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

Mahmudul Hasan¹, Shirin Sultana², Laek Sazzad Andallah¹, Tauhedul Azam¹

¹Department of Mathematics, Jahangirnagar University, Dhaka, Bangladesh.
²Department of Natural Sciences, Daffodil International University, Dhaka, Bangladesh.

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.

Keywords

Traffic Flow, Traffic Velocity, Traffic Density, Lax-Friedrich’s Scheme

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Klar, A., Kuhne, R.D. and Wegener, R. (1996) Mathematical Models for Vehicular Traffic. Technical University of Kaiserslautern, Kaiserslautern.
[2]	Bretti, G., Natalini, R. and Piccoli, B. (2007) A Fluid-Dynamic Traffic Model on Road Networks. Compute Methods Eng., 14, 139-172.
[3]	Lighthill, M.J. and Whitham, G.B. (1955) On Kinematic Waves, I, Flood Movement in Long Rivers. Proceedings of the Royal Society of London A, 229, 281-316. http://dx.doi.org/10.1098/rspa.1955.0088
[4]	Lighthill, M.J. and Whitham, G.B. (1955) A Theory of Traffic Flow on Long Crowded Roads. Proceedings of the Royal Society of London A, 229, 317-345. http://dx.doi.org/10.1098/rspa.1955.0089
[5]	Haberman, R. (1977) Mathematical Models. Prentice-Hall, Inc., Delhi.
[6]	Wegener, R. and Klar, A. (1995) A Kinetic Model for Vehicular Traffic Derived from a Stochastic Microscopic Model. Berichte der Arbeitsgruppe Technomathematik 138, Universitat, Kaiserslautern.
[7]	Andallah, L.S., Ali, S., Gani, M.O., Pandit, M.K. and Akhter, J. (2009) A Finite Difference Scheme for a Traffic Flow Model Based on a Linear Velocity-Density Function. Jahangirnagar University Journal of Science, 32, 61-71.
[8]	Kabir, M.H., Gani, M.O. and Andallah, L.S. (2010) Numerical Simulation of a Mathematical Traffic Flow Model Based on a Non-Linear Velocity-Density Function. Journal of Bangladesh Academy of Sciences, 34, 15-22.
[9]	Leveque, R.J. (1992) Numerical Methods for Conservation Laws. 2nd Edition, Springer, Berlin.
[10]	Daganzo, C.F. (1995) A Finite Difference Approximation of the Kinematic Wave Model of Traffic Flow. Transportation Research Part B: Methodological, 29, 261-276.