Numerical modeling of flocking dynamics with topological interactions

TL;DR

This work numerically investigates flocking dynamics with topological interactions by transitioning from microscopic agent-based models to kinetic and macroscopic (pressureless Euler) descriptions, focusing on how the topological rank $M^N_{i,j}$ and kernel $K$ influence convergence to consensus. It develops a positivity-preserving upwind scheme for the kinetic equation and a second-order finite-volume scheme for the macroscopic system, incorporating non-conservative source terms via a global flux approach. The study analyzes cross-scale consistency under monokinetic and non-monokinetic initial data, demonstrating strong agreement in the monokinetic case while revealing deviations for non-monokinetic data, especially over longer times. Importantly, it reveals sensitive dependence on initial conditions, including non-uniqueness phenomena in 1D and 2D scenarios, highlighting the need for careful mathematical treatment of irregular configurations. Overall, the paper provides concrete numerical tools and insights into how topological interactions shape consensus formation, robustness, and potential non-uniqueness in collective dynamics across scales.

Abstract

In this paper, we propose a numerical investigation of topological interactions in flocking dynamics. Starting from a microscopic description of the phenomena, mesoscopic and macroscopic models have been previously derived under specific assumptions. We explore the role of topological interactions by describing the convergence speed to consensus in both microscopic and macroscopic dynamics, considering different forms of topological interactions. Additionally, we compare mesoscopic and macroscopic dynamics for monokinetic and non-monokinetic initial data. Finally, we illustrate with some simulations in one- and two-dimensional domains the sensitive dependence of solutions on initial conditions, including the case where the system exhibits two solutions starting with the same initial data.

Figures (13)

• Figure 1: Test 1: initial condition $N=100$ agents. Each position is marked with colors based on its initial velocity: the velocities of the agents located in $[-3,-1]$ are randomly chosen in $[-2,0]$, whereas the velocities of the agents located in $[1,3]$ are randomly chosen in $[0,2]$.
• Figure 2: Test 1: plot of $V_i(t)$, $i=1,...,N$, solution to \ref{['sec2:CSmodel_pij']} with $K=\mathds{1}_{[0, \overline{M}/N ]}$ for different values of $\overline{M}$.
• Figure 3: Test 2: plot of the microscopic velocity $V_i(t)$, $i=1,...,N$, solution to \ref{['sec2:CSmodel_pij']} with a) $K$ linear b) $K$ convex function and the comparison between c) $\bar{V}(t)$ and d) $\bar{u}(t)$, for different $K$ functions.
• Figure 4: Test 3: screenshots of numerical simulation of \ref{['sec2:CSmodel_pij']} with $K=K_1$ as in \ref{['function_K1']}.
• Figure 5: Test 4: approximation of monokinetic initial data. Comparison of $\nu_t^0$ and $\rho_t$ (first line) and $\nu_t^1$ and $Q_t$ (second line) at different time steps.

Theorems & Definitions (1)

• Remark 1.1