The Riemann problem for a traffic flow model

Abstract

A traffic flow model describing the formation and dynamics of traffic jams was introduced by Berthelin et al. [5], which consists of a pressureless gas dynamics system under a maximal constraint on the density and can be derived from the Aw-Rascle model under the constraint condition $\rho\leq\rho^*$ by letting the traffic pressure vanish. In this paper, we give up this constraint condition and consider the following form $$ \left\{\begin{array}{ll}\rho_t+(\rho u)_x=0,\(\rho u+\varepsilon p(\rho))_t+(\rho u^2+\varepsilon up(\rho))_x=0,\end{array}\right .\eqno{} $$ in which $p(\rho)=-\frac{1}{\rho} . $ The Riemann problem for the above traffic flow model are solved constructively. The delta shock wave arises in the Riemann solutions, although the system is strictly hyperbolic, its first eigenvalue is genuinely nonlinear and the second eigenvalue is linearly degenerate. Furthermore, we clarify the generalized Rankine-Hugoniot relations and $\delta$-entropy condition. The position, strength and propagation speed of the delta shock wave are obtained from the generalized Rankine-Hugoniot 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 above traffic flow model are exactly those of the pressureless gas dynamics system with the same Riemann initial data as the traffic pressure vanishes.

On the Riemann problem and interaction of elementary waves for two‐layered blood flow model through arteries

“…Shao [39] investigated whether the Riemann solutions for linearly degenerate hyperbolic conservation laws preserve their global structural stability after minor BV perturbations of the initial data. Furthermore, Shao [40,41] established the existence and uniqueness of delta shock wave solution for the Riemann problem of traffic flow model and relativistic full Euler system. The results of single-wave interaction have been used by Jiang et al [42] to derive the weak solution for the Aw-Rascle-Zhang model with a non-genuinely nonlinear field, and their investigation gives a relief from the traffic congestion.…”