Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A nonlocal traffic flow model with stochastic velocity

Timo Böhme, Simone Göttlich, Andreas Neuenkirch

TL;DR

The paper develops a stochastic nonlocal traffic flow model by perturbing the nonlocal velocity in a scalar conservation law with kernel $W_\eta$ and velocity $v(\rho)$. It establishes existence, uniqueness, and BV bounds for weak entropy solutions of the resulting sNV PDE, accompanied by a robust Godunov-type numerical scheme that incorporates discrete-time noise with a CFL-stable update. The analysis shows the stochastic flux raises the mean downstream velocity ($\mathbb{E}[V_\epsilon] \ge V$) and explores how the mean behavior relates to the deterministic NV model, particularly noting deviations at high densities and the influence of parameters $\tau$ and $\eta$. Numerical experiments via Monte Carlo simulations reveal how the stochastic perturbations distribute densities and how the mean behavior can approximate or diverge from the deterministic baseline, providing a practical framework for incorporating randomness in nonlocal traffic dynamics.

Abstract

In this paper, we investigate a nonlocal traffic flow model based on a scalar conservation law, where a stochastic velocity function is assumed. In addition to the modeling, theoretical properties of the stochastic nonlocal model are provided, also addressing the question of well-posedness. A detailed numerical analysis offers insights how the stochasticity affects the evolution of densities. Finally, numerical examples illustrate the mean behavior of solutions and the influence of parameters for a large number of realizations.

A nonlocal traffic flow model with stochastic velocity

TL;DR

The paper develops a stochastic nonlocal traffic flow model by perturbing the nonlocal velocity in a scalar conservation law with kernel and velocity . It establishes existence, uniqueness, and BV bounds for weak entropy solutions of the resulting sNV PDE, accompanied by a robust Godunov-type numerical scheme that incorporates discrete-time noise with a CFL-stable update. The analysis shows the stochastic flux raises the mean downstream velocity () and explores how the mean behavior relates to the deterministic NV model, particularly noting deviations at high densities and the influence of parameters and . Numerical experiments via Monte Carlo simulations reveal how the stochastic perturbations distribute densities and how the mean behavior can approximate or diverge from the deterministic baseline, providing a practical framework for incorporating randomness in nonlocal traffic dynamics.

Abstract

In this paper, we investigate a nonlocal traffic flow model based on a scalar conservation law, where a stochastic velocity function is assumed. In addition to the modeling, theoretical properties of the stochastic nonlocal model are provided, also addressing the question of well-posedness. A detailed numerical analysis offers insights how the stochasticity affects the evolution of densities. Finally, numerical examples illustrate the mean behavior of solutions and the influence of parameters for a large number of realizations.
Paper Structure (23 sections, 9 theorems, 123 equations, 7 figures, 1 table)

This paper contains 23 sections, 9 theorems, 123 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Nonlocal velocity model (NV)
  3. Modeling of stochastic nonlocal velocity
  4. Derivation
  5. The stochastic nonlocal velocity model (sNV)
  6. Analysis of random velocities
  7. Mean and variance of random velocities
  8. Characteristics
  9. Numerical scheme for the sNV model
  10. Properties, existence and uniqueness
  11. Properties of the sNV model
  12. Existence and Uniqueness of the sNV model
  13. Numerical results
  14. Probabilistic densities and their mean behavior
  15. Influence of parameters
  16. ...and 8 more sections

Key Result

Lemma 3.3

Let the Remarks rem:ass_W, rem:ass_v_ND hold, then

Figures (7)

  • Figure 1: Realization of $\epsilon$ as constructed in (\ref{['eq:construction_eps']}) with $\tau=1$ and numerical evaluations.
  • Figure 2: $v(\rho)=1-\rho^2$, the expectation ($\overline{v}_\epsilon$) of $v_\epsilon$ and its variance, given $\tau=0.8$.
  • Figure 3: Characteristics of Example \ref{['ex:standart_ex']} to (\ref{['eq:sNV']}).
  • Figure 4: Characteristics of Example \ref{['ex:enalarged_ex']} to (\ref{['eq:sNV']}) in grey and (\ref{['eq:NV']}) in blue.
  • Figure 5: Characteristics of Example \ref{['ex:enalarged_ex']} to (\ref{['eq:sNV']}) in grey and (\ref{['eq:NV']}) using $\bar{v}_\epsilon$ in green.
  • ...and 2 more figures

Theorems & Definitions (43)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.3: Nonlocal weak entropy solution
  • Remark 2.4
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • Definition 3.4: Characteristics of (\ref{['eq:sNV']})
  • Example 3.5
  • Example 3.6
  • ...and 33 more