American Institute of Mathematical Sciences

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December  2011, 6(4): 681-694. doi: 10.3934/nhm.2011.6.681

Shock formation in a traffic flow model with Arrhenius look-ahead dynamics

Dong Li 1, and  Tong Li 2,

1. 

Department of Mathematics, University of Iowa, Iowa City, IA 52242, United States
2. 

Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, IA 52242-1419

Received  January 2011 Revised  October 2011 Published  December 2011

We consider a nonlocal traffic flow model with Arrhenius look-ahead dynamics. We provide a complete local theory and give the blowup alternative of solutions to the conservation law with a nonlocal flux. We show that the finite time blowup of solutions must occur at the level of the first order derivative of the solution. Furthermore, we prove that finite time singularities do occur for several types of physical initial data by analyzing the solutions on different characteristic lines. These results are new and are consistent with the blowups observed in previous numerical simulations on the nonlocal traffic flow model [6].
Keywords: Traffic flow, nonlocal flux, shock waves, Arrhenius look-ahead dynamics, blowup criteria..
Mathematics Subject Classification: Primary: 35B44, 35L65, 35L67, 35Q35, 76L05, 90B2.
Citation: Dong Li, Tong Li. Shock formation in a traffic flow model with Arrhenius look-ahead dynamics. Networks & Heterogeneous Media, 2011, 6 (4) : 681-694. doi: 10.3934/nhm.2011.6.681
References:
[1]

M. Bando, K. Hasebe, A. Nakayama, A. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation, Phys. Rev. E, 51 (1995), 1035-1042. doi: 10.1103/PhysRevE.51.1035.  Google Scholar

[2]

D. Helbing, Traffic and related self-driven many-particle systems, Rev. Modern Phy., 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067.  Google Scholar

[3]

W. L. Jin and H. M. Zhang, The formation and structure of vehicle clusters in the Payne-Whitham traffic flow model, Transportation Research, B., 37 (2003), 207-223. doi: 10.1016/S0191-2615(02)00008-5.  Google Scholar

[4]

B. S. Kerner and P. Konhäuser, Structure and parameters of clusters in traffic flow, Physical Review E, 50 (1994), 54-83. doi: 10.1103/PhysRevE.50.54.  Google Scholar

[5]

A. Klar and R. Wegener, Kinetic derivation of macroscopic anticipation models for vehicular traffic, SIAM J. Appl. Math., 60 (2000), 1749-1766. doi: 10.1137/S0036139999356181.  Google Scholar

[6]

A. Kurganov and A. Polizzi, Non-oscillatory central schemes for traffic flow models with Arrhenius look-ahead dynamics, Netw. Heterog. Media, 4 (2009), 431-451. doi: 10.3934/nhm.2009.4.431.  Google Scholar

[7]

H. Y. Lee, H.-W. Lee and D. Kim, Steady-state solutions of hydrodynamic traffic models, Phys. Rev. E, 69 (2004), 016118-1-016118-7. Google Scholar

[8]

T. Li, Nonlinear dynamics of traffic jams, Physica D, 207 (2005), 41-51. doi: 10.1016/j.physd.2005.05.011.  Google Scholar

[9]

T. Li, Stability of traveling waves in quasi-linear hyperbolic systems with relaxation and diffusion, SIAM J. Math. Anal., 40 (2008), 1058-1075. doi: 10.1137/070690638.  Google Scholar

[10]

A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Cambridge Univ. Press, 2002. Google Scholar

[11]

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.  Google Scholar

[12]

T. Nagatani, The physics of traffic jams, Rep. Prog. Phys., 65 (2002), 1331-1386. doi: 10.1088/0034-4885/65/9/203.  Google Scholar

[13]

K. Nagel, Particle hopping models and traffic flow theory, Phys. Rev. E, 53 (1996), 4655-4672. doi: 10.1103/PhysRevE.53.4655.  Google Scholar

[14]

I. Prigogine and R. Herman, "Kinetic Theory of Vehicular Traffic," American Elsevier Publishing Company Inc., New York, 1971. Google Scholar

[15]

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

[16]

A. Sopasakis and M. Katsoulakis, Stochastic modeling and simulation of traffic flow: Asymmetric single exclusion process with Arrhenius look-ahead dynamics, SIAM J. Appl. Math., 6 (2006), 921-944. doi: 10.1137/040617790.  Google Scholar

[17]

G. B. Whitham, "Linear and Nonlinear Waves," Wiley, New York, 1974. Google Scholar

show all references

References:
[1]

M. Bando, K. Hasebe, A. Nakayama, A. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation, Phys. Rev. E, 51 (1995), 1035-1042. doi: 10.1103/PhysRevE.51.1035.  Google Scholar

[2]

D. Helbing, Traffic and related self-driven many-particle systems, Rev. Modern Phy., 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067.  Google Scholar

[3]

W. L. Jin and H. M. Zhang, The formation and structure of vehicle clusters in the Payne-Whitham traffic flow model, Transportation Research, B., 37 (2003), 207-223. doi: 10.1016/S0191-2615(02)00008-5.  Google Scholar

[4]

B. S. Kerner and P. Konhäuser, Structure and parameters of clusters in traffic flow, Physical Review E, 50 (1994), 54-83. doi: 10.1103/PhysRevE.50.54.  Google Scholar

[5]

A. Klar and R. Wegener, Kinetic derivation of macroscopic anticipation models for vehicular traffic, SIAM J. Appl. Math., 60 (2000), 1749-1766. doi: 10.1137/S0036139999356181.  Google Scholar

[6]

A. Kurganov and A. Polizzi, Non-oscillatory central schemes for traffic flow models with Arrhenius look-ahead dynamics, Netw. Heterog. Media, 4 (2009), 431-451. doi: 10.3934/nhm.2009.4.431.  Google Scholar

[7]

H. Y. Lee, H.-W. Lee and D. Kim, Steady-state solutions of hydrodynamic traffic models, Phys. Rev. E, 69 (2004), 016118-1-016118-7. Google Scholar

[8]

T. Li, Nonlinear dynamics of traffic jams, Physica D, 207 (2005), 41-51. doi: 10.1016/j.physd.2005.05.011.  Google Scholar

[9]

T. Li, Stability of traveling waves in quasi-linear hyperbolic systems with relaxation and diffusion, SIAM J. Math. Anal., 40 (2008), 1058-1075. doi: 10.1137/070690638.  Google Scholar

[10]

A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Cambridge Univ. Press, 2002. Google Scholar

[11]

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.  Google Scholar

[12]

T. Nagatani, The physics of traffic jams, Rep. Prog. Phys., 65 (2002), 1331-1386. doi: 10.1088/0034-4885/65/9/203.  Google Scholar

[13]

K. Nagel, Particle hopping models and traffic flow theory, Phys. Rev. E, 53 (1996), 4655-4672. doi: 10.1103/PhysRevE.53.4655.  Google Scholar

[14]

I. Prigogine and R. Herman, "Kinetic Theory of Vehicular Traffic," American Elsevier Publishing Company Inc., New York, 1971. Google Scholar

[15]

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

[16]

A. Sopasakis and M. Katsoulakis, Stochastic modeling and simulation of traffic flow: Asymmetric single exclusion process with Arrhenius look-ahead dynamics, SIAM J. Appl. Math., 6 (2006), 921-944. doi: 10.1137/040617790.  Google Scholar

[17]

G. B. Whitham, "Linear and Nonlinear Waves," Wiley, New York, 1974. Google Scholar

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