The Hamilton-Jacobi partial differential equation and the three representations of traffic flow

Transportation Research Part B: Methodological

Volumn 52, Issue , 2013, Pages 17-30

  Laval, Jorge A ^a   Leclercq, Ludovic ^b  

^a Georgia Institute of Technology   (United States)

^b UNIVERSITÉ DE LYON   (France)

Author keywords

Hamilton Jacobi partial differential equation; Kinematic wave model; Stochastic traffic flow

Indexed keywords

BOUNDARY VALUE PROBLEMS; PARTIAL DIFFERENTIAL EQUATIONS; STOCHASTIC SYSTEMS;

CO-ORDINATE SYSTEM; EXPLICIT SOLUTIONS; FLOW DENSITY; HAMILTON-JACOBI; KINEMATIC WAVE MODELS; STOCHASTIC TRAFFIC FLOWS; TRAFFIC FLOW MODELS; VEHICLE NUMBERS;

STOCHASTIC MODELS;

FLOW MODELING; NUMERICAL MODEL; STOCHASTICITY; TRAFFIC MANAGEMENT; TRANSPORT VEHICLE; TRANSPORTATION PLANNING;

EID: 84875851701     PISSN: 01912615     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.trb.2013.02.008     Document Type: Article

References (29)

1. Ahn S., Cassidy M., Laval J.A. Verification of a simplified car-following theory. Transportation Research Part B 2003, 38(5):431-440.
2. Banks J.H. Freeway speed-flow-concentration relationships: more evidence and interpretations. Transportation Research Record 1989, 1225:53-60.
3. Cassidy M., Coifman B. The relation between average speed, flow and occupancy and the analogous relation between density and occupancy. Transportation Research Record 1997, (1591):1-6.
4. Claudel C., Bayen A. Lax-hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part ii: computational methods. IEEE Transactions on Automatic Control 2010, 55(5):1158-1174.
5. Daganzo C. On the variational theory of traffic flow: well-posedness, duality and applications. Networks and Heterogeneous Media 2006, 1(4):601-619.
6. Daganzo C.F. A variational formulation of kinematic wave theory: basic theory and complex boundary conditions. Transportation Research Part B 2005, 39(2):187-196.
7. Daganzo C.F. A variational formulation of kinematic waves: solution methods. Transportation Research Part B 2005, 39(10):934-950.
8. Daganzo C.F. In traffic flow, cellular automata=kinematic waves. Transportation Research Part B 2006, 40(5):396-403.
9. Evans L.C. Partial Differential Equations (Graduate Studies in Mathematics, V. 19) GSM/19 1998, American Mathematical Society.
10. Hofleitner, A., Claudel, C., Bayen, A., 2012. Probabilistic formulation of estimation problems for a class of Hamilton-Jacobi equations. In: 51st IEEE Conference on Decision and Control.
11. Hopf E. On the right weak solution of the cauchy problem for a quasilinear equation of first order. Indiana University Mathematics Journal 1970, 19:483-487.
12. Laval J., He J., Castrillon F. Stochastic extension of Newell's three-detector method. Transportation Research Record: Journal of the Transportation Research Board 2012, 2315:73-80.
13. Laval, J.A., Chilukuri, B.R., 2013. Distribution of congestion on a class of stochastic kinematic wave model. Transportation Science in press, http://dx.doi.org/10.1287/trsc.2013.0462.
14. Laval J.A., Daganzo C.F. Lane-changing in traffic streams. Transportation Research Part B 2006, 40(3):251-264.
15. Lax P.D. Hyperbolic systems of conservation laws ii. Communications on Pure and Applied Mathematics 1957, 537-566. Wiley Periodicals. P. Sarnak, A. Majda (Eds.).
16. Leclercq, L., Becarie, C., 2012. A meso lighthill-whitham and richards model designed for network applications. In: Proceedings of the 91st Transportation Research Board Annual Meeting (TRB), 21-26 January, Washington, (USA) [CDROM]. Transportation Research Board, Washington, 2012, 10p.
17. Leclercq L., Chanut S., Lesort J. Moving bottlenecks in the LWR model: a unified theory. Transportation Research Record 2004, 1883:3-13.
18. Leclercq L., Laval J., Chevallier E. The Lagrangian coordinate system and what it means for first order traffic flow models. 17th International Symposium on Transportation and Traffic Theory 2007, Elsevier, New York. B. Heydecker, M. Bell, R. Allsop (Eds.).
19. LeVeque R.L. Numerical Methods for Conservation Laws 1993, Birkhauser Verlag.
20. Lighthill M.J., Whitham G. On kinematic waves. I Flow movement in long rivers. II A theory of traffic flow on long crowded roads. Proceedings of the Royal Society of London 1955, 229(A):281-345.
21. Mahut, M., 2000. Discrete Flow Model for Dynamic Network Loading. Ph.D. Thesis, Dpartement dinformatique et de recherhe oprationelle, Universit de Montral. (Also published as Report CRT-2001-04, Center for Research on Transportation, University of Montreal.).
22. Mahut, M., Florian, M., Tremblay, N., 2003. Space-time queues and dynamic traffic assignment: a model, algorithm and applications. In: Proceedings of the Transportation Research Board 82nd Annual Meeting. TRB (DVD-Rom), Washington.
23. Makigami Y., Newell G., Rothery R. Three-dimensional representation of traffic flow. Transportation Science 1971, 5(3):302-313.
24. Mazar P., Dehwah A., Claudel C., Bayen A. Analytical and grid-free solutions to the lighthillwhithamrichards traffic flow model. Transportation Research Part B 2011, 45:1727-1748.
25. Newell G.F. A simplified theory of kinematic waves in highway traffic, I general theory, II queuing at freeway bottlenecks, III multi-destination flows. Transportation Research Part B 1993, 27(4):281-313.
26. Newell G.F. A moving bottleneck. Transportation Research Part B 1998, 32(8):531-537.
27. Newell G.F. A simplified car-following theory: a lower order model. Transportation Research Part B 2002, 36(3):195-205.
28. Olejnik O. Discontinuous solutions of non-linear differential equations. Translated by George Biriuk American Mathematical Society Translation: Series II 1957, 26:95-172.
29. Richards P.I. Shockwaves on the highway. Operations Research 1956, (4):42-51.