A hybrid numerical method for a microscopic and macroscopic traffic flow model

Abstract

In this paper, we introduce a traffic flow model based on a microscopic follow-the-leader model, while enforcing maximal constraints on the density and velocity of the flow. The related macroscopic model can be represented in conservative formulation. By introducing an advected variable up with the flow, where p is the velocity offset, and u is the relative velocity, we reformulate the classical Aw-Rascle-Zhang (ARZ) model and the modified Aw-Rascle model to describe a realistic fundamental diagrams. The elementary waves are derived, and the Riemann problem is solved to validate the model's theoretical consistency. We further extend to a two-dimensional model. Numerical simulations are given for both one-and two-dimensional case by using the hybrid Godunov-Glimm scheme to verify the model's performance.