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Fluid Dynamic Approach to Traffic Flow

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Nicole Fiorentino

on 8 May 2014

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Transcript of Fluid Dynamic Approach to Traffic Flow

Motivation
Mathematical modeling of physical systems is essential to the development of an engineering field, and traffic engineering is no exception.

Model System
Lighthill, Whitham, Richards
LWR Model
First macroscopic model of traffic flow
Identical to the first order fluid dynamics model of water flow in a river and pipe
Accurate in moderate to heavy traffic, but fails in light traffic
Assumes velocity changes instantaneously
Fails to depict shocks
Conclusions
A comparison of our results with the results of the LWR model and published results was performed
Second Order Finite Difference accurately models traffic flow
Our models agreed with the published results
Second order finite difference had a clear advantage over the LWR model (Godunov method) in the prediction of traffic flow verses flux
Numerical Techniques
Results
Fluid Dynamic Approach to Traffic Flow:
Our 2D Difference Solution vs. Published Results

Nicole Fiorentino & Emily Helbling

It is proposed that different fluid dynamic approaches to macroscopic traffic flow models may be used and compared in order to analyze the traffic flow in terms of density, velocity, flux, time and space.
Hypothesis
References
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8. Haut, B., G. Bastin, and Y. Chitour. "A macroscopic traffic model for road networks with representation of the capacity drop phenomenon at the junctions." Proceedings of the 16th IFAC World Congress. 2005.
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16. Moin, Parviz. Fundamentals of Engineering Numerical Analysis. Cambridge, UK: Cambridge UP, 2001. Print.
17. Obertscheider, Christof. "Burgers' Equation."
18. Papageorgiou, Markos. "Some Remarks on Macroscopic Traffic Flow Modelling." Transportation Research Part A: Policy and Practice 32.5 (1998): 323-29. Print.
19. Papageorgiou, Markos, Jean-Marc Blosseville, and Habib Hadj-Salem. "Macroscopic Modelling of Traffic Flow on the Boulevard Périphérique in Paris." Transportation Research Part B: Methodological 23.1 (1989): 29-47. Print.
20. Sun, Dazhi, Jinpeng Lv, and S. Travis Waller. "In-depth Analysis of Traffic Congestion Using Computational Fluid Dynamics (CFD) Modeling Method." Journal of Modern Transportation 19.1 (2011): 58-67. Print.
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23. Zhu, Zuojin, and Tongqiang Wu. "Two-Phase Fluids Model for Freeway Traffic Flow and Its Application to Simulate Evolution of Solitons in Traffic." Journal of Transportation Engineering 129.1 (2003): 51. Print.

Shockwave traffic jams recreated for first time
LWR
Conservation law
Assume vehicular speeds change instantaneously: f(ρ)=ρV(ρ)
Greensheild
Linear flow-density relationship
Second Order Finite Difference
Leapfrog time advancement and the second-order central difference for the spatial derivative
Central Difference
Explicit Euler
Leapfrog
Finding Flux Using Density
q is flux--vehicles per unit time
Maximum Values
Density vs Flux
Density vs Velocity
Density vs Position
v is velocity--in miles per minute
ρ is density--cars per unit area
LWR Model
Published Results
Our Finite Difference Model
Published Results
Published Results
Our Finite Difference Model
Our Finite Difference Model
Questions?
Time to take a pit stop
Fasten your seat belts and enjoy the ride!
and
Final Equation (extension of Burger's Equation)
BC: Gaussian Density Distribution
Full transcript