Uniform-in-time error estimate of random batch method with replacement for the Cucker-Smale model

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.