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

TL;DR

The paper analyzes the discrete Motsch-Tadmor model obtained from a first-order Euler discretization of the continuous MT dynamics, addressing mono-cluster flocking, uniform-in-time convergence to the continuum, and uniform-in-time stability. It develops a nonlinear functional framework based on spatial and velocity diameters to derive recursive bounds that yield conditional mono-cluster flocking under explicit thresholds ${M = \frac{1}{4N\|\phi\|_{\text{Lip}}}}$ and $\psi(s) = 1 - \|\phi\|_{\text{Lip}} Ns$, with exponential decay of velocity differences. The work then proves that the discrete solution converges to the continuous MT solution uniformly in time as the time step $h$ tends to zero in the sense of shape discrepancies, and establishes a uniform-in-time stability estimate for the shape discrepancy between two discrete MT solutions, up to a residual term with a well-defined limit. Numerical experiments corroborate the theoretical findings, illustrate the influence of the decay rate parameter $\beta$ on the threshold $M$, and demonstrate robustness of the stability results, while highlighting the open question of unconditional flocking in the discrete setting and the potential for higher-order schemes.

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.

Figures (10)

• Figure 1: Bound $M$ as a function of $\beta$ for the communication weight $a$ as defined in \ref{['eq:a-Sec5']} with $c_1=0.1$ and $c_2=0.5$.
• Figure 2: Deviation of position (left) and velocity (right) for $\beta = 0.001, 0.005, 0.01$ and $d=4$. The behavior is as predicted in \ref{['thm: flocking']}.
• Figure 3: Convergence of $\log(\|\Delta^v (n)\|_F)$ for $\beta = 0.001, 0.005, 0.01$ and $d=4$.
• Figure 4: Distributions of particles and their velocity vectors at time $t= 0, 1.25, 10$ for $\beta = 0.001$ and $d=2$ for a setting not satisfying \ref{['eq: initial']}.
• Figure 5: $\|\Delta^x(n)\|_F$ (left) and $\|\Delta^v(n)\|_F$ (right) corresponding to the particle evolutions from Figure \ref{['fig:dynamics_b = 0.001']} for $\beta = 0.001$ and $d=2$. Flocking emerges even though the conditions of \ref{['thm: flocking']} are not satisfied.

Theorems & Definitions (46)

• remark 2.2
• definition 2.3
• theorem 2.4
• corollary 2.5
• proof
• remark 2.6
• remark 2.7
• lemma 3.1
• proof
• lemma 3.2