Modelling Traffic Flow with Constant Speed using the Galerkin Finite Element Method

Abstract

At macroscopic level, traffic can he described as a continuum flow. Lighthill Witham anti Richards (LWR) have developed a traffic flow model based on the fluid dynamics continuity equation, which is known as the first order LWR traffic flow model The resulting first order partial differential equation (PDE) can be analytically solved for some special cases, given initial and boundary conditions, and numerically using for example the finite element method (FEM). This paper makes use of the Galerkin FEM to solve the LWR model with constant speed. The road is divided into it number of road segments (elements) using the Galerkin FEM. Each element consists of two nodes. Each mule hits one degree (if freedom (d o f), namely the traffic density. The FEM provides a solution for the degrees of freedom, i.e. traffic densities of each node. The resulting simultaneous equations arc solved at different tinge steps using the Euler backward tinge-integration algorithm. In Belgium and also in the Netherlands, there is a special technique that can be used in order to prevent traffic jams anti increasing safety in situations with high volume of cars on the roads, i.e. block driving. It is a technique where cars drive in groups by order of the police when the roads are crowded. In this paper block driving is used as a practical example of the LWR model with constant speed. Thereby, it is simulated using the Galerkin FEM and the results tire compared with the analytical solution. The FIRM gives good results providing that: the road segments and time steps are small enough. A road with length 5000 m, constant speed of 25 m/s, segment length of 100 m anti time steps of 1 s gives good results for the studied case. At points of traffic density rate discontinuities, depending on the road segment size and time step size, the Galerkin FEM is accurate anti requires reasonable computational effort. From the research work carried out in this paper, it is found that the Galerkin FEM is suitable for modelling traffic flow at macroscopic level. The element size and tinge step size tire important parameters in determining the convergence of the solution in case of discontinuities in traffic density rate. Although this paper considers the case of constant speed, the technique can be extended in the future to include the case of non-constant speed, i.e. speed as a function of traffic density.