American Institute of Mathematical Sciences

Advanced
2015, 12(2): 393-413. doi: 10.3934/mbe.2015.12.393

A hybrid model for traffic flow and crowd dynamics with random individual properties

Veronika Schleper 1,

1. 

Institute for Applied Analysis and Numerical Simulation, University of Stuttgart, D-70569 Stuttgart, Germany

Received  April 2014 Revised  June 2014 Published  December 2014

Based on an established mathematical model for the behavior of large crowds, a new model is derived that is able to take into account the statistical variation of individual maximum walking speeds. The same model is shown to be valid also in traffic flow situations, where for instance the statistical variation of preferred maximum speeds can be considered. The model involves explicit bounds on the state variables, such that a special Riemann solver is derived that is proved to respect the state constraints. Some care is devoted to a valid construction of random initial data, necessary for the use of the new model. The article also includes a numerical method that is shown to respect the bounds on the state variables and illustrative numerical examples, explaining the properties of the new model in comparison with established models.
Keywords: random initial data., traffic flow, non-local conservation laws, Crowd dynamics.
Mathematics Subject Classification: Primary: 35L40, 35L65, 90B20, 35R6.
Citation: Veronika Schleper. A hybrid model for traffic flow and crowd dynamics with random individual properties. Mathematical Biosciences & Engineering, 2015, 12 (2) : 393-413. doi: 10.3934/mbe.2015.12.393
References:
[1]

P. Amorim, R. M. Colombo and A. Teixeira, On the numerical integration of scalar conservation laws, preprint, 2013.

[2]

A. Aw and M. Rascle, Resurrection of "second order'' models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938 (electronic). doi: 10.1137/S0036139997332099.

[3]

N. Bellomo and C. Dogbé, On the modelling crowd dynamics from scaling to hyperbolic macroscopic models, Math. Models Methods Appl. Sci., 18 (2008), 1317-1345. doi: 10.1142/S0218202508003054.

[4]

N. Bellomo and C. Dogbé, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Review, 53 (2011), 409-463. doi: 10.1137/090746677.

[5]

R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of non-local models for pedestrian traffic, Mathematical Models and Methods in the Applied Sciences, 22 (2012), 1150023, 34 p. doi: 10.1142/S0218202511500230.

[6]

R. M. Colombo and N. Pogodaev, Confinement strategies in a model for the interaction between individuals and a continuum, SIAM J. Appl. Dyn. Syst., 11 (2012), 741-770. doi: 10.1137/110854321.

[7]

M. Crandall and A. Majda, The method of fractional steps for conservation laws, Numer. Math., 34 (1980), 285-314. doi: 10.1007/BF01396704.

[8]

E. Cristiani, B. Piccoli and A. Tosin, Multiscale modeling of granular flows with application to crowd dynamics, Multiscale Model. Simul., 9 (2011), 155-182. doi: 10.1137/100797515.

[9]

C. F. Daganzo, Requiem for second-order fluid approximations to traffic flow, Transp. Res. B, 29 (1995), 277-286. doi: 10.1016/0191-2615(95)00007-Z.

[10]

G. Dal Maso, P. G. Lefloch and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. (9), 74 (1995), 483-548.

[11]

D. Helbing, Derivation of non-local macroscopic traffic equations and consistent traffic pressures from microscopic car-following models, The European Physical Journal B, 69 (2009), 539-548. doi: 10.1140/epjb/e2009-00192-5.

[12]

D. Helbing and A. F. Johansson, On the controversy around Daganzo's requiem for and Aw-Rascle's resurrection of second-order traffic flow models, Modelling and Optimisation of Flows on Networks, (2013), 271-302. doi: 10.1007/978-3-642-32160-3_4.

[13]

R. L. Hughes, A continuum theory for the flow of pedestrians, Transportation Research Part B: Methodological, 36 (2002), 507-535. doi: 10.1016/S0191-2615(01)00015-7.

[14]

S. N. Kružhkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255.

[15]

M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089.

[16]

S. Mishra and C. Schwab, Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random initial data, Math. Comp., 81 (2012), 1979-2018. doi: 10.1090/S0025-5718-2012-02574-9.

[17]

P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.

[18]

M. Sabry Hassouna and A. A. Farag, Multistencils fast marching methods: A highly accurate solution to the eikonal equation on cartesian domains, IEEE Transactions on Pattern Analysis and Machine Intelligence, 29 (2007), 1563-1574. doi: 10.1109/TPAMI.2007.1154.

[19]

Y. Sugiyama, M. Fukui, M. Kikuchi, K. Hasebe, A. Nakayama, K. Nishinari, S.-i. Tadaki and S. Yukawa, Traffic jams without bottlenecks-experimental evidence for the physical mechanism of the formation of a jam, New Journal of Physics, 10 (2008), 033001. doi: 10.1088/1367-2630/10/3/033001.

[20]

S.-i. Tadaki, M. Kikuchi, F. Minoru, A. Nakayama, K. Nishinari, A. Shibata, Y. Sugiyama, T. Yosida and S. Yukawa, Phase transition in traffic jam experiment on a circuit, New Journal of Physics, 15 (2013), 103034. doi: 10.1088/1367-2630/15/10/103034.

show all references

References:
[1]

P. Amorim, R. M. Colombo and A. Teixeira, On the numerical integration of scalar conservation laws, preprint, 2013.

[2]

A. Aw and M. Rascle, Resurrection of "second order'' models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938 (electronic). doi: 10.1137/S0036139997332099.

[3]

N. Bellomo and C. Dogbé, On the modelling crowd dynamics from scaling to hyperbolic macroscopic models, Math. Models Methods Appl. Sci., 18 (2008), 1317-1345. doi: 10.1142/S0218202508003054.

[4]

N. Bellomo and C. Dogbé, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Review, 53 (2011), 409-463. doi: 10.1137/090746677.

[5]

R. M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of non-local models for pedestrian traffic, Mathematical Models and Methods in the Applied Sciences, 22 (2012), 1150023, 34 p. doi: 10.1142/S0218202511500230.

[6]

R. M. Colombo and N. Pogodaev, Confinement strategies in a model for the interaction between individuals and a continuum, SIAM J. Appl. Dyn. Syst., 11 (2012), 741-770. doi: 10.1137/110854321.

[7]

M. Crandall and A. Majda, The method of fractional steps for conservation laws, Numer. Math., 34 (1980), 285-314. doi: 10.1007/BF01396704.

[8]

E. Cristiani, B. Piccoli and A. Tosin, Multiscale modeling of granular flows with application to crowd dynamics, Multiscale Model. Simul., 9 (2011), 155-182. doi: 10.1137/100797515.

[9]

C. F. Daganzo, Requiem for second-order fluid approximations to traffic flow, Transp. Res. B, 29 (1995), 277-286. doi: 10.1016/0191-2615(95)00007-Z.

[10]

G. Dal Maso, P. G. Lefloch and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. (9), 74 (1995), 483-548.

[11]

D. Helbing, Derivation of non-local macroscopic traffic equations and consistent traffic pressures from microscopic car-following models, The European Physical Journal B, 69 (2009), 539-548. doi: 10.1140/epjb/e2009-00192-5.

[12]

D. Helbing and A. F. Johansson, On the controversy around Daganzo's requiem for and Aw-Rascle's resurrection of second-order traffic flow models, Modelling and Optimisation of Flows on Networks, (2013), 271-302. doi: 10.1007/978-3-642-32160-3_4.

[13]

R. L. Hughes, A continuum theory for the flow of pedestrians, Transportation Research Part B: Methodological, 36 (2002), 507-535. doi: 10.1016/S0191-2615(01)00015-7.

[14]

S. N. Kružhkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255.

[15]

M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089.

[16]

S. Mishra and C. Schwab, Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random initial data, Math. Comp., 81 (2012), 1979-2018. doi: 10.1090/S0025-5718-2012-02574-9.

[17]

P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.

[18]

M. Sabry Hassouna and A. A. Farag, Multistencils fast marching methods: A highly accurate solution to the eikonal equation on cartesian domains, IEEE Transactions on Pattern Analysis and Machine Intelligence, 29 (2007), 1563-1574. doi: 10.1109/TPAMI.2007.1154.

[19]

Y. Sugiyama, M. Fukui, M. Kikuchi, K. Hasebe, A. Nakayama, K. Nishinari, S.-i. Tadaki and S. Yukawa, Traffic jams without bottlenecks-experimental evidence for the physical mechanism of the formation of a jam, New Journal of Physics, 10 (2008), 033001. doi: 10.1088/1367-2630/10/3/033001.

[20]

S.-i. Tadaki, M. Kikuchi, F. Minoru, A. Nakayama, K. Nishinari, A. Shibata, Y. Sugiyama, T. Yosida and S. Yukawa, Phase transition in traffic jam experiment on a circuit, New Journal of Physics, 15 (2013), 103034. doi: 10.1088/1367-2630/15/10/103034.

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