We present a uniform-in-time (and in particle numbers as well) error estimate for 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] to the Cucker–Smale (CS) model. The uniform-in-time error estimates of the RBM have been obtained for various interacting particle systems, when corresponding flow generates a contraction semigroup. In this paper, we derive a uniform-in-time error estimate for RBM-approximation to the CS model in which the corresponding flow does not generate contractive semigroup. To derive uniform error estimate, we use asymptotic flocking estimate of the RBM-approximated CS model which yields the decay of relative velocities to zero, at least in the order of exp(−Ct1−β), while velocities of the original system decay exponentially. Here, β ∈ [0, 1) is the decay rate of the communication weight with respect to the distance between particles in the CS model. We also provide several numerical simulations to confirm the analytical results.

中文翻译：

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Cucker-Smale 模型的随机批处理方法的均匀时间误差估计

我们提出了随机批处理方法 (RBM) [S. Jin、L. Li 和 J.-G。Liu，用于交互粒子系统的随机批处理方法 (RBM)，J.计算机。物理。 400(2020) 108877] 到 Cucker-Smale (CS) 模型。当相应的流生成收缩半群时，已经为各种相互作用的粒子系统获得了 RBM 的时间均匀误差估计。在本文中，我们推导出了一个时间均匀误差估计，用于 RBM 逼近 CS 模型，其中相应的流不会生成收缩半群。为了得出统一的误差估计，我们使用 RBM 近似 CS 模型的渐近植绒估计，该模型产生相对速度衰减为零，至少大约为经验(-C吨1-β)，而原始系统的速度呈指数衰减。这里，β ∈ [0, 1)是通信权重相对于 CS 模型中粒子间距离的衰减率。我们还提供了几个数值模拟来确认分析结果。