Asymptotic theory of traffic jams

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 55, Issue 4, 1997, Pages 4200-4206

  Kerner, B S ^a   Klenov, S L ^b   Konhauser P ^a  

^a DAIMLER AG   (Germany)

^b MOSCOW INSTITUTE OF PHYSICS AND TECHNOLOGY   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001364580     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.56.4200     Document Type: Article

References (63)

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50. (Formula presented) where (Formula presented) (Formula presented) Using the designations (Formula presented) (Formula presented) the solution of this equation can be written as (Formula presented)(Formula presented) where (Formula presented) (Formula presented) Hence the dispersion equation has a solution with the upper sign + in the formulas for ω and λ which would correspond to small-amplitude, nonhomogeneous perturbations propagating toward the downstream side of the source of perturbations. However, all these nonhomogeneous perturbations rapidly attenuate with higher decrement (Formula presented) In other words, local perturbations being considered in a system of coordinates moving at velocity (Formula presented) can indeed propagate in proper manner solely toward the upstream side of the source of perturbations.
51. R. E. O’Malley, Jr., Introduction to Singular Perturbation (Academic, New York, 1974);
52. D. R. Smith, Singular-Perturbation Theory. An Introduction with Applications (Cambridge University Press, Cambridge, 1985).
53. A characteristic of this self-organizing process determining the velocity of the downstream front of a wide jam has recently been proposed by B. S. Kerner [B. S. Kerner, in Transportation Systems, edited by M. Papageorgiou and A. Pouliezos (Technical University of Crete, Chania, Greece, 1997), Vol. 2, p. 793.
54. Note that, as follows from Eqs. (59), (60), (62), and (63), formulas (67) and (68) for narrow jams transform to the formulas for wide jams if the values (Formula presented) (Formula presented) (Formula presented) (Formula presented) and (Formula presented) are replaced by (Formula presented) (Formula presented) (Formula presented) (Formula presented) and (Formula presented) respectively. In particular, as follows from Eqs. (60) and (63), the formula for the value (Formula presented) can be written as (Formula presented)
55. Note that there may be functions (Formula presented) and values (Formula presented) that (Formula presented) in formula (74) exactly equals the flux out from a wide jam. The latter case may be realized for those functions (Formula presented) in which a monotonous increase in parameter (Formula presented) brings about a monotonous decrease in the value (Formula presented) in Eq. (62) in such a way that this value passes through the point of contact of the line (Formula presented) and the function (Formula presented) at some (Formula presented) At (Formula presented) the line (Formula presented) is obviously tangent to the curve (Formula presented) i.e., the value (Formula presented) corresponds to a solution of the set of equations (Formula presented) (Formula presented) (Formula presented) and (Formula presented) Then, for (Formula presented) when Eq. (6) is still valid, (Formula presented) i.e., the jam is wide at the boundary point (Formula presented) In this case, for each density (Formula presented) in the range (Formula presented) a solution for a narrow unstable jam can be found in addition to a solution for a wide stable jam.
56. were used (Fig. 77). This did not affect the results of numerical calculations, because the jams did not reach the ends of the road ((Formula presented) and (Formula presented)) during the numerical calculations.
57. In the examples shown in Figs. 99 and 1111, the cyclic boundary condition (Formula presented) is used, whereas the examples shown in Figs. 1010 and 1212 correspond to the boundary conditions (Formula presented) and (Formula presented) where (Formula presented) and (Formula presented) are constants. A numerical investigation of the LW model showed that the qualitative behavior of jams in the LW model does not depend on the type of boundary conditions.
58. A. Brenneis, P. Konhäuser, B. S. Kerner, and A. Eberle, 26th AIAA Fluid Dynamics Conference, San Diego, 1995 (American Institute of Aeronautics and Astronautics, Washington, D.C., 1995), Reprint AIAA 99-2230.
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63. Each of these two shock waves corresponds to a tangent to the fundamental diagram (Formula presented) tangent 1 is related to the upstream front, and tangent 2 in Figs. 1111(b) and 1111(c) is related to the downstream front of the jam. One can see that at some time tangent 2 has a more negative slope in comparison with tangent 1 [Fig. 1111(c), (Formula presented) Therefore, the downstream front of the jam overtakes the upstream front, and the jam disappears.