Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Scalar Approach to ARZ-Type Systems of Conservation Laws

Abraham Sylla

TL;DR

The paper develops a rigorous, splitting-based framework for GARZ-type 2×2 conservation-law systems modeling self-organized traffic, combining discontinuous-flux theory for the density with Panov's renormalized transport for the order parameter $w$. A robust finite-volume scheme is shown to converge to an entropy solution under a CFL condition, with compensated compactness handling the density's nonlinear flux and Panov theory controlling the transport component. The analysis yields existence and well-posedness results, BV propagation for $w$, and explicit entropy inequalities, while the numerical experiments illustrate Riemann-type wave structures and two-phase traffic behavior. The results provide a solid mathematical basis for analyzing GARZ-type traffic models and for implementing convergent numerical schemes in regimes with discontinuous flux and phase transitions.

Abstract

We are interested in 2x2 systems of conservation laws of special structure, including generalized Aw-Rascle and Zhang (GARZ) models for road traffic. The simplest representative is the Keyfitz-Kranzer system, where one equation is nonlinear and not coupled to the other, and the second equation is a linear transport equation which coefficients depend on the solution of the first equation. In GARZ systems, the coupling is stronger, they do not have the triangular structure of Keyfitz-Kranzer. In our setting, we claim that it makes sense to address these systems via a kind of splitting approach. Indeed, [E. Y. Panov, Instability in models connected with fluid flows II, 2008] proposes a robust framework for solving linear transport equations with divergence free coefficients. Our idea is to use this theory for the second equation of GARZ systems, and to exploit discontinuous flux theory advances for the first equation of the system.

Scalar Approach to ARZ-Type Systems of Conservation Laws

TL;DR

The paper develops a rigorous, splitting-based framework for GARZ-type 2×2 conservation-law systems modeling self-organized traffic, combining discontinuous-flux theory for the density with Panov's renormalized transport for the order parameter . A robust finite-volume scheme is shown to converge to an entropy solution under a CFL condition, with compensated compactness handling the density's nonlinear flux and Panov theory controlling the transport component. The analysis yields existence and well-posedness results, BV propagation for , and explicit entropy inequalities, while the numerical experiments illustrate Riemann-type wave structures and two-phase traffic behavior. The results provide a solid mathematical basis for analyzing GARZ-type traffic models and for implementing convergent numerical schemes in regimes with discontinuous flux and phase transitions.

Abstract

We are interested in 2x2 systems of conservation laws of special structure, including generalized Aw-Rascle and Zhang (GARZ) models for road traffic. The simplest representative is the Keyfitz-Kranzer system, where one equation is nonlinear and not coupled to the other, and the second equation is a linear transport equation which coefficients depend on the solution of the first equation. In GARZ systems, the coupling is stronger, they do not have the triangular structure of Keyfitz-Kranzer. In our setting, we claim that it makes sense to address these systems via a kind of splitting approach. Indeed, [E. Y. Panov, Instability in models connected with fluid flows II, 2008] proposes a robust framework for solving linear transport equations with divergence free coefficients. Our idea is to use this theory for the second equation of GARZ systems, and to exploit discontinuous flux theory advances for the first equation of the system.

Paper Structure

This paper contains 18 sections, 17 theorems, 111 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Numerical integrations
  4. Analytical Proofs
  5. Preliminaries
  6. Step 1: approximation.
  7. Step 2: compactness/convergence.
  8. Discontinuous Flux
  9. Convergence of the numerical scheme
  10. Stability and BV propagation
  11. Compensated compactness
  12. Convergence
  13. Step 1.
  14. Step 2: $(\rho, f(\rho, w))$ is divergence free.
  15. Step 3: Item \ref{['item:WeakFormulation']} holds.
  16. ...and 3 more sections

Key Result

Theorem 2.2

Assume that $V_{\min}, V_{\max} \in \mathbf{Lip}([0, 1], \mathbb{R}^+)$ satisfy eq:VelocitiesAssumption-eq:StrictConcavity-eq:vacuum. Fix $\rho_o \in \mathbf{L}^{\infty}(\mathbb{R}, [\mathrm{\varepsilon}, 1])$ and $w_o \in \mathbf{BV}(\mathbb{R}, [0, 1])$ piecewise constant. Then, as $(\Delta x, \De

Figures (4)

  • Figure 1: Illustration of the phase transition.
  • Figure 2: Shock of the first family followed by a contact discontinuity.
  • Figure 3: Rarefaction of the first family followed by a contact discontinuity.
  • Figure 4: $w_o$ piecewise constant. We see the constant propagation along the characteristics.

Theorems & Definitions (21)

  • Definition 2.1
  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 4.1
  • Theorem 4.2
  • Definition 4.3
  • Proposition 4.4
  • Proposition 4.5
  • Theorem 4.6
  • ...and 11 more