### Finite volume method for simulations of traffic dynamics with exits and entrances

#### Abstract

We discuss a macroscopic study of vehicle traffic dynamics in which traffic flow is considered as a continuum governed by the kinematic Lighthill--Whitham--Richard model, with a source term accounting for traffic entries and exits through a junction. The kinematic model is solved using finite volume method, with the flux function is approximated using the upwind method. In order to prevent flows which exceed road capacity and to preserve the positivity of traffic density in simulations with entries and exits, a stop-go procedure is adopted. The resulting scheme is used to simulate the responses of traffic density to light and heavy entrances from a junction, and the dynamics predicted by the simulations are shown to conform well to analytical solutions. With a validated numerical algorithm at hand, we simulate traffic dynamics for several scenarios on a roadway with entrances and exits. Depending on the rate of exit or entrance and the initial condition, density profiles in the form of shock wave and rarefaction wave may appear.

**References**- P. Bagnerini, R. M. Colombo, A. Corli, S. Pedretti, Conservation Versus Balance Laws in Traffic Flow,
*Traffic and Granular Flow '03*, 235–240, (2005). doi:10.1007/3-540-28091-X_22 - C. F. Daganzo, The cell transmission model: a dynamic representation of highway traffic consistent with hydrodynamic theory,
*Transportation Research B*,**28**(4), 269–287 (1994). doi:10.1016/0191-2615(94)90002-7 - C. F. Daganzo, The cell transmission model II: Network traffic,
*Transportation Research B*,**29**(2), 79–93, (1995). doi:10.1016/0191-2615(94)00022-R - N. D. Fowkes, J. J. Mahony,
*An Introduction to Mathematical Modelling*. John Wiley and Sons, England, (1994). - R. Haberman,
*Mathematical Model*, Prentice–Hall, Inc., New York, (1977). doi:10.1137/1.9781611971156 - H. Holden and N. H. Risebro. A mathematical model of traffic flow on a network of unidirectional roads,
*SIAM Journal on Mathematical Analysis*,**26**(4):999–1017, (1995). doi:10.1137/S0036141093243289 - J. P. Lebacque, The godunov scheme and what it means for first order traffic flow models, In
*The International Symposium on Transportation and Traffic Theory*, Lyon, France, (1996). http://worldcat.org/isbn/0080425860 - R. M. M. Mattheij, ten Thije Boonkkamp, S. W. Rienstra, Partial Differential Equation: Modelling, Analysis, Computation,
*The Netherlands Society for Industrial and Applied Mathematics*, Philadelphia, (2005). doi:10.1137/1.9780898718270 - M. Mercier, Traffic flow modelling with junctions,
*Jour. of Math. Analysis and Applications*,**350**(1), 369–383, (2009). doi:10.1016/j.jmaa.2008.09.040 - G. Orosz, R. E. Wilson, G. Stepana, Traffic jams: dynamics and control,
*Phil. Trans. R. Soc. A*368, 2010, pp. 4455–4479. doi:10.1098/rsta.2010.0205 - M. Papageorgiou, Dynamic modelling, assignment and route guidance in traffic networks,
*Transportation Research B*, 24(6), (1990), pp. 471–495. doi:10.1016/0191-2615(90)90041-V - S.R. Pudjaprasetya, J. Bunawan, C. Novtiar, Traffic light or roundabout? Analysis using the modified kinematic LWR model,
*East Asian Journal of Applied Mathematics*,**6**(1), 80–88, (2016). doi:10.4208/eajam.210815.281215a

#### Keywords

Hyperbolic Conservation Laws, Continuum Traffic Model, finite volume method

DOI: http://dx.doi.org/10.21914/anziamj.v60i0.12435

**Remember,**for most actions you have to record/upload into this online system

**and then**inform the editor/author via clicking on an email icon or Completion button.

ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.