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

Abstract

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. © 2021 World Scientific Publishing Company.