The Random Batch Method (RBM), proposed by Jin et al. in 2020, is an efficient algorithm for simulating interacting particle systems. The uniform-in-time error estimates of the RBM without replacement have been obtained for various interacting particle systems, while the analysis of the RBM with replacement is just considered in (Cai et al., 2024) recently for the first-order systems governed by Langevin dynamics. In this work, we present the error estimate for the RBM with replacement applied to a second-order system known as the Cucker-Smale model. By introducing a crucial auxiliary system and leveraging the intrinsic characteristics of the Cucker-Smale model, we derive an estimate that is uniform in both time and particle numbers. Additionally, we provide numerical simulations to validate the analytical results.

翻译：随机批处理方法（RBM）由Jin等人于2020年提出，是模拟相互作用粒子系统的一种高效算法。对于无替换的RBM，已在多种相互作用粒子系统中获得了一致时间误差估计；而带替换的RBM分析则近期首次在（Cai等人，2024）中针对朗之万动力学控制的一阶系统展开。本工作中，我们提出了将带替换的RBM应用于二阶系统（即Cucker-Smale模型）的误差估计。通过引入关键辅助系统并利用Cucker-Smale模型的内在特性，我们推导出了在时间和粒子数上均具有一致性的估计。此外，我们提供了数值模拟以验证分析结果。