Stability of a traffic flow model with nonconvex relaxation. (English) Zbl 1099.35156

Li, Tong; Liu, Hailiang

Commun. Math. Sci. 3, No. 2, 101-118 (2005).

Summary: This paper is concerned with the nonlinear stability of traveling wave solutions for a quasi-linear relaxation model with a nonconvex equilibrium flux. The study is motivated by and the results are applied to the well-known dynamic continuum traffic flow model, the Payne and Whitham (PW) model with a nonconcave fundamental diagram. The PW model is the first of its kind and it has been widely adopted by traffic engineers in the study of stability and instability phenomena of traffic flow. The traveling wave solutions are shown to be asymptotically stable under small disturbances and under the sub-characteristic condition using a weighted energy method. The analysis applies to both non-degenerate case and the degenerate case where the traveling wave has exponential decay rates at infinity and has an algebraic decay rate at infinity, respectively.

MSC:

35Q80	Applications of PDE in areas other than physics (MSC2000)
35B35	Stability in context of PDEs
90B20	Traffic problems in operations research
35B40	Asymptotic behavior of solutions to PDEs
35L65	Hyperbolic conservation laws
76L05	Shock waves and blast waves in fluid mechanics

Keywords:

nonlinear stability; traveling wave solutions; quasi-linear relaxation model; nonconvex equilibrium flux; nonconcave fundamental diagram; algebraic decay