Mono-cluster flocking and uniform-in-time stability of the discrete Motsch-Tadmor model

TL;DR

This paper analyzes the discrete Motsch-Tadmor flocking model, establishing conditions for mono-cluster flocking, uniform-in-time stability, and the transition to the continuous model, supported by numerical comparisons.

Contribution

It provides a comprehensive framework for mono-cluster flocking, uniform-in-time stability, and the discrete-to-continuous transition in the Motsch-Tadmor model, addressing normalization challenges.

Findings

Conditions for mono-cluster flocking established

Uniform-in-time stability estimates derived

Numerical examples confirm analytical results

Abstract

The Motsch-Tadmor (MT) model is a variant of the Cucker-Smale model with a normalized communication weight function. The normalization poses technical challenges in analyzing the collective behavior due to the absence of conservation of momentum. We study three quantitative estimates for the discrete-time MT model considering the first-order Euler discretization. First, we provide a sufficient framework leading to the asymptotic mono-cluster flocking. The proposed framework is given in terms of coupling strength, communication weight function, and initial data. Second, we show that the continuous transition from the discrete MT model to the continuous MT model can be made uniformly in time using the finite-time convergence result and asymptotic flocking estimate. Third, we present uniform-in-time stability estimates for the discrete MT model. We also provide several numerical examples and compare them with analytical results.