This
paper investigates the effect of inflow, outflow and shock waves in a single lane
highway traffic flow problem. A constant source term has been introduced to demonstrate
the inflow and outflow. The classical Lighthill Whitham and Richards (LWR) model
combined with the Greenshields model is used to obtain analytical and numerical
solutions. The model is treated as an IBVP and numerical solutions are presented
using Lax Friedrichs scheme. Godunov method is also used to present shock wave analysis.
The numerical procedures adopted in this investigation yield results which are very
much consistent with real life scenario in terms of traffic density and velocity.
References
[1]
Daganzo, C.F. (1995) A Finite Difference Approximation of the Kinematic Wave Model of Traffic Flow. Transportation Research Part B: Methodological, 29, 261-276.
[2]
Zhang, H.M. (2001) A Finite Difference Approximation of a Non-Equilibrium Traffic Flow Model. Transportation Research Part B: Methodological, 35, 337-365.
Ali, A., Andallah, L.S. and Hossain, Z. (2015) Numerical Solution of a Fluid Dynamic Traffic Flow Model Associated with a Constant Rate Inflow. American Journal of Computational and Applied Mathematics, 5, 18-26.
[5]
Bagnerini, P., Colombo, R.M. and Corli, A. (2006) On the Role of Source Terms in Continuum Traffic Flow Models. Mathematical and Computer Modelling, 44, 917-930. http://dx.doi.org/10.1016/j.mcm.2006.02.019
[6]
Lighthill, M.J. and Whitham, G.B. (1955) On Kinematic Waves. II. A Theory of Traffic Flow on Long Crowded Roads. Proceedings of the Royal Society of London. Series A, 229, 317-345. http://dx.doi.org/10.1098/rspa.1955.0089
[7]
Greenshields, B.D. (1935) A Study of Traffic Capacity. Highway Research Board, 14, 448-477.
[8]
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, 33, 61-71.
[9]
Haberman, R. (1977) Mathematical Models: Mechanical Vibrations, Population Dynamics and Traffic Flow. Prentice-Hall, Inc., Englewood Cliffs.
[10]
Le Veque, R.J. (1992) Numerical Methods for Conservation Laws. Birkhauser, Berlin.
[11]
Hoogendoorn, S.P., Luding, S., Bovy, P.H.L., Schreckberg, M. and Wolf, D.E. (2005) Traffic and Granular Flow ’03. Springer. http://dx.doi.org/10.1007/3-540-28091-x
[12]
Trangenstein, J.A. (2007) Numerical Solution of Hyperbolic Conservation Laws. Department of Mathematics, Duke University, Durham, NC 27708-0320. https://services.math.duke.edu/~jliu/math226/
[13]
Hasan, M., Sultana, S., Andallah, L. and Azam, T. (2015) Lax-Friedrich Scheme for the Numerical Simulation of a Traffic Flow Model Based on a Nonlinear Velocity Density Relation. American Journal of Computational Mathematics, 5, 186-194. http://dx.doi.org/10.4236/ajcm.2015.52015
[14]
Tabak, E.G. (2004) Notes for PDE I (Traffic Flow) Spring 2004 [PDF Document]. Retrieved from Lecture Notes Online Website. http://math.nyu.edu/faculty/tabak/PDEs/
