Numerical Modeling of Flocking Dynamics with Topological Interactions

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.