Qualitative analysis of some PDE models of traffic flow

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September 2013, 8(3): 773-781. doi: 10.3934/nhm.2013.8.773

Qualitative analysis of some PDE models of traffic flow

1.	Department of Mathematics, University of Iowa, Iowa City, IA 52242

Received February 2012 Revised June 2013 Published October 2013

Abstract
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We review our previous results on partial differential equation(PDE) models of traffic flow. These models include the first order PDE models, a nonlocal PDE traffic flow model with Arrhenius look-ahead dynamics, and the second order PDE models, a discrete model which captures the essential features of traffic jams and chaotic behavior. We study the well-posedness of such PDE problems, finite time blow-up, front propagation, pattern formation and asymptotic behavior of solutions including the stability of the traveling fronts. Traveling wave solutions are wave front solutions propagating with a constant speed and propagating against traffic.

Keywords: traveling waves, nonlocal flux, blowup criteria, Arrhenius look-ahead dynamics, Traffic flow, shock waves, asymptotic behavior, chaos..

Mathematics Subject Classification: Primary: 35B44, 35L65, 35L67, 35Q35, 76L05, 90B2.

Citation: Tong Li. Qualitative analysis of some PDE models of traffic flow. Networks and Heterogeneous Media, 2013, 8 (3) : 773-781. doi: 10.3934/nhm.2013.8.773

References:

[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 ().
[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.

show all references

References:

[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 ().
[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.

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