American Institute of Mathematical Sciences

Advanced
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.  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.  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, 37 (2003), 207.  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.  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.  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.  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.   Google Scholar

[8]

T. Li, Nonlinear dynamics of traffic jams,, Physica D, 207 (2005), 41.  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.  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., 229 (1955), 317.   Google Scholar

[12]

T. Nagatani, The physics of traffic jams,, Rep. Prog. Phys., 65 (2002), 1331.  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.  doi: 10.1103/PhysRevE.53.4655.  Google Scholar

[14]

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

[15]

P. I. Richards, Shock waves on highway,, Operations Research, 4 (1956), 42.  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.  doi: 10.1137/040617790.  Google Scholar

[17]

G. B. Whitham, "Linear and Nonlinear Waves,", Wiley, (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.  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.  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, 37 (2003), 207.  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.  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.  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.  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.   Google Scholar

[8]

T. Li, Nonlinear dynamics of traffic jams,, Physica D, 207 (2005), 41.  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.  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., 229 (1955), 317.   Google Scholar

[12]

T. Nagatani, The physics of traffic jams,, Rep. Prog. Phys., 65 (2002), 1331.  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.  doi: 10.1103/PhysRevE.53.4655.  Google Scholar

[14]

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

[15]

P. I. Richards, Shock waves on highway,, Operations Research, 4 (1956), 42.  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.  doi: 10.1137/040617790.  Google Scholar

[17]

G. B. Whitham, "Linear and Nonlinear Waves,", Wiley, (1974).   Google Scholar

[1]

Yongki Lee, Hailiang Liu. Thresholds for shock formation in traffic flow models with Arrhenius look-ahead dynamics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 323-339. doi: 10.3934/dcds.2015.35.323
[2]

Alexander Kurganov, Anthony Polizzi. Non-oscillatory central schemes for traffic flow models with Arrhenius look-ahead dynamics. Networks & Heterogeneous Media, 2009, 4 (3) : 431-451. doi: 10.3934/nhm.2009.4.431
[3]

Wen Shen. Traveling waves for conservation laws with nonlocal flux for traffic flow on rough roads. Networks & Heterogeneous Media, 2019, 14 (4) : 709-732. doi: 10.3934/nhm.2019028
[4]

Johanna Ridder, Wen Shen. Traveling waves for nonlocal models of traffic flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4001-4040. doi: 10.3934/dcds.2019161
[5]

Michael Herty, S. Moutari, M. Rascle. Optimization criteria for modelling intersections of vehicular traffic flow. Networks & Heterogeneous Media, 2006, 1 (2) : 275-294. doi: 10.3934/nhm.2006.1.275
[6]

Peter Howard, K. Zumbrun. The Evans function and stability criteria for degenerate viscous shock waves. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 837-855. doi: 10.3934/dcds.2004.10.837
[7]

Wen Shen, Karim Shikh-Khalil. Traveling waves for a microscopic model of traffic flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2571-2589. doi: 10.3934/dcds.2018108
[8]

Xiao-Biao Lin, Stephen Schecter. Traveling waves and shock waves. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : i-ii. doi: 10.3934/dcds.2004.10.4i
[9]

Jan Friedrich, Oliver Kolb, Simone Göttlich. A Godunov type scheme for a class of LWR traffic flow models with non-local flux. Networks & Heterogeneous Media, 2018, 13 (4) : 531-547. doi: 10.3934/nhm.2018024
[10]

James K. Knowles. On shock waves in solids. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 573-580. doi: 10.3934/dcdsb.2007.7.573
[11]

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
[12]

Yuri Gaididei, Anders Rønne Rasmussen, Peter Leth Christiansen, Mads Peter Sørensen. Oscillating nonlinear acoustic shock waves. Evolution Equations & Control Theory, 2016, 5 (3) : 367-381. doi: 10.3934/eect.2016009
[13]

Piotr Biler, Grzegorz Karch, Jacek Zienkiewicz. Morrey spaces norms and criteria for blowup in chemotaxis models. Networks & Heterogeneous Media, 2016, 11 (2) : 239-250. doi: 10.3934/nhm.2016.11.239
[14]

John M. Hong, Cheng-Hsiung Hsu, Bo-Chih Huang, Tzi-Sheng Yang. Geometric singular perturbation approach to the existence and instability of stationary waves for viscous traffic flow models. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1501-1526. doi: 10.3934/cpaa.2013.12.1501
[15]

Chunlai Mu, Zhaoyin Xiang. Blowup behaviors for degenerate parabolic equations coupled via nonlinear boundary flux. Communications on Pure & Applied Analysis, 2007, 6 (2) : 487-503. doi: 10.3934/cpaa.2007.6.487
[16]

Frederike Kissling, Christian Rohde. The computation of nonclassical shock waves with a heterogeneous multiscale method. Networks & Heterogeneous Media, 2010, 5 (3) : 661-674. doi: 10.3934/nhm.2010.5.661
[17]

Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115
[18]

Piotr Biler, Tomasz Cieślak, Grzegorz Karch, Jacek Zienkiewicz. Local criteria for blowup in two-dimensional chemotaxis models. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1841-1856. doi: 10.3934/dcds.2017077
[19]

Yacine Chitour, Benedetto Piccoli. Traffic circles and timing of traffic lights for cars flow. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 599-630. doi: 10.3934/dcdsb.2005.5.599
[20]

Denis Serre, Alexis F. Vasseur. The relative entropy method for the stability of intermediate shock waves; the rich case. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4569-4577. doi: 10.3934/dcds.2016.36.4569

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences