American Institute of Mathematical Sciences

Advanced
September  2013, 8(3): 745-772. doi: 10.3934/nhm.2013.8.745

Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models

Benjamin Seibold 1, , Morris R. Flynn 2, , Aslan R. Kasimov 3, and  Rodolfo R. Rosales 4,

1. 

Temple University, Department of Mathematics, 1805 North Broad Street Philadelphia, PA 19122
2. 

Department of Mechanical Engineering, University of Alberta, Edmonton, AB, T6G 2G8, Canada
3. 

4700 King Abdullah University of, Science and Technology, Thuwal 23955-6900, Saudi Arabia
4. 

Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139

Received  April 2012 Revised  June 2013 Published  October 2013

Fundamental diagrams of vehicular traffic flow are generally multi-valued in the congested flow regime. We show that such set-valued fundamental diagrams can be constructed systematically from simple second order macroscopic traffic models, such as the classical Payne-Whitham model or the inhomogeneous Aw-Rascle-Zhang model. These second order models possess nonlinear traveling wave solutions, called jamitons, and the multi-valued parts in the fundamental diagram correspond precisely to jamiton-dominated solutions. This study shows that transitions from function-valued to set-valued parts in a fundamental diagram arise naturally in well-known second order models. As a particular consequence, these models intrinsically reproduce traffic phases.
Keywords: traffic model, sensor data, Payne-Whitham, jamiton, Aw-Rascle-Zhang, Second order, traveling wave, effective flow rate., fundamental diagram.
Mathematics Subject Classification: Primary: 35B40, 35C07, 35L65, 35Q91; Secondary: 76L05, 91C9.
Citation: Benjamin Seibold, Morris R. Flynn, Aslan R. Kasimov, Rodolfo R. Rosales. Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models. Networks & Heterogeneous Media, 2013, 8 (3) : 745-772. doi: 10.3934/nhm.2013.8.745
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show all references

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J. Stat. Phys, 133 (2008), 1083-1105.  Google Scholar

[2]

SIAM J. Appl. Math., 60 (2000), 916-938. doi: 10.1137/S0036139997332099.  Google Scholar

[3]

Arch. Ration. Mech. Anal., 187 (2008), 185-220. doi: 10.1007/s00205-007-0061-9.  Google Scholar

[4]

SIAM J. Appl. Math., 71 (2011), 107-127. doi: 10.1137/090754467.  Google Scholar

[5]

Comm. Pure Appl. Math., 47 (1994), 787-830. doi: 10.1002/cpa.3160470602.  Google Scholar

[6]

Traffic flow—modelling and simulation. Math. Comput. Modelling, 35 (2002), 683-688. doi: 10.1016/S0895-7177(02)80029-2.  Google Scholar

[7]

SIAM J. Appl. Math., 63 (2003), 708-721. doi: 10.1137/S0036139901393184.  Google Scholar

[8]

Transp. Res. B, 28 (1994), 269-287. doi: 10.1016/0191-2615(94)90002-7.  Google Scholar

[9]

Transp. Res. B, 29 (1995), 79-93. doi: 10.1016/0191-2615(94)00022-R.  Google Scholar

[10]

Transp. Res. B, 29 (1995), 277-286. doi: 10.1016/0191-2615(95)00007-Z.  Google Scholar

[11]

Transp. Res. B, 40 (2006), 396-403. doi: 10.1016/j.trb.2005.05.004.  Google Scholar

[12]

Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.  Google Scholar

[13]

in preparation, 2012. Google Scholar

[14]

Univ. of California Press, Berkeley, CA, 1979. Google Scholar

[15]

Phys. Rev. E, 79 (2009), 056113, 13 pp. doi: 10.1103/PhysRevE.79.056113.  Google Scholar

[16]

Oper. Res., 7 (1959), 79-85. doi: 10.1287/opre.7.1.79.  Google Scholar

[17]

SIAM J. Appl. Math., 62 (2001), 729-745. doi: 10.1137/S0036139900378657.  Google Scholar

[18]

SIAM J. Appl. Math., 64 (2004), 1175-1185(electronic). doi: 10.1137/S0036139903431737.  Google Scholar

[19]

Proceedings of the Highway Research Record, 14 (1935), 448-477. Google Scholar

[20]

D. Helbing, Video of traffic waves,, Website. , ().   Google Scholar

[21]

Reviews of Modern Physics, 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067.  Google Scholar

[22]

Elsevier, New York, 1971. Google Scholar

[23]

Commun. Math. Sci., 1 (2003), 1-12.  Google Scholar

[24]

in preparation, 2013. Google Scholar

[25]

Phys. Rev. Lett., 81 (1998), 3797-3800. doi: 10.1103/PhysRevLett.81.3797.  Google Scholar

[26]

Phys. Rev. E, 56 (1997), 4200-4216. doi: 10.1103/PhysRevE.56.4200.  Google Scholar

[27]

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[28]

Phys. Rev. E, 50 (1994), 54-83. doi: 10.1103/PhysRevE.50.54.  Google Scholar

[29]

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[30]

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[32]

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[33]

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[35]

Comm. Math. Sci., 3 (2005), 101-118.  Google Scholar

[36]

Discrete Contin. Dyn. Syst., 24 (2009), 511-521. doi: 10.3934/dcds.2009.24.511.  Google Scholar

[37]

Proc. Roy. Soc. A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089.  Google Scholar

[38]

Comm. Math. Phys., 108 (1987), 153-175. doi: 10.1007/BF01210707.  Google Scholar

[39]

Traffic Engrg. Control, 31 (1990), 466-470. Google Scholar

[40]

J. Phys. I France, 2 (1992), 2221-2229. doi: 10.1051/jp1:1992277.  Google Scholar

[41]

In A. Ceder, editor, Proceedings of the 14th International Symposium on Transportation and Trafic Theory, pages 51-79, Jerusalem, 1999. Google Scholar

[42]

Operations Research, 9 (1961), 209-229. doi: 10.1287/opre.9.2.209.  Google Scholar

[43]

Transp. Res. B, 27 (1993), 289-303. doi: 10.1016/0191-2615(93)90039-D.  Google Scholar

[44]

Minnesota Department of Transportation, Mn/DOT traffic data,, Website. , ().   Google Scholar

[45]

Proc. Simulation Council, 1 (1971), 51-61. Google Scholar

[46]

Transp. Res. Rec., 722 (1979), 68-77. Google Scholar

[47]

Transportation Planning and Technology, 5 (1979), 131-138. doi: 10.1080/03081067908717157.  Google Scholar

[48]

Journal of Applied Physics, 24 (1953), 274-281. doi: 10.1063/1.1721265.  Google Scholar

[49]

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[50]

in preparation, 2013. Google Scholar

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SIAM J. Appl. Math., 66 (2006), 1150-1162(electronic). doi: 10.1137/050627113.  Google Scholar

[52]

New Journal of Physics, 10 (2008), 033001. doi: 10.1088/1367-2630/10/3/033001.  Google Scholar

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Technical report, Yale Bureau of Highway Traffic, 1961. Google Scholar

[54]

Federal Highway Administration US Department of Transportation, Next generation simulation (NGSIM),, Website. , ().   Google Scholar

[55]

Eur. J. Control, 11 (2005), 301-309. doi: 10.3166/ejc.11.301-309.  Google Scholar

[56]

Transp. Res. B, 39 (2005), 141-167. doi: 10.1016/j.trb.2004.03.003.  Google Scholar

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Proc. Instn. Civ. Engrs., 3 (1954), 158-171. doi: 10.1680/ipeds.1954.11628.  Google Scholar

[58]

Comm. Pure Appl. Math., 12 (1959), 113-158. doi: 10.1002/cpa.3160120107.  Google Scholar

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Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. xvi+636 pp.  Google Scholar

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Transp. Res. B, 36 (2002), 275-290. doi: 10.1016/S0191-2615(00)00050-3.  Google Scholar

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