Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling

Interfaces and Free Boundaries

Volumn 10, Issue 2, 2008, Pages 197-221

  Chalons, Christophe ^a   Goatin, Paola ^a,b  

^a UNIVERSITÉ PARIS DESCARTES   (France)

^b UNIVERSITÉ DE TOULON   (France)

Author keywords

Continuous traffic models; Godunov scheme; Hyperbolic conservation laws; Phase transitions; Sampling

Indexed keywords

EID: 46949106911     PISSN: 14639963     EISSN: None     Source Type: Journal    
DOI: 10.4171/IFB/186     Document Type: Article

References (24)

1. ABGRALL, R. Generalization of Roe scheme for the computation of mixture of perfect gases. Rech. Aerosp. 6 (1988), 31-43. Zbl 0662.76097
2. BELLOMO, N., COSCIA, V., & DELITALA, M. On the mathematical theory of vehicular traffic flow I. Fluid dynamic and kinetic modelling. Math. Models Methods Appl. Sci. 12 (2002), 1801-1844. Zbl 1041.76061 MR 1946724
3. 1-stable. Numer. Math. 104 (2006), 27-46. Zbl 1106.65079 MR 2232001
4. CHALONS, C. Transport-equilibrium schemes for computing nonclassical shocks. Scalar conservation laws. Numer. Methods Partial Differential Equations 24 (2008), 1127-1147.
5. CHALONS, C. Numerical approximation of a macroscopic model of pedestrian flows, SIAM J. Sci. Comput. 29 (2007), 539-555.
6. CHALONS, C., & COQUEL, F. Capturing infinitely sharp discrete shock profiles with the Godunov scheme. Proc. 11th International Conference on Hyperbolic Problems: Theory, Numerics, Applications, S. Benzoni-Gavage and D. Serre (eds.), Springer (2008), 363-370.
7. CHALONS, C., & LEFLOCH, P. G. Computing undercompressive waves with the random choice scheme. Nonclassical shock waves. Interfaces Free Bound. 5 (2003), 129-158. Zbl 1038.35134 MR 1979428
8. COLELLA, P. Glimm's method for gas dynamics. SIAM J. Sci. Statist. Comput. 3 (1982), 76-110. Zbl 0502.76073 MR 0651869
9. COLOMBO, R. M. A 2 × 2 hyperbolic traffic flow model. Math. Comput. Modelling 35 (2002), 683-688. Zbl 0994.90021 MR 1884026
10. COLOMBO, R. M. Hyperbolic phase transitions in traffic flow. SIAM J. Appl. Math. 63 (2002), 708-721. Zbl 1037.35043 MR 1951956
11. COLOMBO, R. M., GOATIN, P., & PRIULI, F. S. Global well posedness of traffic flow models with phase transitions. Nonlinear Anal. Ser. A 66 (2007), 2413-2426. Zbl 1113.35123 MR2312597
12. DRAKE, J. S., SCHOFER, J. L., & MAY, A. D. A statistical analysis of speed density hypothesis. Highway Research Record 154 (1967), 53-87.
13. GOATIN, P. The Aw-Rascle vehicular traffic flow model with phase transitions. Math. Comput. Modelling 44 (2006), 287-303. Zbl pre05248498 MR 2239057
14. GODLEWSKY, E., & RAVIART, P. A. Hyperbolic Systems of Conservation Laws. Ellipse (1991). Zbl 0768.35059 MR 1304494
15. HELBING, D. Traffic and related self-driven many-particle systems. Rev. Modern Phys. 73 (2001), 1067-1141.
16. HOFF, D. Invariant regions for systems of conservation laws. Trans. Amer. Math. Soc. 289 (1985), 591-610. Zbl 0535.35056 MR 0784005
17. KERNER, B. The Physics of Traffic. Springer (2004).
18. LEFLOCH, P. G. Hyperbolic Systems of Conservation Laws: The theory of classical and nonclassical shock waves. E.T.H. Lecture Notes Ser., Birkhäuser (2002). Zbl 1019.35001 MR 1927887
19. LIGHTHILL, M. J., & WHITHAM, G. B. On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. Roy. Soc. London Ser. A 229 (1955), 317-345. Zbl 0064.20906 MR 0072606
20. RICHARDS, P. I. Shock waves on the highway. Operations Res. 4 (1956), 42-51. MR 0075522
21. TEMPLE, B. Systems of conservation laws with coinciding shock and rarefaction curves. Nonlinear Partial Differential Equations (Durham, NH, 1982), Contemp. Math. 17, Amer. Math. Soc. (1983), 143-151. Zbl 0538.35050
22. TORO, E. F. Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer (1999). Zbl 0923.76004
23. VAN LEER, B. Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method. J. Comput. Phys. 32 (1979), 101-136. Zbl 0939.76063 MR 1486274
24. ZHONG, X.-G., HOU, T. Y., & LEFLOCH, P. G. Computational methods for propagating phase boundaries. J. Comput. Phys. 124 (1996), 192-216. Zbl 0855.73080 MR 1378558