ஜர்னல் ஆஃப் அப்ளைடு & கம்ப்யூட்டேஷனல் கணிதம்

Simulation and Analysis of Diffusion Type Traffic Flow Model Using Linear and Exponential Velocity Density Relationship

Abstract

Mojammel Haque1*, Most Halimatuj Sadia2 and Laek Sazzad Andallah3

In our modern era, traffic congestion in big cities has become threat for our daily life. It is one of the greatest problems in many developed countries of the world like Bangladesh. Various mathematical models have been employed to address the issue of traffic congestion. Traffic flow models assume that density and velocity are related. Bruce Greenshields first introduced traffic density-velocity connection. In this paper, we would like to study with linear velocity density relationship (Greenshields Model) and exponential velocity density relationship (Underwood Model) using the model for the flow of traffic based on diffusion. To make a comparative study between Greenshields and Underwood Models we solve the traffic flow model of the diffusion type as analytically as well as numerically. Due to analytical solution complexity, we use the finite difference method to solve the model numerically. We use explicit upwind, explicit centered, and explicit Lax-Wendrooff schemes to solve the model numerically. For each of the schemes, we present a comparison between linear and exponential velocity-density relationships. From this comparison, we can say that the exponential velocity- density relationship is appropriate to overcome the traffic congestion problem for each of the schemes.