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

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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

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

Abstract
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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. doi: 10.1103/PhysRevE.51.1035.
[2]	D. Helbing, Traffic and related self-driven many-particle systems,, Rev. Modern Phy., 73 (2001), 1067. doi: 10.1103/RevModPhys.73.1067.
[3]	W. L. Jin and H. M. Zhang, The formation and structure of vehicle clusters in the Payne-Whitham traffic flow model,, Transportation Research, 37 (2003), 207. doi: 10.1016/S0191-2615(02)00008-5.
[4]	B. S. Kerner and P. Konhäuser, Structure and parameters of clusters in traffic flow,, Physical Review E, 50 (1994), 54. doi: 10.1103/PhysRevE.50.54.
[5]	A. Klar and R. Wegener, Kinetic derivation of macroscopic anticipation models for vehicular traffic,, SIAM J. Appl. Math., 60 (2000), 1749. doi: 10.1137/S0036139999356181.
[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. doi: 10.3934/nhm.2009.4.431.
[7]	H. Y. Lee, H.-W. Lee and D. Kim, Steady-state solutions of hydrodynamic traffic models,, Phys. Rev. E, 69 (2004), 016118.
[8]	T. Li, Nonlinear dynamics of traffic jams,, Physica D, 207 (2005), 41. doi: 10.1016/j.physd.2005.05.011.
[9]	T. Li, Stability of traveling waves in quasi-linear hyperbolic systems with relaxation and diffusion,, SIAM J. Math. Anal., 40 (2008), 1058. doi: 10.1137/070690638.
[10]	A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow,", Cambridge Univ. Press, (2002).
[11]	M. J. Lighthill and G. B. Whitham, On kinematic waves: II. A theory of traffic flow on long crowded roads,, Proc. Roy. Soc., 229 (1955), 317.
[12]	T. Nagatani, The physics of traffic jams,, Rep. Prog. Phys., 65 (2002), 1331. doi: 10.1088/0034-4885/65/9/203.
[13]	K. Nagel, Particle hopping models and traffic flow theory,, Phys. Rev. E, 53 (1996), 4655. doi: 10.1103/PhysRevE.53.4655.
[14]	I. Prigogine and R. Herman, "Kinetic Theory of Vehicular Traffic,", American Elsevier Publishing Company Inc., (1971).
[15]	P. I. Richards, Shock waves on highway,, Operations Research, 4 (1956), 42. doi: 10.1287/opre.4.1.42.
[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. doi: 10.1137/040617790.
[17]	G. B. Whitham, "Linear and Nonlinear Waves,", Wiley, (1974).

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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. doi: 10.1103/PhysRevE.51.1035.
[2]	D. Helbing, Traffic and related self-driven many-particle systems,, Rev. Modern Phy., 73 (2001), 1067. doi: 10.1103/RevModPhys.73.1067.
[3]	W. L. Jin and H. M. Zhang, The formation and structure of vehicle clusters in the Payne-Whitham traffic flow model,, Transportation Research, 37 (2003), 207. doi: 10.1016/S0191-2615(02)00008-5.
[4]	B. S. Kerner and P. Konhäuser, Structure and parameters of clusters in traffic flow,, Physical Review E, 50 (1994), 54. doi: 10.1103/PhysRevE.50.54.
[5]	A. Klar and R. Wegener, Kinetic derivation of macroscopic anticipation models for vehicular traffic,, SIAM J. Appl. Math., 60 (2000), 1749. doi: 10.1137/S0036139999356181.
[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. doi: 10.3934/nhm.2009.4.431.
[7]	H. Y. Lee, H.-W. Lee and D. Kim, Steady-state solutions of hydrodynamic traffic models,, Phys. Rev. E, 69 (2004), 016118.
[8]	T. Li, Nonlinear dynamics of traffic jams,, Physica D, 207 (2005), 41. doi: 10.1016/j.physd.2005.05.011.
[9]	T. Li, Stability of traveling waves in quasi-linear hyperbolic systems with relaxation and diffusion,, SIAM J. Math. Anal., 40 (2008), 1058. doi: 10.1137/070690638.
[10]	A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow,", Cambridge Univ. Press, (2002).
[11]	M. J. Lighthill and G. B. Whitham, On kinematic waves: II. A theory of traffic flow on long crowded roads,, Proc. Roy. Soc., 229 (1955), 317.
[12]	T. Nagatani, The physics of traffic jams,, Rep. Prog. Phys., 65 (2002), 1331. doi: 10.1088/0034-4885/65/9/203.
[13]	K. Nagel, Particle hopping models and traffic flow theory,, Phys. Rev. E, 53 (1996), 4655. doi: 10.1103/PhysRevE.53.4655.
[14]	I. Prigogine and R. Herman, "Kinetic Theory of Vehicular Traffic,", American Elsevier Publishing Company Inc., (1971).
[15]	P. I. Richards, Shock waves on highway,, Operations Research, 4 (1956), 42. doi: 10.1287/opre.4.1.42.
[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. doi: 10.1137/040617790.
[17]	G. B. Whitham, "Linear and Nonlinear Waves,", Wiley, (1974).

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