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

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 [S.-Y. Ha, S. Jin, D. Kim and D. Ko, Uniform-in-time error estimate of the random batch method for the Cucker–Smale model, Math. Models Methods Appl. Sci. 31 (2021) 1099–1135.] Additionally, we provide numerical simulations to validate our analytical results.