TY - EJOU
AU - Vikram, Durgesh
AU - Mittal, Sanjay
AU - Chakroborty, Partha
TI - A Stabilized Finite Element Formulation for Continuum Models of Traffic Flow
T2 - Computer Modeling in Engineering \& Sciences
PY - 2011
VL - 79
IS - 3&4
SN - 1526-1506
AB - 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.
KW - Traffic flow
KW - Finite element method
KW - (IEPG) Inter-Equation/Petrov-Galerkin stabilization
KW - SUPG
KW - Shock wave
KW - Expansion wave
DO - 10.3970/cmes.2011.079.237