, we made the choice of working in Lagrangian coordinates, which greatly simplifies computations. The same procedure could be carried out for the original problem in Eulerian coordinates. Besides, as mentioned in the introduction, the choice of the WFT scheme for constructing approximate solutions was motivated by the lack of T V bounds on classical Godunov approximations. This was already pointed out in [37] for system (1.4), but remains valid in Lagrangian coordinates

D. Amadori and A. Corli, Global existence of BV solutions and relaxation limit for a model of multiphase reactive flow, Nonlinear Anal, vol.72, issue.5, pp.2527-2541, 2010.

D. Amadori and G. Guerra, Global BV solutions and relaxation limit for a system of conservation laws, Proc. Roy. Soc. Edinburgh Sect. A, vol.131, issue.1, pp.1-26, 2001.

F. Ancona and P. Goatin, Uniqueness and stability of L 8 solutions for Temple class systems with boundary and properties of the attainable sets, SIAM J. Math. Anal, vol.34, issue.1, pp.28-63, 2002.

A. Aw, A. Klar, T. Materne, and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math, vol.63, issue.1, pp.259-278, 2002.

A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math, vol.60, issue.3, pp.916-938, 2000.

P. Bagnerini and M. Rascle, A multiclass homogenized hyperbolic model of traffic flow, SIAM J. Math. Anal, vol.35, issue.4, pp.949-973, 2003.

P. Baiti and A. Bressan, The semigroup generated by a Temple class system with large data, Differential Integral Equations, vol.10, issue.3, pp.401-418, 1997.

A. Bressan, Hyperbolic systems of conservation laws, Oxford Lecture Series in Mathematics and its Applications, vol.20, 2000.

A. Bressan and R. M. Colombo, Decay of positive waves in nonlinear systems of conservation laws, Ann. Scuola Norm. Sup. Pisa Cl. Sci, vol.26, issue.4, pp.133-160, 1998.

A. Bressan and P. Goatin, Oleinik type estimates and uniqueness for n ? n conservation laws, J. Differential Equations, vol.156, issue.1, pp.26-49, 1999.

A. Bressan and P. Goatin, Stability of L 8 solutions of Temple class systems, Differential Integral Equations, vol.13, pp.1503-1528, 2000.

A. Bressan and T. Yang, A sharp decay estimate for positive nonlinear waves, SIAM J. Math. Anal, vol.36, issue.2, pp.659-677, 2004.

C. Chalons and P. Goatin, Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling, Interfaces Free Bound, vol.10, issue.2, pp.197-221, 2008.

G. Q. Chen, C. D. Levermore, and T. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math, vol.47, issue.6, pp.787-830, 1994.

G. Q. Chen and T. Liu, Zero relaxation and dissipation limits for hyperbolic conservation laws, Comm. Pure Appl. Math, vol.46, issue.5, pp.755-781, 1993.

C. Christoforou and K. Trivisa, Sharp decay estimates for hyperbolic balance laws, J. Differential Equations, vol.247, issue.2, pp.401-423, 2009.

C. Christoforou and K. Trivisa, Decay of positive waves of hyperbolic balance laws, Acta Math. Sci. Ser. B Engl. Ed, vol.32, issue.1, pp.352-366, 2012.

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, 1948.

G. Crasta and B. Piccoli, Viscosity solutions and uniqueness for systems of inhomogeneous balance laws, Discrete Contin. Dynam. Systems, vol.3, issue.4, pp.477-502, 1997.

C. M. Dafermos, Hyperbolic conservation laws in continuum physics, Grundlehren der Mathematischen Wissenschaften, vol.325
DOI : 10.1007/978-3-662-49451-6

. Springer-verlag, , 2000.

C. F. Daganzo, Requiem for second-order fluid approximations of traffic flow, Transportation Research Part B: Methodological, vol.29, issue.4, pp.277-286, 1995.

P. Goatin and L. Gosse, Decay of positive waves for n ? n hyperbolic systems of balance laws, Proc. Amer. Math. Soc, vol.132, issue.6, pp.1627-1637, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00419731

J. M. Greenberg, Extensions and amplifications of a traffic model of Aw and Rascle, SIAM J. Appl. Math, vol.62, issue.3, p.2, 2001.

D. Hoff, Invariant regions for systems of conservation laws, Trans. Amer. Math. Soc, vol.289, issue.2, pp.591-610, 1985.

H. Holden and N. H. Risebro, Front tracking for hyperbolic conservation laws, Applied Mathematical Sciences, vol.152, 2002.

H. K. Jenssen and C. Sinestrari, On the spreading of characteristics for non-convex conservation laws, Proc. Roy. Soc. Edinburgh Sect. A, vol.131, issue.4, pp.909-925, 2001.

C. Lattanzio and P. Marcati, The zero relaxation limit for the hydrodynamic Whitham traffic flow model, J. Differential Equations, vol.141, issue.1, pp.150-178, 1997.
DOI : 10.1006/jdeq.1997.3311

URL : https://doi.org/10.1006/jdeq.1997.3311

T. Li, Global solutions and zero relaxation limit for a traffic flow model, SIAM J. Appl. Math, vol.61, issue.3, pp.1042-1061, 2000.
DOI : 10.1137/s0036139999356788

T. Li, L 1 stability of conservation laws for a traffic flow model, Electron. J. Differential Equations, vol.18, issue.14, 2001.

T. Li, Well-posedness theory of an inhomogeneous traffic flow model, Discrete Contin. Dyn. Syst. Ser. B, vol.2, issue.3, pp.401-414, 2002.

T. Li, Global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow, J. Differential Equations, vol.190, issue.1, pp.131-149, 2003.

T. Li and H. Liu, Critical thresholds in a relaxation model for traffic flows, Indiana Univ. Math. J, vol.57, issue.3, pp.1409-1430, 2008.

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, vol.229, pp.317-345, 1955.

O. A. Ole?-i-nik, Discontinuous solutions of non-linear differential equations. Uspehi Mat. Nauk, N.S.), vol.12, issue.3, pp.3-73, 1957.

O. A. Ole?-i-nik, Discontinuous solutions of non-linear differential equations, Amer. Math. Soc. Transl, vol.26, issue.2, pp.95-172, 1963.

H. J. Payne, Models of freeway traffic and control. Mathematical models of public systems, 1971.

M. Rascle, An improved macroscopic model of traffic flow: derivation and links with the Lighthill-Whitham model, Math. Comput. Modelling, vol.35, issue.5-6, pp.581-590, 2002.

P. I. Richards, Shock waves on the highway, Operations Res, vol.4, pp.42-51, 1956.
DOI : 10.1287/opre.4.1.42

D. Serre, Systèmes de lois de conservation II. structures géométriques, oscillation et probì emes mixtes, 1996.

B. Temple, Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc, vol.280, issue.2, pp.781-795, 1983.

D. H. Wagner, Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions, J. Differential Equations, vol.68, issue.1, pp.118-136, 1987.

G. B. Whitham, Linear and nonlinear waves. pages xvi+636, Pure and Applied Mathematics, 1974.

H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B: Methodological, vol.36, issue.3, pp.275-290, 2002.