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

TL;DR

The work addresses the limitation of classical ARZ-type models in enforcing physical bounds on density $\rho$ and velocity $u$ by introducing a refined microscopic acceleration law and a conserved advected quantity $\rho\widetilde{u}p$ with $\widetilde{u}=(\frac{1}{u}-\frac{1}{u^*})^{-1}$ and $p=(\frac{1}{\rho}-\frac{1}{\rho^*})^{-\gamma}$. The authors derive a macroscopic system $\rho_t +(\rho u)_x=0$, $(\rho \widetilde{u} p)_t+(\rho u \widetilde{u} p)_x=0$, analyze characteristics, and solve Riemann problems to validate theoretical consistency, then extend to 2D. A hybrid Godunov–Glimm scheme is developed to accurately resolve contact discontinuities and nonlinear waves, with Strang splitting used for the 2D extension and HLL flux in the split steps. Numerical tests in 1D and 2D demonstrate that the model yields realistic fundamental diagrams, enhanced jam-avoidance behavior, and robust wave-resolution, offering a physically consistent and computationally efficient framework for complex traffic dynamics in networks.

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.

Figures (23)

• Figure 1: Comparison of fundamental diagrams among the three models.
• Figure 2: Comparison of fundamental diagrams with different $\gamma$.
• Figure 3: Shock and rarefaction wave curves
• Figure 4: The Riemann solution of case 1 and case 2.
• Figure 5: The Riemann solution in $\left[x_{j-1}, x_{j+1}\right]$.