Asymptotic uniform estimate of random batch method with replacement for the Cucker-Smale model

TL;DR

This paper analyzes the asymptotic behavior and error estimates of the Random Batch Method variants applied to the Cucker-Smale flocking model, demonstrating uniformity in time and particle number with numerical validation.

Contribution

It establishes uniform error estimates for RBM variants on the Cucker-Smale model, improving upon previous results and providing rigorous analytical and numerical validation.

Findings

Global flocking emerges asymptotically under RBM.

Error estimates are uniform in time and particle number.

Numerical simulations confirm theoretical results.

Abstract

The Random Batch Method (RBM) [S. Jin, L. Li and J.-G. Liu, Random Batch Methods (RBM) for interacting particle systems, J. Comput. Phys. 400 (2020) 108877] is not only an efficient algorithm for simulating interacting particle systems, but also a randomly switching networked model for interacting particle system. This work investigates two RBM variants (RBM-r and RBM-1) applied to the Cucker-Smale flocking model. We establish the asymptotic emergence of global flocking and derive corresponding error estimates. By introducing a crucial auxiliary system and leveraging the intrinsic characteristics of the Cucker-Smale model, and under suitable conditions on the force, our estimates are uniform in both time and particle numbers. In the case of RBM-1, our estimates are sharper than those in Ha et al. (2021). Additionally, we provide numerical simulations to validate our analytical results.