- D. Amadori, Initial-boundary value problems for nonlinear systems of conservation laws. NoDEA 4 (1997) 1–42. [CrossRef] [MathSciNet]
- D. Amadori and R.M. Colombo, Continuous dependence for 2×2 conservation laws with boundary. J. Differ. Equ. 138 (1997) 229–266. [CrossRef]
- F. Ancona and A. Marson, Scalar non-linear conservation laws with integrable boundary data. Nonlinear Anal. 35 (1999) 687–710. [CrossRef] [MathSciNet]
- B. Andreianov, P. Goatin and N. Seguin, Finite volume schemes for locally constrained conservation laws. Numer. Math. 115 (2010) 609–645. [CrossRef] [MathSciNet]
- A. Aw and M. Rascle, Resurrection of “second order” models of traffic flow. SIAM J. Appl. Math. 60 (2000) 916–938. [CrossRef] [MathSciNet]
- C. Bardos, A.Y. le Roux and J.-C. Nédélec, First order quasilinear equations with boundary conditions. Comm. Partial Differential Equations 4 (1979) 1017–1034. [CrossRef] [MathSciNet]
- S. Blandin, D. Work, P. Goatin, B. Piccoli and A. Bayen, A general phase transition model for vehicular traffic. SIAM J. Appl. Math. (to appear).
- A. Bressan, Hyperbolic systems of conservation laws – The one-dimensional Cauchy problem,Oxford Lecture Series in Mathematics and its Applications 20. Oxford University Press, Oxford (2000).
- W. Chen, S.C. Wong, C.W. Shu and P. Zhang, Front tracking algorithm for the Lighthill-Whitham-Richards traffic flow model with a piecewise quadratic, continuous, non-smooth and non-concave fundamental diagram. Int. J. Numer. Anal. Model. 6 (2009) 562–585. [MathSciNet]
- R.M. Colombo, Hyperbolic phase transitions in traffic flow. SIAM J. Appl. Math. 63 (2002) 708–721. [CrossRef] [MathSciNet]
- R.M. Colombo and P. Goatin, A well posed conservation law with a variable unilateral constraint. J. Differ. Equ. 234 (2007) 654–675. [CrossRef]
- R.M. Colombo and A. Groli, Minimising stop and go waves to optimise traffic flow. Appl. Math. Lett. 17 (2004) 697–701. [CrossRef] [MathSciNet]
- R.M. Colombo, P. Goatin, G. Maternini and M.D. Rosini, Conservation laws with unilateral constraints in traffic modeling, in Transport Management and Land-Use Effects in Presence of Unusual Demand, L. Mussone and U. Crisalli Eds., Atti del convegno SIDT 2009 (2009).
- R.M. Colombo, P. Goatin and B. Piccoli, Road networks with phase transitions. J. Hyperbolic Differ. Equ. 7 (2010) 85–106. [CrossRef] [MathSciNet]
- R.M. Colombo, F. Marcellini and M. Rascle, A 2-phase traffic model based on a speed bound. SIAM J. Appl. Math. 70 (2010) 2652–2666. [CrossRef] [MathSciNet]
- C.M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law. J. Math. Anal. Appl. 38 (1972) 33–41. [CrossRef] [MathSciNet]
- C. Daganzo, The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory. Transp. Res. B 28B (1994) 269–287. [CrossRef]
- F. Dubois and P. LeFloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. Differential Equations 71 (1988) 93–122. [CrossRef] [MathSciNet]
- P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions. Math. Comput. Model. 44 (2006) 287–303. [CrossRef]
- J. Goodman, Initial Boundary Value Problems for Hyperbolic Systems of Conservation Laws. Ph.D. thesis, California University (1982).
- H. Greenberg, An analysis of traffic flow. Oper. Res. 7 (1959) 79–85. [CrossRef]
- B. Greenshields, A study of traffic capacity. Proceedings of the Highway Research Board 14 (1935) 448–477.
- B. Haut, G. Bastin and Y. Chitour, A macroscopic traffic model for road networks with a representation of the capacity drop phenomenon at the junctions, in Proceedings 16th IFAC World Congress, Prague, Czech Republic, July (2005) Tu-M01-TP/3.
- D. Helbing, S. Lämmer and J.-P. Lebacque, Self-Organized Control of Irregular or Perturbed Network Traffic, in Optimal Control and Dynamic Games, Advances in Computational Management Science 7, Springer (2005) 239–274.
- J.C. Herrera and A.M. Bayen, Incorporation of lagrangian measurements in freeway traffic state estimation. Transp. Res. Part B: Methodol. 44 (2010) 460–481. [CrossRef]
- H. Holden and N.H. Risebro, Front tracking for hyperbolic conservation laws, Applied Mathematical Sciences 152. Springer-Verlag, New York (2002).
- W.-L. Jin, Continuous kinematic wave models of merging traffic flow. Transp. Res. Part B: Methodol. 44 (2010) 1084–1103. [CrossRef]
- W.L. Jin and H.M. Zhang, The formation and structure of vehicle clusters in the Payne-Whitham traffic flow model. Transp. Res. B 37 (2003) 207–223. [CrossRef]
- W.L. Jin and H.M. Zhang, On the distribution schemes for determining flows through a merge. Transp. Res. Part B: Methodol. 37 (2003) 521–540. [CrossRef]
- B.S. Kerner and P. Konhäuser, Cluster effect in initially homogeneous traffic flow. Phys. Rev. E 48 (1993) R2335–R2338. [CrossRef]
- B.S. Kerner and P. Konhäuser, Structure and parameters of clusters in traffic flow. Phys. Rev. E 50 (1994) 54–83. [CrossRef]
- B.S. Kerner and H. Rehborn, Experimental features and characteristics of traffic jams. Phys. Rev. E 53 (1996) R1297–R1300. [CrossRef]
- A. Klar, Kinetic and Macroscopic Traffic Flow Models. School of Computational Mathematics: Computational aspects in kinetic models, XXth edition (2002).
- S.N. Kružhkov, First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81 (1970) 228–255. [MathSciNet]
- L. Leclercq, Bounded acceleration close to fixed and moving bottlenecks. Transp. Res. Part B: Methodol. 41 (2007) 309–319. [CrossRef]
- L. Leclercq, Hybrid approaches to the solutions of the Lighthill-Whitham-Richards model. Transp. Res. Part B: Methodol. 41 (2007) 701–709. [CrossRef]
- H. Lee, H.-W. Lee and D. Kim, Empirical phase diagram of traffic flow on highways with on-ramps, in Traffic and Granular Flow '99, M.S.D.W.D. Helbing and H.J. Herrmann Eds. (2000).
- R.J. LeVeque, Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge (2002).
- J. Li, Q. Chen, H. Wang and D. Ni, Analysis of LWR model with fundamental diagram subject to uncertainties, in TRB 88th Annual Meeting Compendium of Papers, number 09-1189 in TRB (2009) 14.
- M.J. Lighthill and G.B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. Roy. Soc. London. Ser. A. 229 (1955) 317–345. [CrossRef] [MathSciNet]
- H.X. Liu, X. Wu, W. Ma and H. Hu, Real-time queue length estimation for congested signalized intersections. Transp. Res. Part C 17 (2009) 412–427. [CrossRef]
- S. Mammar, J.-P. Lebacque and H.H. Salem, Riemann problem resolution and Godunov scheme for the Aw-Rascle-Zhang model. Transp. Sci. 43 (2009) 531–545. [CrossRef]
- G. Newell, A simplified theory of kinematic waves in highway traffic, part II. Transp. Res. B 27 B (1993) 289–303.
- E.Y. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws. J. Hyperbolic Differ. Equ. 4 (2007) 729–770. [CrossRef] [MathSciNet]
- B. Piccoli and M. Garavello, Traffic flow on networks – Conservation laws models, AIMS Series on Applied Mathematics 1. American Institute of Mathematical Sciences (AIMS), Springfield, MO (2006).
- P.I. Richards, Shock waves on the highway. Oper. Res. 4 (1956) 42–51. [CrossRef] [MathSciNet]
- D. Serre, Systems of conservation laws 1 & 2. Cambridge University Press, Cambridge (1999).
- C. Tampere, S. Hoogendoorn and B. van Arem, A behavioural approach to instability, stop & go waves, wide jams and capacity drop, in Proceedings of 16th International Symposium on Transportation and Traffic Theory (ISTTT), Maryland (2005).
- B. Temple, Global solution of the Cauchy problem for a class of 2×2 nonstrictly hyperbolic conservation laws. Adv. Appl. Math. 3 (1982) 335–375. [CrossRef] [MathSciNet]
- E. Tomer, L. Safonov, N. Madar and S. Havlin, Optimization of congested traffic by controlling stop-and-go waves. Phys. Rev. E 65 (2002) 4.
- M. Treiber, A. Hennecke and D. Helbing, Congested traffic states in empirical observations and microscopic simulation. Phys. Rev. E 62 (2000) 1805–1824. [CrossRef]
Volume 45, Number 5, September-October 2011
|Page(s)||853 - 872|
|Published online||23 February 2011|