Lax-Friedrich Scheme for the Numerical Simulation of a Traffic Flow Model Based on a Nonlinear Velocity Density Relation

Mahmudul Hasan,Shirin Sultana,Tauhedul Azam,Laek Sazzad Andallah

doi:10.4236/ajcm.2015.52015

Abstract

A fluid dynamic traffic flow model based on a non-linear velocity-density function is considered. The model provides a quasi-linear first order hyperbolic partial differential equation which is appended with initial and boundary data and turns out an initial boundary value problem (IBVP). A first order explicit finite difference scheme of the IBVP known as Lax-Friedrich’s scheme for our model is presented and a well-posedness and stability condition of the scheme is established. The numerical scheme is implemented in order to perform the numerical features of error estimation and rate of convergence. Fundamental diagram, density, velocity and flux profiles are presented.

Highlights

With the increasingly rapid economic globalization and urbanization, more problems are brought to our attention
At the core of traffic congestion, development of traffic management is the need of time
An efficient traffic control and management is essential in order to get rid of such huge traffic congestion

Summary

Introduction

With the increasingly rapid economic globalization and urbanization, more problems are brought to our attention. Traffic jams are a major problem in most of the cities. At the core of traffic congestion, development of traffic management is the need of time. Modeling and computer simulation play an increasing role in the flow management. Many scientists have been working to develop various mathematical models [1] [2] in order to describe traffic flow. (2015) Lax-Friedrich Scheme for the Numerical Simulation of a Traffic Flow Model Based on a Nonlinear Velocity Density Relation. S. Andallah have used Non-Linear Velocity-Density Function for the development of Traffic Flow Model. We have used a non-linear velocity-density relationship but we have presented the Lax-Friedrich’s scheme for the development of our model. Fundamental diagram, density, velocity and flux profiles are presented

General Feature of the Model

Exact Solution of the Non-Linear PDE

A Finite Difference Scheme for the Model of IBVP

Stability Condition and Physical Constraint Conditions

Error Estimation of the Numerical Scheme

Conclusion