Semi-linear Poisson-mediated Flocking in a Cucker-Smale Model

TL;DR

This paper introduces a novel approach to flocking dynamics by transforming non-local integro-differential equations into a local PDE system using semi-linear Poisson equations, enabling efficient computation of flock behavior.

Contribution

It develops a new class of interaction functions as Green's functions for semi-linear Poisson equations, simplifying the analysis and simulation of flocking models.

Findings

Transformation of mean-field equations to PDEs in a compact set

Efficient finite difference solution with tridiagonal system

Applicable to flock-centered dynamics without loss of generality

Abstract

We propose a family of compactly supported parametric interaction functions in the general Cucker-Smale flocking dynamics such that the mean-field macroscopic system of mass and momentum balance equations with non-local damping terms can be converted from a system of partial integro-differential equations to an augmented system of partial differential equations in a compact set. We treat the interaction functions as Green's functions for an operator corresponding to a semi-linear Poisson equation and compute the density and momentum in a translating reference frame, i.e. one that is taken in reference to the flock's centroid. This allows us to consider the dynamics in a fixed, flock-centered compact set without loss of generality. We approach the computation of the non-local damping using the standard finite difference treatment of the chosen differential operator, resulting in a tridiagonal system which can be solved quickly.