The Inhomogeneous Kinematic Wave Traffic Flow Model as a Resonant Nonlinear System

The kinematic wave traffic flow model for an inhomogeneous road is studied as a resonant nonlinear system, where an additional conservation law is introduced to model time-invariant road inhomogeneities such as changes in grades or number of lanes. This resonant system has two families of waves, one of which is a standing wave originated at the inhomogeneity. The nature of these waves are examined and their time-space structures are studied under Riemann initial conditions and proper entropy conditions. Moreover, the system is solved numerically with Godunov’s method, and the solutions are found to be consistent with those of Daganzo (1995) and Lebacque (1996) for the same initial conditions. Finally, the numerical approximation is applied to model traffic flow on a ring road with a bottleneck and the results conform to expectations. Introduction The kinematic wave traffic flow model of LWR was introduced by Lighthill and Whitham (1955) and Richards (1956) for modeling dense traffic flow on crowded roads, where the evolution of density 1 ρ(x, t) and flow-rate q(x, t) over time is described by equation, ρt + qx = 0. (1) This equation follows conservation of traffic that vehicles are neither generated nor destroyed on a road section with no entries and exits. The conservation equation alone is not sufficient to describe traffic evolution, because it does not capture the unique character of vehicular flow—drivers slow down when their front spacing is reduced to affect safety. The LWR model addresses this issue by assuming a functional relationship between local flow-rate and density, i.e., q = f(x, ρ). This flow-density relation, also known as the fundamental diagram of traffic flow, is often assumed to be concave in ρ and is a function of the local characteristics of a road location, such as the number of lanes, curvature, grades, and speed limit, as well as vehicle and driver composition. When a piece of roadway is homogeneous; i.e., the aforementioned characteristics of the road are uniform throughout the road section, the fundamental diagram is invariant to location x and the LWR model becomes ρt + f(ρ)x = 0. (2) In contrast, if a section of a roadway is inhomogeneous, the LWR model is ρt + f(x, ρ)x = 0. (3) Here we introduce a more explicit notation, an inhomogeneity factor a(x), into the flux function f(x, ρ) and obtain the following equivalent LWR model for an inhomogeneous road ρt + f(a, ρ)x = 0. (4) This equation is particularly suited for our later analysis of the LWR model for inhomogeneous roads (We shall hereafter call (2) the homogeneous LWR model and (4) the inhomogeneous LWR model). Both the homogeneous and inhomogeneous LWR models have been studied by researchers and applied by practitioners in the transportation community. Note that the homogeneous version (2) is nothing more than a scalar conservation law. Therefore, its wave solutions exist and are unique under the so-called “Lax entropy condition” (Lax, 1972). These solutions are formed by basic solutions to the Riemann problem of (2), in which the initial conditions jump at a boundary and are constant both upstream and downstream of the jump spot. Nevertheless, because analytical solutions are difficult to obtain for (2) with arbitrary initial/boundary conditions, numerical solutions have to be computed in most cases. The most often used approximation of (2) is perhaps 2 that of Godunov. In the Godunov method, a roadway is partitioned into a number of cells; and the change of the number of vehicles in each cell during a time interval is the net inflow of vehicles from its boundaries. The rate of traffic flowing through a boundary is obtained by solving a Riemann problem at this boundary. Besides the Godunov method, there are other types of approximations of the homogeneous LWR model, and some of them are shown to be variants of Godunov’s method (Lebacque 1996). In contrast to the well researched homogeneous LWR model, the inhomogeneous model is less studied and less understood. Of the few efforts to rigorously solve the inhomogeneous LWR model, the works of Daganzo (1995) and Lebacque (1996) should be mentioned. In his cell transmission model, Daganzo started with a discrete form of the conservation equation and suggested that the flow through a boundary connecting two cells of a homogeneous road is the minimum of the “sending flow” from the upstream cell and the “receiving flow” of the downstream cell. The “sending flow” is equal to the upstream flow-rate if the upstream traffic is undercritical (UC) or the capacity of the upstream section if the upstream traffic is overcritical (OC); on the other hand, the “receiving flow” is equal to the capacity of the downstream section if the downstream traffic is UC or the downstream flow-rate if the downstream traffic is OC. In the homogeneous case, the boundary flux computed from the “sending flow” and the “receiving flow” is the same as that computed from solutions of the associated Riemann problem. Since the definitions of “sending flow” and “receiving flow” can be extended to inhomogeneous sections, Daganzo’s method can also be applied to the inhomogeneous LWR model. Different from Daganzo, Lebacque started his method with the solution of the “generalized” Riemann problem for (3). In this work, Lebacque came up with some rules for solving the “generalized” Riemann problem. These rules play the same role as entropy conditions. Moreover, Lebacque found that the boundary flux obtained from solving the Riemann problem is consistent with that from Daganzo’s method, and he called Daganzo’s “sending flow” demand and “receiving flow” supply. The methods of Daganzo and Lebacque are streamlined versions of Godunov’s method for the inhomogeneous LWR model. They hinge upon the definitions of the demand and supply functions, which can be obtained unambiguously when f(a, ρ) is unimodal. When f(a, ρ) has multiple local maximum, or when the traffic flow model is of higher order, it is yet to be determined if equivalent demand/supply functions exist. Thus, these two methods may not be applicable to solve the LWR model that has multiple critical points on its fundamental diagram, nor higher-order models of traffic flow, such as the Payne-Whitham (Payne, 1971; Whitham, 1974) model and Zhang’s (1998, 1999, 2000, 2001) model. Note that, however, these higher-order models for homogeneous roads can still be solved with Godunov’s method (Zhang, 2001). 3 In this paper, we present a new method for solving the Riemann problem for (4), which can be extended to solve higher-order models. By introducing an additional conservation law for a(x), we consider the inhomogeneous LWR model as a resonant nonlinear system and study its properties (Section 1). We also solve the Riemann problem for (4) and show that the boundary flux at the location of the inhomogeneity is consistent with the one given by Lebacque and Daganzo for the same initial condition (Section 2). Finally, we demonstrate our method through solving an initial value problem on a ring road with a bottleneck, and draw some conclusions from our analyses. 1 Properties of the inhomogeneous LWR model as a resonant nonlinear system Instead of directly study the inhomogeneous LWR model described by (4), we augment (4) into a system of conservation laws through the introduction of an additional conservation law at = 0 for the inhomogeneity factor a(x), which leads to Ut + F (U)x = 0, (5) where U = (a, ρ), F (U) = (0, f(a, ρ)), x ∈ R, t ≥ 0. Without loss of generality, we assume the inhomogeneity is the drop/increase of lanes at a particular location, and write the fundamental diagram as f(a, ρ) = ρv∗( ρ a), where v = v∗( ρ a) is the speed-density relation. The results obtained hereafter apply to other types of inhomogeneities, such as changes in grades. The inhomogeneous LWR model (5) can be linearized as Ut + ∂F (U)Ux = 0, (6) where the differential ∂F (U) of the flux vector F (U) is ∂F =   0 0 − ρ 2 a v ′ ∗ ( a) v∗( ρ a) + ρ av ′ ∗ ( a)  . (7) The two eigenvalues of ∂F are λ0 = 0, λ1 = v∗( ρ a ) + ρ a v ∗ ( ρ a ). (8) The corresponding right eigenvectors are R0 =   v∗( ρ a) + ρ av ′ ∗ ( a) ( a) 2v′ ∗ ( a) 