [45]: practically, it is helpful for efficient implementations of ramp metering [41], evacuation [46], signal control, and other management and control strategies; theoretically, it can yield better network loading. Since the kinematic wave model of network traffic flow is a system of hyperbolic conservation laws on a network structure, solutions to the Riemann problem at a junction, in which all links carry constant initial conditions, but discontinuities can occur at the junction, are of physical, analytical, and numerical importance: physically, they can define physical merging, diverging, and other behavioral rules; analytically, a system of hyperbolic conservation laws is well-defined if and only if the Riemann problem is uniquely solved [6]; and numerically, they can be incorporated into the Godunov finite difference equations [19]. Different from that in [25], the Riemann solver in this study uses a discrete flux function, which is defined in terms of upstream demands and downstream supplies, as an entropy condition Such a discrete flux function is originally developed within the framework of CTM, models conflicts among merging and diverging traffic streams at the aggregate level, and is a natural choice as the entropy condition to pick out unique physical solutions. With (2) in the following sense [9]:
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