Modelling Traffic Flow with the Nonlinear LWR Scheme using the High Order Discontinuous Galerkin Discretization
Alberto Costa Nogueira Junior1; Jonathan Bernardo2; Stephen Moore3
1 IBM Reasearch Brazil; 2 IBM Research Brazil; 3 IBM Research Australia
doi:10.20906/CPS/CILAMCE2015-0232
Resumo
In this article, we propose a high-order discontinuous Galerkin finite element (DG-FEM) discretization to solve the classical LWR traffic flow model. The monotone Lax-Friedrichs flux was chosen to compute numerical fluxes. Numerical stability was added to the scheme through over integration of the nonlinear physical flux. A sub-cell shock capturing mechanism with localized artificial viscosity was also used to stabilize solutions in the presence of shock discontinuities. To deal with the 2nd order differential operator associated to the artificial dissipation, the LDG (Local Discontinuous Galerkin) method was employed. Numerical examples were computed for smooth and discontinuous initial conditions. The DG-FEM scheme was compared to the following standard discretizations: the 2nd order MacCormack finite difference (FD) scheme, the standard 1st order finite volume (FV) method and the 2nd order MUSCL scheme. The sub-cell shock capturing mechanism, commonly used in aerospace applications, proved to be very sharp and reliable to stabilize solutions which develop shock discontinuities although it can over smear the shock profile when the approximate solution evolves for long time periods. Overall, DG-FEM approach revealed to be more accurate, robust and flexible to solve the LWR traffic equation when compared to their above mentioned competitors.
Palavras-chave: High order; Discontinuous Galerkin; Finite Element; Shock capturing; Traffic flow model