Random-batch method for multi-species stochastic interacting particle systems

A random-batch method for interacting particle systems is proposed, extending the method of S. Jin, L. Li, and J.-G. Liu (2020) [16] to multicomponent systems with and without multiplicative noise. The idea of the algorithm is to randomly divide, at each time step, the ensemble of particles into small batches and then to evolve the interaction of each particle within the batches until the next time step. This reduces the computational cost by one order of magnitude, while keeping a certain accuracy. It is proved that the L2 error of the error process behaves like the square root of the time step size, uniformly in time, thus providing the convergence of the scheme. The numerical efficiency is tested for some examples, and numerical simulations for a Poisson–Boltzmann model as well as of the segregation of two populations and the opinion formation in a hierarchical company are presented.