A discontinuous Galerkin method for the Aw-Rascle traffic flow model on networks

Abstract

Macroscopic models for flows strive to depict the physical world by considering quantities of interest at the aggregate level versus focusing on each discrete particle in the system. Many practical problems of interest such as the blood flow in the circulatory system, irrigation channels, supply chains, and vehicular traffic on freeway systems can all be modeled using hyperbolic conservation laws that track macroscopic quantities through a network. In this paper we consider the latter, specifically the second-order Aw-Rascle (AR) traffic flow model on a network, and propose a discontinuous Galerkin (DG) method for solving the AR system of hyperbolic partial differential equations with appropriate coupling conditions at the junctions. On each road, the standard DG method with Lax-Friedrichs flux is employed, and at the junction, we solve an optimization problem to evaluate the numerical flux of the DG method. As the choice of well-posed coupling conditions for the AR model is not unique, we test different coupling conditions at the junctions. Numerical examples are provided to demonstrate the high-order accuracy, and comparison of results between the first-order Lighthill-Whitham-Richards model and the second-order AR model. The ability of the model to capture the capacity drop phenomenon is also explored.