Lie Symmetry Analysis of the Aw-Rascle-Zhang Model for Traffic State Estimation

We extend our analysis on the Lie symmetries in fluid dynamics to the case of macroscopic traffic estimation models. In particular we study the Aw-Rascle-Zhang model for traffic estimation, which consists of two hyperbolic first-order partial differential equations. The Lie symmetries, the one-dimensional optimal system and the corresponding Lie invariants are determined. Specifically, we find that the admitted Lie symmetries form the four-dimensional Lie algebra A(4,12). The resulting one-dimensional optimal system is consisted by seven one-dimensional Lie algebras. Finally, we apply the Lie symmetries in order to define similarity transformations and derive new analytic solutions for the traffic model. The qualitative behaviour of the solutions is discussed.

Automated summary

We extend our analysis on the Lie symmetries in fluid dynamics to macroscopic traffic estimation models. Specifically, we study the Aw-Rascle-Zhang model, which consists of two hyperbolic first-order partial differential equations. We determine the Lie symmetries, the one-dimensional optimal system, and the corresponding Lie invariants. We find that the admitted Lie symmetries form the four-dimensional Lie algebra A(4,12). The resulting one-dimensional optimal system is composed of seven one-dimensional Lie algebras. We use the Lie symmetries to define similarity transformations and derive new analytic solutions for the traffic model, discussing the qualitative behavior of the solutions.