The design of simulations of free evolution in dipolar-coupled nuclear-spin systems using low-order correlations in Liouville space (LCL) is discussed, and a computational scheme relying on the Suzuki-Trotter algorithm and involving minimal memory requirements is described. The unusual nature of the approximation introduced by Liouville-space reduction in a spinning solid is highlighted by considering the accuracy of LCL simulations at different spinning frequencies, the quasiequilibria achieved by spin systems in LCL simulations, and the growth of high-order coherences in the exact dynamics. In particular, it is shown that accurate LCL simulations of proton spin diffusion occur in a regime where the reduced space excludes the coherences that make the dominant contribution to parallel to sigma parallel to(2), the norm-squared of the density matrix. (C) 2010 American Institute of Physics. [doi:10.1063/1.3505455]