Skip to main content Accessibility help

A Diffusively Corrected Multiclass Lighthill-Whitham-Richards Traffic Model with Anticipation Lengths and Reaction Times

  • Raimund Bürger (a1), Pep Mulet (a2) and Luis M. Villada (a1)


Multiclass Lighthill-Whitham-Richards traffic models [Benzoni-Gavage and Colombo, Euro. J. Appl. Math., 14 (2003), pp. 587–612; Wong and Wong, Transp. Res. A, 36 (2002), pp. 827–841] give rise to first-order systems of conservation laws that are hyperbolic under usual conditions, so that their associated Cauchy problems are well-posed. Anticipation lengths and reaction times can be incorporated into these models by adding certain conservative second-order terms to these first-order conservation laws. These terms can be diffusive under certain circumstances, thus, in principle, ensuring the stability of the solutions. The purpose of this paper is to analyze the stability of these diffusively corrected models under varying reaction times and anticipation lengths. It is demonstrated that instabilities may develop for high reaction times and short anticipation lengths, and that these instabilities may have controlled frequencies and amplitudes due to their nonlinear nature.


Corresponding author

Corresponding author. Email:


Hide All
[1]Abeynaike, A., Sederman, A. J., Khan, Y., Johns, M. L., Davidson, J. F. and Mackley, M. R., The experimental measurement and modelling of sedimentation and creaming for glyc-erol/biodiesel droplet dispersions, Chem. Eng. Sci., 79 (2012), pp. 125137.
[2]Batchelor, G. K. and Rensburg, R.W. Janse Van, Structure formation in bidisperse sedimentation, J. Fluid Mech., 166 (1986), pp. 379407.
[3]Benzoni-Gavage, S. and Colombo, R. M., An n-populations model for traffic flow, Euro. J. Appl. Math., 14 (2003), pp. 587612.
[4]Benzoni-Gavage, S., Colombo, R. M. and Gwiazda, P., Measure valued solutions to conservation laws motivated by traffic modelling, Proc. Royal Soc. A, 462 (2006), pp. 17911803.
[5]Berres, S., Bürger, R., Karlsen, K.H. and Tory, E. M., Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression, SIAM J. Appl. Math., 64 (2003), pp. 4180.
[6]Berres, S., Bürher, R. and Kozakevicius, A., Numerical approximation of oscillatory solutions of hyperbolic-elliptic systems of conservation laws by multiresolution schemes, Adv. Appl. Math. Mech., 1 (2009), pp. 581614.
[7]Bürger, R., García, A., Karlsen, K.H. and Towers, J. D., A family of numerical schemes for kinematic flows with discontinuous flux, J. Eng. Math., 60 (2008), pp. 387425.
[8]Bürger, R. and Karlsen, K.H., On a diffusively corrected kinematic-wave traffic flow model with changing road surface conditions, Math. Models Methods Appl. Sci., 13 (2003), pp. 17671799.
[9]Bürger, R., Karlsen, K.H., Tory, E. M. and Wendland, W. L., Model equations and instability regions for the sedimentation of polydisperse suspensions of spheres, ZAMM Z. Angew. Math. Mech., 82 (2002), pp. 699722.
[10]Bürger, R., Karlsen, K. H. and Towers, J. D., On some difference schemes and entropy conditions for a class of multi-species kinematic flow models with discontinuous flux, Netw. Heterog. Media, 5 (2010), pp. 461485.
[11]Bürger, R. and Kozakevicius, A., Adaptive multiresolution WENO schemes for multi-species kinematic flow models, J. Comput. Phys., 224 (2007), pp. 11901222.
[12]Bürger, R., Mulet, P. and Villada, L. M., Implicit-explicit methods for diffusively corrected multi-species kinematic flow models, Preprint 2012-21, Centro de Investigación en Ingeniería Matemática, Universidad de Concepción, 2012.
[13]Daganzo, C., Requiem for second-order fluid approximations of traffic flow, Transp. Res. B, 29 (1995), pp. 277286.
[14]Dick, A. C., Speed/flow relationships within an urban area, Traffic Eng. Control, 8 (1996), pp. 393396.
[15]Donat, R. and Mulet, P., Characteristic-based schemes for multi-class Lighthill-Whitham-Richards traffic models, J. Sci. Comput., 37 (2008), pp. 233250.
[16]Donat, R. and Mulet, P., A secular equation for the Jacobian matrix of certain multi-species kinematic flow models, Numer. Methods Partial Differential Equations, 26 (2010), pp. 159175.
[17]Greenberg, H., An analysis of traffic flow, Oper. Res., 7 (1959), pp. 7985.
[18]Herty, M., Kirchner, C. and Moutari, S., Multi-class traffic models on road networks, Commun. Math. Sci., 4 (2006), pp. 591608.
[19]Kurganov, A. and Polizzi, A., Non-oscillatory central schemes for traffic flow models with Arrhenius look-ahead dynamics, Netw. Heterog. Media, 4 (2009), pp. 431451.
[20]Kurganov, A. and Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160 (2000), pp. 241282.
[21]Lighthill, M. J. and Whitham, G. B., On kinematic waves: II. A theory of traffic flow on long crowded roads, Proc. Royal Soc. A, 229 (1955), pp. 317345.
[22]Logghe, S. and Immers, L. H., Multi-class kinematic wave theory of traffic flow, Transp. Res. B, 42 (2008), pp. 523541.
[23]Nelson, P., Synchronized traffic flow from a modified Lighthill-Whitman model, Phys. Rev. E, 61 (2000), pp. R6062–R6055.
[24]Nelson, P., Traveling-wave solution of the diffusively corrected kinematic-wave model, Math. Comput. Model., 35 (2002), pp. 561579.
[25]Nelson, P. and Sopasakis, A., The Chapman-Enskog expansion: a novel approach to hierarchical extension of Lighthill-Whitham models, In: Ceder, A. (ed.), Transportation and Traffic Theory: Proceedings of the 14th International Symposium on Transportation and Traffic Theory, Jerusalem, Israel, 20–23 July 1999. Elsevier, Amsterdam, 1999, pp. 5179.
[26]Ngoduy, D., Multiclass first-order modelling of traffic networks using discontinuous flow-density relationships, Transportmetrica, 6 (2010), pp. 121141.
[27]Ngoduy, D., Multiclass first-order traffic model using stochastic fundamental diagrams, Transportmetrica, 7 (2011), pp. 111125.
[28]Ngoduy, D., Effect of driver behaviours on the formation and dissipation of traffic flow instabilities, Nonlin. Dynamics, 69 (2012), pp. 969975.
[29]Ngoduy, D. and Tampere, C., Macroscopic effect of reaction time on traffic flow characteristics, Phys. Scripta, 80 (2009), paper 025802.
[30]Prigogine, I. and Herman, R., Kinetic Theory of Vehicular Traffic, American Elsevier, New York, 1971.
[31]Richards, P. I., Shock waves on the highway, Oper. Res., 4 (1956), pp. 4251.
[32]Rouvre, E. and Gagneux, G., Solution forte entropique de lois scalaires hyperboliques-paraboliques dégénérées, C. R. Acad. Sci. Paris Sér. I, 329 (1999), pp. 599602.
[33]Siebel, F. and Mauser, W., On the fundamental diagram of traffic flow, SIAM J. Appl. Math., 66 (2006), pp. 11501162.
[34]Sopasakis, A. and Katsoulakis, M. A., Stochastic modeling and simulation of traffic flow: asymmetric single exclusion process with Arrhenius look-ahead dynamics, SIAM J. Appl. Math., 66 (2006), pp. 921944.
[35]Treiber, M. and Kesting, A., Verkehrsdynamik und- simulation, Springer-Verlag, Berlin, 2010.
[36]Treiber, M., Kesting, A. and Helbing, D., Influence of reaction times and anticipation on stability of vehicular traffic flow, Transp. Res., Record No. 1999 (2007), pp. 2329.
[37]Wong, G. C. K. and Wong, S. C., A multi-class traffic flow model–an extension of LWR model with heterogeneous drivers, Transp. Res. A, 36 (2002), pp. 827841.
[38]Zhang, M., Shu, C.-W., Wong, G. C. K. and Wong, S. C., A weighted essentially non-oscillatory numerical scheme for a multi-class Lighthill-Whitham-Richards traffic flow model, J. Comput. Phys., 191 (2003), pp. 639659.
[39]Zhang, P., Liu, R.-X., Wong, S. C. and Dai, S. Q., Hyperbolicity and kinematic waves of a class of multi-population partial differential equations, Euro. J. Appl. Math., 17 (2006), 171200.
[40]Zhang, P., Wong, S. C. and Dai, S.-Q., A note on the weighted essentially non-oscillatory numerical scheme for a multi-class Lighthill-Whitham-Richards traffic flow model, Commun. Numer. Meth. Eng., 25 (2009), pp. 11201126.
[41]Zhang, P., Wong, S. C. and shu, C.-W., A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway, J. Comput. Phys., 212 (2006), pp. 739756.
[42]Zhang, P., Wong, S. C. and Xu, Z., A hybrid scheme for solving a multi-class traffic flow model with complex wave breaking, Comput. Meth. Appl. Mech. Eng., 197 (2008), pp. 38163827.



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed