Stability analysis of a hyperbolic stochastic Galerkin formulation for the Aw-Rascle-Zhang model with relaxation

TL;DR

This paper analyzes the stability of a hyperbolic stochastic Galerkin method applied to the Aw-Rascle-Zhang traffic flow model with uncertainties, demonstrating stabilization through relaxation and providing computational validation.

Contribution

It introduces a stochastic Galerkin formulation for the Aw-Rasque-Zhang model, including conservative and non-conservative forms, and proves stability results with relaxation.

Findings

Stability of the stochastic Galerkin system is established.

Relaxation to a first-order model stabilizes the system.

Computational tests confirm theoretical stability results.

Abstract

We investigate the propagation of uncertainties in the Aw-Rascle-Zhang model, which belongs to a class of second order traffic flow models described by a system of nonlinear hyperbolic equations. The stochastic quantities are expanded in terms of wavelet-based series expansions. Then, they are projected to obtain a deterministic system for the coefficients in the truncated series. Stochastic Galerkin formulations are presented in conservative form and for smooth solutions also in the corresponding non-conservative form. This allows to obtain stabilization results, when the system is relaxed to a first-order model. Computational tests illustrate the theoretical results.