A Stabilized Finite Element Formulation for Continuum Models of Traffic Flow

A stabilized finite element formulation is presented to solve the governing equations for traffic flow. The flow is assumed to be one-dimensional. Both, PW-type (Payne-Whitham) 2-equation models and the LWR-type (Lighthill-Whitham-Richards) 1-equation models are considered. The SUPG (Streamline-Upwind/Petrov-Galerkin) and shock capturing stabilizations are utilized. These stabilizations are sufficient for the 1-equation models. However, an additional stabilization is necessary for the 2-equation models. For the first time, such a stabilization is proposed. It arises from the coupling between the two equations and is termed as IEPG (Inter-Equation/Petrov-Galerkin) stabilization. Two behavioral models are studied: Greenshields' (GS) and Greenberg's (GB) models. Numerical tests are carried out for cases involving traffic expansion as well as shock. Excellent agreement with the exact solution is observed. The need of the IEPG stabilization for the 2-equation traffic models is demonstrated. An interesting observation is made for the first time regarding the Greenberg's (GB) model in the presence of a shock. The model is found to be inconsistent in the sense that it leads to different shock speed from the continuity and behavior equations. As a result, the 2-equation model leads to secondary waves in the presence of shocks.