24 Transportation Research Record: Journal of the Transportation Research Board, No. 2260, Transportation Research Board of the National Academies, Washington, D.C., 2011, pp. 24–41. DOI: 10.3141/2260-04 model provides a good approximation of the dynamics of traffic flow that has proved to be useful for most practical purposes. Although the ability to directly measure different traffic parame- ters has dramatically increased over the years, the measurements are still widely different in their accuracy and reliability. For instance, flow rate and density can be obtained as a result of simple counting processes performed at a single point or a pair of points, respec- tively. However, in practice, large inconsistencies between counts in consecutive stations (with no exit or entrance in between) exist. In the case of loop detectors, this drift phenomenon is well known. In addition, at a macroscopic level, the definition of density is ambiguous in that the length over which the concentration of vehi- cles would affect a driver’s behavior under normal conditions is not specified. In contrast, speed measurements, which theoretically are expected to be more difficult to obtain, have proved to be a far more reliable source of traffic data. Thus, in this paper, the focus is on a speed-based equivalent of the LWR model. Work et al. proposed a finite difference method (FDM) to numer- ically solve the speed-based LWR model (4). FDM is a powerful and efficient approximate solution method for partial differential equations (PDEs) such as the LWR model. An alternative numer- ical solution method to solve the LWR model is FEM, which is a powerful general approximate solution method for PDEs. In this method, the solution domain is divided into smaller and well- defined areas known as elements. The function of interest at each point inside an element is determined by interpolating the function values at elemental nodes located on the boundaries of that element. The weights used in the interpolation process are called shape functions and are a characteristic of the element type. As a result, FEM essentially transforms a PDE into a system of equations with a vector of unknowns that represent function values at the elemental nodes. Beskos and Michalopoulos reported the first application of FEM in solving the traffic model of a signalized intersection (5). Beskos et al. (6) and Okutani et al. (7 ) applied the Galerkin FEM method combined with the Newton–Raphson and Lax–Wendroff methods, respectively, to numerically solve the freeway dynamics modeling problem. An average absolute error in speed estimates of roughly more than 6 mph using a simple continuum traffic model, in comparison with simulation results, is reported in these studies. Wong and Wong represented the spatial variations of density in an element by using a set of wavelets (8). They used a domain trans- formation technique to maintain the nonnegativity of the solutions. Their FEM set of equations was also based on the Galerkin method. They did not compare the numerical solutions with exact solu- tions obtained from method of characteristics directly, and there- Real-Time Solution of Velocity-Based First-Order Continuum Traffic Model with Finite Element Method Kaveh F. Sadabadi and Ali Haghani Two approximate solution methods are investigated for the velocity- based version of a first-order continuum traffic flow model commonly known as the Lighthill–Whitham–Richards model. The finite difference and finite element methods are commonly used to numerically solve par- tial differential equations. The finite difference method adopted uses a standard Godunov scheme to solve the continuum model. The finite ele- ment method uses one-dimensional simplex elements with first-order interpolation function along with a Galerkin scheme to derive the element characteristic matrices and vectors. A high-resolution, real-world data set from the Next Generation Simulation program is used to evaluate the two solution methods. Results show that both methods provide accurate approximations to the observed speeds. The accuracy and quality of solutions and future directions of work in this area are discussed. In this paper, a finite element method (FEM) for numerically solving the velocity-based equivalent of the first-order continuum traffic flow model is proposed. This model provides a theoretical framework to understand and analyze traffic processes on a variety of roadway facilities. This model is widely used in an array of traffic operation and control applications worldwide. Therefore, both efficient and accurate solution methods are needed for this model. The perfor- mance of the proposed FEM for a detailed real-world data set from the Next Generation Simulation (NGSIM) program is compared with an existing numerical method based on the finite difference approach. Continuum traffic flow theory is a powerful tool for describing the evolution of macroscopic traffic parameters over time and space. This is in contrast to microscopic models of traffic flow, which gen- erally require meticulous handling of the movements of individual vehicles in the traffic stream. The most basic continuum traffic flow theory builds on two basic physical principles: the conservation of vehicles and the fundamental relationship between flow rate, den- sity, and speed. The conservation principle states that no vehicle is added or lost in traffic at any time other than the ones that enter or exit through the boundaries. This basic continuum theory was first proposed by Lighthill and Whitham (1) and Richards (2). Despite its simplicity, and therefore its inherent limitations (3), the so-called kinematic wave theory, or Lighthill–Whitham–Richards (LWR), Department of Civil and Environmental Engineering, University of Maryland, 1173 Glenn L. Martin Hall, College Park, MD 20742. Corresponding author: A. Haghani, haghani@umd.edu.