Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hydrodynamic traffic flow models including random accidents: A kinetic derivation

Felisia Angela Chiarello, Simone Göttlich, Thomas Schilliger, Andrea Tosin

TL;DR

A kinetic derivation of a second order macroscopic traffic model from a stochastic particle model is presented, giving a system of hyperbolic partial differential equations with a discontinuous flux function in which the traffic density and the headway are the averaged quantities.

Abstract

We present a formal kinetic derivation of a second order macroscopic traffic model from a stochastic particle model. The macroscopic model is given by a system of hyperbolic partial differential equations (PDEs) with a discontinuous flux function, in which the traffic density and the headway are the averaged quantities. A numerical study illustrates the performance of the second order model compared to the particle approach. We also analyse numerically uncertain traffic accidents by considering statistical measures of the solution to the PDEs.

Hydrodynamic traffic flow models including random accidents: A kinetic derivation

TL;DR

A kinetic derivation of a second order macroscopic traffic model from a stochastic particle model is presented, giving a system of hyperbolic partial differential equations with a discontinuous flux function in which the traffic density and the headway are the averaged quantities.

Abstract

We present a formal kinetic derivation of a second order macroscopic traffic model from a stochastic particle model. The macroscopic model is given by a system of hyperbolic partial differential equations (PDEs) with a discontinuous flux function, in which the traffic density and the headway are the averaged quantities. A numerical study illustrates the performance of the second order model compared to the particle approach. We also analyse numerically uncertain traffic accidents by considering statistical measures of the solution to the PDEs.
Paper Structure (15 sections, 1 theorem, 91 equations, 17 figures)

This paper contains 15 sections, 1 theorem, 91 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Microscopic dynamics with headway
  3. Enskog-type kinetic description and hydrodynamics
  4. Slow relaxation regime
  5. Hydrodynamic limit
  6. Analytical insights into the macroscopic system
  7. Fast relaxation regime
  8. Hydrodynamic limit
  9. Numerical simulations
  10. Discretisations
  11. Particle-macro and micro-macro comparison
  12. Kinetic and macroscopic descriptions including random accidents
  13. Enskog-type kinetic description and its hydrodynamic limit
  14. Microscopic and macroscopic numerical simulations
  15. Conclusions

Key Result

Theorem 3.2

Let us consider the set $\Omega$ such that for every $(\rho,h,c)\in \Omega$ the following conditions are verified: For every compact $K \subset \Omega$ there exists a constant $\delta>0$ with the following property. For every initial condition $\bar{u}$ with the Cauchy problem eq:macroheadway_conservative_3-eq:initial_cond has a weak entropy solution $u(t,x)=(\rho, h, c)(t,x),$ defined for all $

Figures (17)

  • Figure 1: Comparison of density and headway between the particle model and the second order macroscopic model (left) and the microscopic model together with first and second order macroscopic model (right) without relaxation ($a=0$).
  • Figure 2: Comparison of density and headway between the particle model and the second order macroscopic model (left) and the microscopic model together with first and second order macroscopic model (right) with relaxation parameter $a=1$.
  • Figure 3: Prototypical road capacity function parameterised by the random extent of an accident. Outside the uncertain stretch $[-Y,\,Y]$ we have $c=1$, whereas within the uncertain stretch $[-Y,\,Y]$ we have $c<1$.
  • Figure 4: Mean, median and 90 percent confidence interval of the density evolution of the second order macroscopic model \ref{['eq:random_macro2']} at $T=10$ for uniformly distributed $Y$.
  • Figure 5: Mean, median and 90 percent confidence interval of the density evolution of the microscopic model \ref{['eq:micro_model_Y']} at $T=10$ for uniformly distributed $Y$.
  • ...and 12 more figures

Theorems & Definitions (5)

  • Definition 3.1
  • Theorem 3.2
  • Remark 3.3
  • Definition 3.4
  • Remark 4.1