Send the link below via email or IMCopy
Present to your audienceStart remote presentation
- Invited audience members will follow you as you navigate and present
- People invited to a presentation do not need a Prezi account
- This link expires 10 minutes after you close the presentation
- A maximum of 30 users can follow your presentation
- Learn more about this feature in our knowledge base article
Fluid Dynamic Approach to Traffic Flow
Transcript of Fluid Dynamic Approach to Traffic Flow
Mathematical modeling of physical systems is essential to the development of an engineering field, and traffic engineering is no exception.
Lighthill, Whitham, Richards
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
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
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.
1. Aw, A., and M. Rascle. "Resurrection of "Second Order" Models of Traffic Flow." SIAM Journal on Applied Mathematics 60.3 (2000): 916. Print.
2. Blandin, S., G. Bretti, A. Cutolo, and B. Piccoli. "Numerical Simulations of Traffic Data via Fluid Dynamic Approach." Applied Mathematics and Computation 210.2 (2009): 441-54. Print.
3. Bretti, Gabriella, Roberto Natalini, and Benedetto Piccoli. "A Fluid-Dynamic Traffic Model on Road Networks." Archives of Computational Methods in Engineering 14.2 (2007): 139-72. Print.
4. Coscia, V., M. Delitala, and P. Frasca. "On the Mathematical Theory of Vehicular Traffic Flow II: Discrete Velocity Kinetic Models." International Journal of Non-Linear Mechanics 42.3 (2007): 411-21. Print.
5. Daganzo, Carlos F. "A Finite Difference Approximation of the Kinematic Wave Model of Traffic Flow." Transportation Research Part B: Methodological 29.4 (1995): 261-76. Print.
6. Daganzo, Carlos F. "Requiem for Second-order Fluid Approximations of Traffic Flow." Transportation Research Part B: Methodological 29.4 (1995): 277-86. Print.
7. Gani, Mo, Mm Hossain, and Ls Andallah. "A Finite Difference Scheme for a Fluid Dynamic Traffic Flow Model Appended with Two-point Boundary Condition." GANIT: Journal of Bangladesh Mathematical Society 31.0 (2012): n. pag. Print.
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.
9. Helbing, Dirk. "From microscopic to macroscopic traffic models." A perspective look at nonlinear media. Springer Berlin Heidelberg, 1998. 122-139.
10. Hoogendoorn, Serge P., and Piet H.l. Bovy. "Continuum Modeling of Multiclass Traffic Flow." Transportation Research Part B: Methodological 34.2 (2000): 123-46. Print.
11. Jiang, Rui, Qing-Song Wu, and Zuo-Jin Zhu. "A New Continuum Model for Traffic Flow and Numerical Tests." Transportation Research Part B: Methodological 36.5 (2002): 405-19. Print.
12. Jin, Wenlong. Kinematic Wave Models of Network Vehicular Traffic. N.p.: n.p., 2003. Print.
13. Kabir, Mh, A. Afroz, and Ls Andallah. "A Finite Difference Scheme for a Macroscopic Traffic Flow Model Based on a Nonlinear Density-velocity Relationship." Bangladesh Journal of Scientific and Industrial Research 47.3 (2012): n. pag. Print.
14. Leo, Chin Jian, and Robert L. Pretty. "Numerical Simulation of Macroscopic Continuum Traffic Models." Transportation Research Part B: Methodological 26.3 (1992): 207-20. Print.
15. Logghe, S., and L.h. Immers. "Multi-class Kinematic Wave Theory of Traffic Flow." Transportation Research Part B: Methodological 42.6 (2008): 523-41. Print.
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.
21. Williams, JAMES C. "Macroscopic flow models." Revised monograph on traffic flow theory (1997): 6-1.
22. Zhang, H.m. "A Theory of Nonequilibrium Traffic Flow." Transportation Research Part B: Methodological 32.7 (1998): 485-98. Print.
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
Assume vehicular speeds change instantaneously: f(ρ)=ρV(ρ)
Linear flow-density relationship
Second Order Finite Difference
Leapfrog time advancement and the second-order central difference for the spatial derivative
Finding Flux Using Density
q is flux--vehicles per unit time
Density vs Flux
Density vs Velocity
Density vs Position
v is velocity--in miles per minute
ρ is density--cars per unit area
Our Finite Difference Model
Our Finite Difference Model
Our Finite Difference Model
Time to take a pit stop
Fasten your seat belts and enjoy the ride!
Final Equation (extension of Burger's Equation)
BC: Gaussian Density Distribution