A continuum model for two-directional traffic flow
Authors:
J. H. Bick and G. F. Newell
Journal:
Quart. Appl. Math. 18 (1960), 191-204
DOI:
https://doi.org/10.1090/qam/99969
MathSciNet review:
QAM99969
Full-text PDF Free Access
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Abstract: The flow of traffic in two directions of an undivided highway is investigated using the equations of continuity and assumed empirical relations between the average velocities and densities in both lanes. These lead to a pair of quasi-linear partial differential equations. Even if the velocity in one lane depends only very weakly on the density in the other lane, it is found that for a certain small range of densities the equations are of elliptic rather than the expected hyperbolic type. For densities outside this range, solutions of the equations can be found for various special types of initial conditions.
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K. Ostwatitsch, Über die Charakteristikenverfahren der Hydrodynamik, Z. angew. Math. Mech. 25, 195–208 1947)
P. D. Lax, Non-linear hyperbolic equations, Communs. Pure and Appl. Math. 6, 231–258 (1953)
- R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, N. Y., 1948. MR 0029615
- E. Kamke, Differentialgleichungen. Lösungsmethoden und Lösungen. Band I. Gewöhnliche Differentialgleichungen, Mathematik und ihre Anwendungen in Physik und Technik, Band $18_1$, Akademische Verlagsgesellschaft, Leipzig, 1944 (German). 3d ed. MR 0021170
J. H. Bick and G. F. Newell, A continuum model for traffic flow on an undivided highway, Brown University report IBM-22 (mimeographed), 1958
- Garrett Birkhoff, Hydrodynamics. A study in logic, fact, and similitude, Dover Publications, Inc., New York, 1955. MR 0070349
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M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffiic flow on long crowded roads, Proc. Roy. Soc. A229, 317–345 (1955)
P. I. Richards, Shock waves on the highway, J. Opns. Research Soc. Amer. 4, 42–51 (1956)
S. C. De, Kinematic wave theory of bottlenecks of varying capacity, Proc. Cambridge Phil. Soc. 52, 564–572 (1956)
K. Ostwatitsch, Über die Charakteristikenverfahren der Hydrodynamik, Z. angew. Math. Mech. 25, 195–208 1947)
P. D. Lax, Non-linear hyperbolic equations, Communs. Pure and Appl. Math. 6, 231–258 (1953)
R. Courant and K. O. Friedrichs, Supersonic flow and shock waves, Interscience Publishers, New York, 1948, Chaps. II and III
E. Kamke, Differentialgleichungen, LÖsungsmethoden und LÖsungen, Akademische Verlagsgesellschaft, Leipzig, 1944, p. 32
J. H. Bick and G. F. Newell, A continuum model for traffic flow on an undivided highway, Brown University report IBM-22 (mimeographed), 1958
G. Birkhoff, Hydrodynamics, Dover Publications, New York, 1955, Chap. IV
I. Petrovsky, Lectures on partial differential equations, Interscience Publishers, New York, 1954, p. 68
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Article copyright:
© Copyright 1960
American Mathematical Society
