Citation: |
[1] |
A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938.doi: 10.1137/S0036139997332099. |
[2] |
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. |
[3] |
N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463.doi: 10.1137/090746677. |
[4] |
N. Bellomo and V. Coscia, First order models and closure of the mass conservation equation in the mathematical theory of vehicular traffic flow, C. R. Mecanique, 333 (2005), 843-851.doi: 10.1016/j.crme.2005.09.004. |
[5] |
F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams, Arch. Rational Mech. Anal., 187 (2008), 185-220.doi: 10.1007/s00205-007-0061-9. |
[6] |
F. Berthelin, P. Degond, V. Le Blanc, S. Moutari, M. Rascle and J. Royer, A traffic-flow model with constraints for the modelling of traffic jams, Math. Mod. Meth. Appl. Sci., 18 (2008), (Supplement), 1269-1298.doi: 10.1142/S0218202508003030. |
[7] |
V. Coscia, On a closure of mass conservation equation and stability analysis in the mathematical theory of vehicular traffic flow, C. R. Mecanique, 332 (2004), 585-590.doi: 10.1016/j.crme.2004.03.016. |
[8] |
C. Daganzo, Requiem for second-order approximations of traffic flow, Transportation Research, B, 29 (1995), 277-286.doi: 10.1016/0191-2615(95)00007-Z. |
[9] |
P. Degond and M. Delitala, Modelling and simulation of vehicular traffic jam formation, Kinetic Related Models, 1 (2008), 279-293.doi: 10.3934/krm.2008.1.279. |
[10] |
D. Helbing, Improved fluid-dynamic model for vehicular traffic, Physical Review E, 51 (1995), 3154-3169.doi: 10.1103/PhysRevE.51.3164. |
[11] |
D. Helbing, Traffic and related self-driven many-particle systems, Rev. Modern Phy., 73 (2001), 1067-1141.doi: 10.1103/RevModPhys.73.1067. |
[12] |
D. Helbing, Derivation of non-local macroscopic traffic equations and consistent traffic pressures from microscopic car-following models, Eur. Phys. J. B, 69 (2009), 539-548. |
[13] |
S. Jin and Jian-Guo Liu, Relaxation and diffusion enhanced dispersive waves, Proceedings: Mathematical and Physical Sciences, 446 (1994), 555-563. |
[14] |
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. |
[15] |
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. |
[16] |
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. |
[17] |
R. D. Kühne, Macroscopic freeway model for dense traffic-stop-start waves and incident detection, Ninth International Symposium on Transportation and Traffic Theory, VNU Science Press, Ultrecht, 1984, 21-42. |
[18] |
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. |
[19] |
D. A. Kurtze and D. C. Hong, Traffic jams, granular flow, and soliton selection, Phys. Rev. E, 52 (1995), 218-221.doi: 10.1103/PhysRevE.52.218. |
[20] |
H. Y. Lee, H.-W. Lee and D. Kim, Steady-state solutions of hydrodynamic traffic models, Phys. Rev. E, 69 (2004), 016118.doi: 10.1103/PhysRevE.69.016118. |
[21] |
Dong Li and Tong Li, Shock formation in a traffic flow model with arrhenius look-ahead dynamics, Networks and Heterogeneous Media, 6 (2011), 681-694.doi: 10.3934/nhm.2011.6.681. |
[22] |
Tong Li, Global solutions and zero relaxation limit for a traffic flow model, SIAM J. Appl. Math., 61 (2000), 1042-1061.doi: 10.1137/S0036139999356788. |
[23] |
Tong Li, $L^1$ stability of conservation laws for a traffic flow model, Electron. J. Diff. Eqns., 2001 18 pp. (electronic). |
[24] |
Tong Li, Well-posedness theory of an inhomogeneous traffic flow model, Discrete and Continuous Dynamical Systems, Series B, 2 (2002), 401-414.doi: 10.3934/dcdsb.2002.2.401. |
[25] |
Tong Li, Global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow, J. Diff. Eqns., 190 (2003), 131-149.doi: 10.1016/S0022-0396(03)00014-7. |
[26] |
Tong Li, Mathematical modelling of traffic flows, Hyperbolic problems: Theory, numerics, applications, 695-704, Springer, Berlin, 2003. |
[27] |
Tong Li, Modelling traffic flow with a time-dependent fundamental diagram, Math. Methods Appl. Sci., 27 (2004), 583-601.doi: 10.1002/mma.470. |
[28] |
Tong Li, Nonlinear dynamics of traffic jams, Physica D, 207 (2005), 41-51.doi: 10.1016/j.physd.2005.05.011. |
[29] |
Tong Li, Instability and formation of clustering solutions of traffic flow, Bulletin of the Institute of Mathematics, Academia Sinica (New Series), 2 (2007), 281-295. |
[30] |
Tong 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. |
[31] |
Tong Li and Hailiang Liu, Stability of a traffic flow model with nonconvex relaxation, Comm. Math. Sci., 3 (2005), 101-118. |
[32] |
Tong Li and Hailiang Liu, Critical thresholds in hyperbolic relaxation systems, J. Diff. Eqns., 247 (2009), 33-48.doi: 10.1016/j.jde.2009.03.032. |
[33] |
Tong Li and Hailiang Liu, Critical thresholds in a relaxation system with resonance of characteristic speeds, Discrete and Continuous Dynamical Systems - Series A, 24 (2009), 511-521.doi: 10.3934/dcds.2009.24.511. |
[34] |
Tong Li and Hailiang Liu, Critical thresholds in a relaxation model for traffic flows, Indiana Univ. Math. J., 57 (2008), 1409-1430.doi: 10.1512/iumj.2008.57.3215. |
[35] |
Tong Li and Yaping Wu, Linear and nonlinear exponential stability of traveling waves for hyperbolic systems with relaxation, Comm. Math. Sci., 7 (2009), 571-593. |
[36] |
Tong Li and H. M. Zhang, The mathematical theory of an enhanced nonequilibrium traffic flow model, Network and Spatial Economics, A Journal of Infrastructure Modeling and Computation, Special Double Issue on Traffic Flow Theory, 1&2 (2001), 167-177. |
[37] |
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., A229 (1955), 317-345.doi: 10.1098/rspa.1955.0089. |
[38] |
T. Nagatani, The physics of traffic jams, Rep. Prog. Phys., 65 (2002), 1331-1386.doi: 10.1088/0034-4885/65/9/203. |
[39] |
K. Nagel, Particle hopping models and traffic flow theory, Phys. Rev. E, 53 (1996), 4655-4672. |
[40] |
O. A. Oleinik, Discontinuous solutions of non-linear differential equations, (Russian) Uspehi. Mat. Nauk., 12 (1957), 3-73. |
[41] |
H. J. Payne, Models of freeway traffic and control, "Simulation Councils Proc. Ser. : Mathematical Models of Public Systems," 1 (1971), 51-61, Editor G.A. Bekey, La Jolla, CA. |
[42] |
I. Prigogine and R. Herman, "Kinetic Theory of Vehicular Traffic," American Elsevier Publishing Company Inc., New York, 1971. |
[43] |
P. I. Richards, Shock waves on highway, Operations Research, 4 (1956), 42-51.doi: 10.1287/opre.4.1.42. |
[44] |
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., 66 (2006), 921-944.doi: 10.1137/040617790. |
[45] |
Lina Wang, Yaping Wu and Tong Li, Exponential stability of large-amplitude traveling fronts for quasi-linear relaxation systems with diffusion, Physica D, 240 (2011), 971-983.doi: 10.1016/j.physd.2011.02.003. |
[46] |
G. B. Whitham, "Linear and Nonlinear Waves," Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. xvi+636 pp. |
[47] |
Lei Yu, Tong Li and Zhong-Ke Shi, Density waves in a traffic flow model with reactive-time delay, Physica A, Statistical Mechanics and its Applications, 389 (2010), 2607-2616.doi: 10.1016/j.physa.2010.03.009. |
[48] |
Lei Yu, Tong Li and Zhong-Ke Shi, The effect of diffusion in a new viscous continuum model, Physics Letters, Section A: General, Atomic and Solid State Physics, 374 (2010), 2346-2355.doi: 10.1016/j.physleta.2010.03.056. |
[49] |
H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research, B., 36 (2002), 275-290.doi: 10.1016/S0191-2615(00)00050-3. |
[50] |
H. M. Zhang, Driver memory, traffic viscosity and a viscous vehicular traffic flow model, Transportation Research, B., 37 (2003), 27-41. |