The Aw–Rascle traffic model with Chaplygin pressure

Abstract In this paper, we introduce the Chaplygin pressure function to the Aw–Rascle (AR) traffic model. The solutions to the Riemann problem of the AR model with Chaplygin pressure are obtained constructively. In particular, the existence and uniqueness of delta shock solutions are established under the generalized Rankine–Hugoniot relation and entropy conditions. The delta shock may be useful for the description of the serious traffic jam. More importantly, it is proved that the limits of the Riemann solutions of the AR traffic model with Chaplygin pressure are exactly those of the pressureless gas dynamics model as the traffic pressure vanishes. Furthermore, we study the initial–boundary value problem of the AR model with Chaplygin pressure. A boundary entropy condition is derived, thanks to the results of Dubois and Le Floch [F. Dubois, P. Le Floch, Boundary conditions of nonlinear hyperbolic systems of conservation laws, Journal of Differential Equations 71 (1988) 93–122]. The application of our results in real traffic problem is also given.

[1]  Lijun Pan,et al.  The generalized Riemann problem for the Aw–Rascle model with phase transitions☆ , 2012 .

[2]  P. Floch,et al.  Boundary conditions for nonlinear hyperbolic systems of conservation laws , 1988 .

[3]  Tong Zhang,et al.  The two-dimensional Riemann problem for isentropicChaplygin gas dynamic system$^*$ , 2009 .

[4]  M J Lighthill,et al.  On kinematic waves II. A theory of traffic flow on long crowded roads , 1955, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[5]  Hanchun Yang.  Riemann Problems for a Class of Coupled Hyperbolic Systems of Conservation Laws , 1999 .

[6]  Meina Sun,et al.  Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model , 2010 .

[7]  D. Helbing.  Traffic and related self-driven many-particle systems , 2000, cond-mat/0012229.

[8]  J. Nédélec,et al.  First order quasilinear equations with boundary conditions , 1979 .

[9]  Tong Zhang,et al.  The Riemann Problem for the Transportation Equations in Gas Dynamics , 1999 .

[10]  P. I. Richards.  Shock Waves on the Highway , 1956 .

[11]  Y. Brenier,et al.  Sticky Particles and Scalar Conservation Laws , 1998 .

[12]  Sergei F. Shandarin,et al.  The large-scale structure of the universe: Turbulence, intermittency, structures in a self-gravitating medium , 1989 .

[13]  Michel Rascle,et al.  Resurrection of "Second Order" Models of Traffic Flow , 2000, SIAM J. Appl. Math..

[14]  Hanchun Yang,et al.  Riemann problem for the relativistic Chaplygin Euler equations , 2011 .

[15]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[16]  Y. Brenier.  Solutions with Concentration to the Riemann Problem for the One-Dimensional Chaplygin Gas Equations , 2005 .

[17]  Nicola Bellomo,et al.  ON THE MATHEMATICAL THEORY OF VEHICULAR TRAFFIC FLOW I: FLUID DYNAMIC AND KINETIC MODELLING , 2002 .