The coarse‐grained Monte Carlo (CGMC) algorithm was originally proposed in the series of works [M. A. Katsoulakis, A. J. Majda, and D. G. Vlachos, J. Comput. Phys., 186 (2003), pp. 250–278; M. A. Katsoulakis, A. J. Majda, and D. G. Vlachos, Proc. Natl. Acad. Sci. USA, 100 (2003), pp. 782–787; M. A. Katsoulakis and D. G. Vlachos, J. Chem. Phys., 119 (2003), pp. 9412–9427]. In this paper we further investigate the approximation properties of the coarse‐graining procedure and provide both analytical and numerical evidence that the hierarchy of the coarse models is built in a systematic way that allows for error control in both transient and long‐time simulations. We demonstrate that the numerical accuracy of the CGMC algorithm as an approximation of stochastic lattice spin flip dynamics is of order two in terms of the coarse‐graining ratio and that the natural small parameter is the coarse‐graining ratio over the range of particle/particle interactions. The error estimate is shown to hold in the weak convergence sense. We employ the derived analytical results to guide CGMC algorithms and demonstrate a CPU speed‐up in demanding computational regimes that involve nucleation, phase transitions, and metastability
