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Congested Crossing Pedestrian Traffic Flow : Dispersion vs. Transport in Crowded Areas

Mariam Al Khatib, Said Gounane, Noureddine Igbida, Ghadir Jradi

Abstract

This study investigates the complex dynamic interactions between two typed populations coexisting within a shared space. We propose both theoretical and numerical study to analyze scenarios where one population (population $1$) must traverse a territory occupied by another (population $2$), necessitating strategies to mitigate overcrowding caused by spatial limitations. To capture these interactions, we model population $1$ using a linear transport equation, while population $2$ is described by a granular diffusion model a la sandpile to represent its internal dynamics and tendency to decongest. Through numerical simulations, we explore how different movement strategies of the traversing population (population $1$) - including directed motion towards a specific destination, internal dispersion to minimize crowding, and uniform dispersal across the space - affects the behavior of population $2$.

Congested Crossing Pedestrian Traffic Flow : Dispersion vs. Transport in Crowded Areas

Abstract

This study investigates the complex dynamic interactions between two typed populations coexisting within a shared space. We propose both theoretical and numerical study to analyze scenarios where one population (population ) must traverse a territory occupied by another (population ), necessitating strategies to mitigate overcrowding caused by spatial limitations. To capture these interactions, we model population using a linear transport equation, while population is described by a granular diffusion model a la sandpile to represent its internal dynamics and tendency to decongest. Through numerical simulations, we explore how different movement strategies of the traversing population (population ) - including directed motion towards a specific destination, internal dispersion to minimize crowding, and uniform dispersal across the space - affects the behavior of population .

Paper Structure

This paper contains 20 sections, 7 theorems, 142 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Transport flow in population $\rho_1$
  3. Reminder on linear transport equation in a bounded domain
  4. Application to directed crowd motion for population $\rho_1$
  5. Congestion management theory
  6. Developing a congestion management model
  7. Theoretical study
  8. Algorithm and numerical computation
  9. Algorithms
  10. Numerical approximation:
  11. Numerical computation:
  12. Example 1:
  13. Example 2:
  14. Example 3:
  15. Example 4:
  16. ...and 5 more sections

Key Result

Theorem 2.1

Assume $V$ satisfies the assumption $(T1)$ and $(T2),$ and $u_0\in L^\infty(\Omega)$, Then, the problem PDEtransportBC has a unique weak solution $u$ in the sense that : $u\in L^\infty(Q)$, $( u\: V)\cdot \nu \in L^\infty(\Sigma)$, $( u\: V)\cdot \nu =0 ,$$\mathcal{H}^ {N-1}-$a.e. on $\Sigma^- ,$ for any $\xi \in \mathcal{C}^\infty_c(\Omega).$ In particular, $( u\: V)\cdot \nu$ is uniquely well

Figures (9)

  • Figure 1: Snapshots of $\rho_2$ (left) moving along the red-patterned field, interacting with $\rho_1$ (moving along the white field, center). The right image visualizes the combined density $\rho_1+ \rho_2$. The red movement of population $\rho_2$ is only triggered when it encounters population $\rho_1$ and $\rho_1+ \rho_2= 1$
  • Figure 2: The absence of $\rho_2$ flux (red arrows) in the bottom block can be attributed to the enforcement of the maximum density constraint throughout the dynamical evolution.
  • Figure 3: Snapshots of $\rho_2$ (left) totating along the red-patterned field, interacting with $\rho_1$ (moving along the white field, center).
  • Figure 4: Snapshots of $\rho_2$ (left) moving along the red-patterned field, interacting with $\rho_1$ (moving along the white field, center). The right image visualizes the combined density $\rho_1+ \rho_2$. Population 2’s red-indicated movement is triggered only when it encounters population $\rho_1$ and$\rho_1+ \rho_2= 1.$ By the end of the simulation, the entire population $\rho_1$ continues to leave the domain without congestion, and population $\rho_2$ is left with its final distribution from the previous time step.
  • Figure 5: In contrast to the earlier scenario in Example $4$, the reflective nature of the boundary conditions serves to preserve congestion within the domain upon population $rho_1$ reaching the boundary.
  • ...and 4 more figures

Theorems & Definitions (20)

  • Theorem 2.1
  • proof
  • Remark 1
  • Remark 2
  • Corollary 2.1
  • proof
  • Remark 3
  • Definition 3.1
  • Remark 4
  • Theorem 3.2
  • ...and 10 more