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

Published online by Cambridge University Press:  03 June 2015

Raimund Bürger ,
Pep Mulet  and
Luis M. Villada
Raimund Bürger*
Affiliation:
CI2MA, and Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción, Chile
Pep Mulet*
Affiliation:
Departament de Matemàtica Aplicada, Universitat de València, Av. Dr. Moliner 50, E-46100 Burjassot, Spain
Luis M. Villada*
Affiliation:
CI2MA, and Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción, Chile
*
Corresponding author. Email: rburger@ing-mat.udec.cl
Email: mulet@uv.es
Email: lmvillada@ing-mat.udec.cl
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Abstract

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.

Keywords

35K6535L6576T9976E99Traffic flowmultispecies modelanticipation lengthreaction timediffusive correctionquasilinear system of second-order PDEsstabilitynumerical simulation
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Issue 5 , October 2013 , pp. 728 - 758
Copyright
Copyright © Global-Science Press 2013

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References

[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. 125–137.CrossRefGoogle Scholar
[2]Batchelor, G. K. and Rensburg, R.W. Janse Van, Structure formation in bidisperse sedimentation, J. Fluid Mech., 166 (1986), pp. 379–407.Google Scholar
[3]Benzoni-Gavage, S. and Colombo, R. M., An n-populations model for traffic flow, Euro. J. Appl. Math., 14 (2003), pp. 587–612.Google Scholar
[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. 1791–1803.CrossRefGoogle Scholar
[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. 41–80.Google Scholar
[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. 581–614.CrossRefGoogle Scholar
[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. 387–425.CrossRefGoogle Scholar
[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. 1767–1799.Google Scholar
[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. 699–722.Google Scholar
[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. 461–485.Google Scholar
[11]Bürger, R. and Kozakevicius, A., Adaptive multiresolution WENO schemes for multi-species kinematic flow models, J. Comput. Phys., 224 (2007), pp. 1190–1222.CrossRefGoogle Scholar
[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.Google Scholar
[13]Daganzo, C., Requiem for second-order fluid approximations of traffic flow, Transp. Res. B, 29 (1995), pp. 277–286.Google Scholar
[14]Dick, A. C., Speed/flow relationships within an urban area, Traffic Eng. Control, 8 (1996), pp. 393–396.Google Scholar
[15]Donat, R. and Mulet, P., Characteristic-based schemes for multi-class Lighthill-Whitham-Richards traffic models, J. Sci. Comput., 37 (2008), pp. 233–250.Google Scholar
[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. 159–175.Google Scholar
[17]Greenberg, H., An analysis of traffic flow, Oper. Res., 7 (1959), pp. 79–85.Google Scholar
[18]Herty, M., Kirchner, C. and Moutari, S., Multi-class traffic models on road networks, Commun. Math. Sci., 4 (2006), pp. 591–608.CrossRefGoogle Scholar
[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. 431–451.Google Scholar
[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. 241–282.CrossRefGoogle Scholar
[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. 317–345.Google Scholar
[22]Logghe, S. and Immers, L. H., Multi-class kinematic wave theory of traffic flow, Transp. Res. B, 42 (2008), pp. 523–541.Google Scholar
[23]Nelson, P., Synchronized traffic flow from a modified Lighthill-Whitman model, Phys. Rev. E, 61 (2000), pp. R6062–R6055.Google Scholar
[24]Nelson, P., Traveling-wave solution of the diffusively corrected kinematic-wave model, Math. Comput. Model., 35 (2002), pp. 561–579.CrossRefGoogle Scholar
[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. 51–79.Google Scholar
[26]Ngoduy, D., Multiclass first-order modelling of traffic networks using discontinuous flow-density relationships, Transportmetrica, 6 (2010), pp. 121–141.Google Scholar
[27]Ngoduy, D., Multiclass first-order traffic model using stochastic fundamental diagrams, Transportmetrica, 7 (2011), pp. 111–125.Google Scholar
[28]Ngoduy, D., Effect of driver behaviours on the formation and dissipation of traffic flow instabilities, Nonlin. Dynamics, 69 (2012), pp. 969–975.CrossRefGoogle Scholar
[29]Ngoduy, D. and Tampere, C., Macroscopic effect of reaction time on traffic flow characteristics, Phys. Scripta, 80 (2009), paper 025802.Google Scholar
[30]Prigogine, I. and Herman, R., Kinetic Theory of Vehicular Traffic, American Elsevier, New York, 1971.Google Scholar
[31]Richards, P. I., Shock waves on the highway, Oper. Res., 4 (1956), pp. 42–51.Google Scholar
[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. 599–602.Google Scholar
[33]Siebel, F. and Mauser, W., On the fundamental diagram of traffic flow, SIAM J. Appl. Math., 66 (2006), pp. 1150–1162.Google Scholar
[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. 921–944.Google Scholar
[35]Treiber, M. and Kesting, A., Verkehrsdynamik und- simulation, Springer-Verlag, Berlin, 2010.Google Scholar
[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. 23–29.Google Scholar
[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. 827–841.Google Scholar
[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. 639–659.CrossRefGoogle Scholar
[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), 171–200.Google Scholar
[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. 1120–1126.CrossRefGoogle Scholar
[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. 739–756.Google Scholar
[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. 3816–3827.Google Scholar