Traffic flow on star graph: Nonlinear diffusion

TL;DR

This paper models urban traffic flow on star graphs using nonlinear diffusion equations, deriving analytical and numerical solutions for vehicle densities and fundamental diagrams at different densities.

Contribution

It introduces a nonlinear diffusion framework for traffic flow on star graphs and provides analytical solutions matching numerical results.

Findings

Analytical solutions for vehicle densities at various densities.

Numerical steady-state densities on different graph types.

Urban-scale fundamental diagrams derived from the model.

Abstract

We study the urban-scale macroscopic traffic flow in city networks. Star graph is considered as traffic network. Star graphs with controlled traffic flow are transformed to various cell-transmission graphs by using the cell transmission method. The dynamic equations of vehicular densities on all nodes (roads) are presented on cell-transmission graphs by using the speed-matching model. The density equations are given by nonlinear-diffusion equations. The traffic flow on star graph is mapped to the nonlinear diffusion process on the cell-transmission graphs. At low mean density, the dynamic equations of densities can be approximated by the conventional diffusion equations. At low and high mean densities, the analytical solutions of densities on all nodes (roads) are obtained on cell-transmission complete, cycle and star graphs. By solving the dynamic equations numerically, the densities on all roads are derived at a steady state. The urban-scale macroscopic fundamental diagrams are obtained numerically on the cell-transmission graphs. The analytical solutions agree with the numerical solutions.