Bale, D., LeVeque, R., Mitran, S. and Rossmanith, J. (2002). A wave propagation method for conservation laws and balance laws with spatially varying flux functions.
Siam Journal on Scientific Computing, 24(3):955–978.
CrossRefBui, D., Nelson, P. and Narasimhan, S. (1992). Computational Realizations of the Entropy Condition in Modeling Congested Traffic Flow. Technical report.
Bultelle, M., Grassin, M. and Serre, D. (1998). Unstable Godunov discrete profiles for steady shock waves. SIAM J. Numer. Anal, 35(6), 2272–2297.
Burger, R., Garcia, A., Karlsen, K., Towers, J., Tosin, A., Ambrosio, L., Crippa, G., LeFloch, P., Donato, P. and Gaveau, F. et al. (2008). Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model. Networks and Heterogeneous Media, 3(1), 1.
Burger, R., Karlsen, K., Mishra, S. and Towers, J. (2005). On conservation laws with discontinuous flux. Trends in Applications of Mathematics to Mechanics, 75–84.
Coclite, G., Garavello, M. and Piccoli, B. (2005). Traffic flow on a road network.
SIAM Journal on Mathematical Analysis, 36(6), 1862–1886.
CrossRefColella, P. and Puckett, E.G. (2004). Modern Numerical Methods for Fluid Flow. In draft.
Courant, R., Friedrichs, K. and Lewy, H. (1928). ber die partiellen differenzengleichungen dermathematischen physik.
Mathematische Annalen, 100, 32–74.
CrossRefDaganzo, C.F. (1994). The cell transmission model: a dynamic representation of highway traffic consistent with hydrodynamic theory.
Transportation Research Part B, 28(4), 269–287.
CrossRefDaganzo, C.F. (1995a). The cell transmission model II: network traffic. Transportation Research Part B, 29(2), 79–93.
Daganzo, C.F. (1995b). A finite difference approximation of the kinematic wave model of traffic flow. Transportation Research Part B, 29(4), 261–276.
Daganzo, C.F. (1997). A continuum theory of traffic dynamics for freeways with special lanes.
Transportation Research Part B, 31(2), 83–102.
CrossRefDaganzo, C.F. (2006). On the variational theory of traffic flow: well-posedness, duality and applications. Networks and Heterogeneous Media, 1(4), 601.
Diehl, S. (1995). On scalar conservation laws with point source and discontinuous flux function.
SIAM Journal on Mathematical Analysis, 26(6), 1425–1451.
CrossRefDiehl, S. (1996a). A conservation law with point source and discontinuous flux function modelling continuous sedimentation. SIAM Journal on Mathematical Analysis, 56, 388–419.
Diehl, S. (1996b). Scalar conservation laws with discontinuous flux function: I. The viscous profile condition. Communications in Mathematical Physics, 176(1), 23–44.
Diehl, S. and Wallin, N. (1996). Scalar conservation laws with discontinuous flux function: II. On the stability of the viscous profiles.
Communications in Mathematical Physics, 176(1), 45–71.
CrossRefEngquist, B. and Osher, S. (1980a). One-Sided Difference Schemes and Transonic Flow. Proceedings of the National Academy of Sciences, 77(6), 3071–3074.
Engquist, B. and Osher, S. (1980b). Stable and entropy satisfying approximations for transonic flow calculations. Proceedings of the National Academy of Sciences, 34(149), 45–75.
Engquist, B. and Osher, S. (1981). One-sided difference approximations for nonlinear conservation laws.
Mathematics of Computation, 36(154), 321–351.
CrossRefGaravello, M., Natalini, R., Piccoli, B. and Terracina, A. (2007). Conservation laws with discontinuous flux. Networks and Heterogeneous Media, 2, 159–179.
Gazis, D.C., Herman, R. and Rothery, R.W. (1961). Nonlinear follow-the-leader models of traffic flow.
Operations Research, 9(4), 545–567.
CrossRefGimse, T. (1993). Conservation Laws with Discontinuous Flux Functions.
SIAM Journal on Mathematical Analysis, 24, 279.
CrossRefGimse, T. and Risebro, N. (1990). Riemann problems with a discontinuous flux function. Theory, Numerical Methods and Applications. I.
Godunov, S.K. (1959). A difference method for numerical calculations of discontinuous solutions of the equations of hydrodynamics. Matematicheskii Sbornik, 47, 271–306.
Greenshields, B.D. (1935). A study in highway capacity. Highway Research Board Proceedings, 14, 448–477.
Herrmann, M. and Kerner, B.S. (1998). Local cluster effect in different traffic flow models.
Physica A, 255, 163–198.
CrossRefHerty, M., Sea¨ıd, M. and Singh, A. (2007). A domain decomposition method for conservation laws with discontinuous flux function.
Applied Numerical Mathematics, 57(4), 361–373.
CrossRefHolden, H. and Risebro, N.H. (1995). A mathematical model of traffic flow on a network of unidirectional roads.
SIAM Journal on Mathematical Analysis, 26(4), 999–1017.
CrossRefIsaacson, E.I. and Temple, J.B. (1992). Nonlinear resonance in systems of conservation laws.
SIAM Journal on Applied Mathematics, 52(5), 1260–1278.
CrossRefJin,W.-L. (2003).
Kinematic Wave Models of Network Vehicular Traffic. PhD thesis, University of California, Davis.
http://arxiv.org/abs/math.DS/0309060.
Jin, W.-L. (2008). Asymptotic traffic dynamics arising in diverge-merge networks with two intermediate links. Transportation Research Part B. In press.
Jin,W.-L. (2009). Continuous kinematic wave models of merging traffic flow. Proceedings of TRB 2009 Annual Meeting.
Jin, W.-L. and Zhang, H.M. (2003a). The inhomogeneous kinematic wave traffic flow model as a resonant nonlinear system. Transportation Science, 37(3), 294–311.
Jin, W.-L. and Zhang, H. M. (2003b). On the distribution schemes for determining flows through a merge. Transportation Research Part B, 37(6), 521–540.
Kerner, B.S. and Konh¨auser, P. (1994). Structure and parameters of clusters in traffic flow.
Physical Review E, 50(1), 54–83.
CrossRefKlingenberg, C. and Risebro, N. (1995). Convex conservation laws with discontinuous coefficients. existence, uniqueness and asymptotic behavior.
Communications in Partial Differential Equations, 20(11), 1959–1990.
CrossRefLebacque, J.P. (1996). The Godunov scheme and what it means for first order traffic flow models. The International Symposium on Transportation and Traffic Theory, Lyon, France.
LeVeque, R.J. (2002). Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge; New York.
Lighthill, M.J. and Whitham, G.B. (1955). On kinematic waves: II. a theory of traffic flow on long crowded roads. Proceedings of the Royal Society of London A, 229(1178):317–345.
CrossRefLin, L., Temple, J.B. and Wang, J. (1995). A comparison of convergence rates for Godunov’s method and Glimm’s method in resonant nonlinear systems of conservation laws. SIAM Journal on Numerical Analysis, 32(3):824–840.
CrossRefMochon, S. (1987). An analysis of the traffic on highways with changing surface conditions. Math. Modelling, 9:1–11.
CrossRefNagel, K. and Schreckenberg, M. (1992). A cellular automaton model for freeway traffic. Journal de Physique I France, 2(2):2221–2229.
CrossRefOsher, S. (1984). Riemann Solvers, The Entropy Condition, and Difference Approximations. SIAM Journal on Numerical Analysis, 21(2):217–235.
CrossRefOsher, S. and Solomon, F. (1982). Upwind Difference Schemes for Hyperbolic Systems of Conservation Laws. Mathematics of Computation, 38(158):339–374.
CrossRefRichards, P.I. (1956). Shock waves on the highway. Operations Research, 4:42–51.
CrossRefSeguin, N. and Vovelle, J. (2003). Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients. Math. Models Methods Appl. Sci, 13(2):221–257.
CrossRefSmoller, J. (1983). Shock waves and reaction-diffusion equations. Springer-Verlag, New York.
van Leer, B. (1984). On the Relation Between the Upwind-Differencing Schemes of Godunov, Engquist–Osher and Roe. SIAM Journal on Scientific and Statistical Computing, 5:1.
CrossRefWong, G.C.K. and Wong, S.C. (2002). A multi-class traffic flow model: an extension of LWR model with heterogeneous drivers. Transportation Research Part A: Policy and Practice, 36(9):827–841.
CrossRefZhang, P. and Liu, R. (2003). Hyperbolic conservation laws with space-dependent flux: I. Characteristics theory and Riemann problem. Journal of Computational and Applied Mathematics, 156(1):1–21.
Zhang, P. and Liu, R. (2005a). Generalization of Runge-Kutta discontinuous Galerkin method to LWR traffic flow model with inhomogeneous road conditions. Numerical Methods for Partial Differential Equations, 21(1):80–88.
Zhang, P. and Liu, R. (2005b). Hyperbolic conservation laws with space-dependent fluxes: II. General study of numerical fluxes. Journal of Computational and Applied Mathematics, 176(1):105–129.
Zhang, P.,Wong, S. and Shu, C. (2006). A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway. Journal of Computational Physics, 212(2):739–756
CrossRef