Lighthill Whitham Richards Model Research Articles

Convergence of a Discretised Travel-Time Model

In network models for dynamic traffic assignment (DTA), the travel time on a link is often treated as a function of the number of vehicles on the link. Instead of applying this model to the whole link, we divide the link into segments, apply the model (suitably adjusted) sequentially to these segments, and investigate how the solution is affected by various levels of discretisation (as the discretisation is refined, the solution converges to the solution of the Lighthill-Whitham-Richards (LWR) model). We also restrict the link (and segment) travel-time function to ensure that it satisfies a first-in-first-out (FIFO) property and explore how this affects and restricts the form of the flow-density functions used in the LWR model. We numerically illustrate the solution of the discretised model for various travel-time functions and patterns of inflows, for both homogeneous and inhomogeneous links. Subject to the above restriction on the flow-density function, the numerical results suggest that dividing “long” links into even a few segments can make the model solution closely approximate the LWR solution, while retaining tractability in the network model. We also observe, for example, that the whole-link (undescretised) travel-time model has a “flattening” effect on the profiles of flows and travel times (this effect can be reduced to any desired extent by using discretisation); and that the travel time and outflow for an inhomogeneous link can be approximated very closely by treating the link as homogeneous, with capacity parameters set equal to the average capacity from the inhomogeneous link.

A weighted essentially non-oscillatory numerical scheme for a multi-class Lighthill–Whitham–Richards traffic flow model

In this paper, we present a high-order weighted essentially non-oscillatory (WENO) scheme for solving a multi-class extension of the Lighthill–Whitham–Richards (LWR) model. We first review the multi-class LWR model and present some of its analytical properties. We then present the WENO schemes, which were originally designed for computational fluid dynamics problems and for solving hyperbolic conservation laws in general, and demonstrate how to apply these to the present model. We found through numerical experiments that the WENO method is vastly more efficient than the low-order Lax–Friedrichs scheme, yet both methods converge to the same solution of the physical model. It is especially interesting to observe the small staircases in the solution which are completely missed out, because of the numerical viscosity, if a lower-order method is used without a sufficiently refined mesh. To demonstrate the applicability of this new, efficient numerical tool, we study the multi-class model under different parameter regimes and traffic stream models. We consider also the convergence of the multi-class LWR model when the number of classes goes to infinity. We show that the solution converges to a smooth profile without staircases when the number of classes increases.

Comparing whole-link travel time models

In a model commonly used in dynamic traffic assignment the link travel time for a vehicle entering a link at time t is taken as a function of the number of vehicles on the link at time t. In an alternative recently introduced model, the travel time for a vehicle entering a link at time t is taken as a function of an estimate of the flow in the immediate neighbourhood of the vehicle, averaged over the time the vehicle is traversing the link. Here we compare the solutions obtained from these two models when applied to various inflow profiles. We also divide the link into segments, apply each model sequentially to the segments and again compare the results. As the number of segments is increased, the discretisation refined to the continuous limit, the solutions from the two models converge to the same solution, which is the solution of the Lighthill, Whitham, Richards (LWR) model for traffic flow. We illustrate the results for different travel time functions and patterns of inflows to the link. In the numerical examples the solutions from the second of the two models are closer to the limit solutions. We also show that the models converge even when the link segments are not homogeneous, and introduce a correction scheme in the second model to compensate for an approximation error, hence improving the approximation to the LWR model.

Mixing Microscopic and Macroscopic Representations of Traffic Flow: Hybrid Model Based on Lighthill–Whitham–Richards Theory

The hybrid traffic flow model, coupling a microscopic (vehicle-based) and a macroscopic (flow-based) representation of traffic flow, may be a useful tool to better understand the relationships between the various types of representation. It can also be a basis for implementing various model extensions, which may be easier with using one type of representation or the other. The hybrid model presented here combines a flow and a vehicular representation of the same model, which is the classical Lighthill–Whitham–Richards model. The use of a simple and unique model makes it possible to focus on the specific problems raised by translating boundary conditions from vehicular to flow formulation and conversely. This translation is made in order to ensure the conservation of flow and proper transmission of information both downstream and upstream and to minimize the perturbations induced by transitions between a continuous and a discrete representation of flow. The resulting model is shown to have good properties, particularly concerning congestion propagation and flow smoothing at the interfaces between the two models.

A theoretical probe of a German experiment on stationary moving traffic jams

Kerner and Rehborn (Phys. Rev. E 53 (2) (1996)) reported on the observation of two stationary moving jams that lasted for about an hour on a 13 km long German highway section. They attributed the phenomenon to intrinsic characteristics of traffic flow, something that would arise spontaneously within the traffic stream due to drivers’ driving behavior. We show in this paper that these moving jams are not particularly peculiar but can be explained with the hydrodynamic theory of traffic flow, or the Lighthill–Whitham–Richards model, and the merge and diverge models in the cell transmission model. In fact, we demonstrate that this stationary jam phenomenon can be replicated with a simple two-wave velocity (or triangular) flow–density relationship in conjunction with the hydrodynamic theory. This finding provides some evidence to support that a triangular flow–density relationship is a good approximation of field observations and that a simple first-order hydrodynamic theory is capable of explaining complex traffic phenomenon.

Driver memory, traffic viscosity and a viscous vehicular traffic flow model

This paper studies the linkage between microscopic car-following and macroscopic fluid-like behavior of traffic flow. We find that driver memory in car-following leads to viscous effects in continuum traffic flow dynamics. This linkage is further exploited to develop a second-order continuum model with viscosity. Further, we show that this viscous model contains practically every well-known continuum model as a special case (e.g., the Lighthill–Whitham–Richards model and the Payne–Whitham model), has a stable wave structure of first- and second-order waves, and controls the extent of non-anisotropic and diffusive influences through a dimensionless parameter called anisotropic factor.

Effects of Relaxation and Anticipation on Riemann Solutions of Payne-Whitham Model

The solutions of Riemann problems of a particular higher-order model—the Payne-Whitham (PW) model—are studied using Roe’s flux splitting scheme as presented by Leo and Pretty. Despite numerous works on higher-order models, little is known about Riemann solutions of these models and how relaxation and anticipation affect these solutions. Riemann solutions of the PW model are computed and compared with those of the Lighthill-Whitham-Richards (LWR) model having the same initial (density) data. It was found that faster relaxation forces the PW model to behave much like the LWR model, that strong anticipation has a stabilizing effect on traffic, and that shock waves travel at different speeds in the PW model than they do in the LWR model. These findings provide a basic checklist for experimental validation of PW-like higher models.