The Buoyancy-Drag model is a simple model, based on ordinary differential equations, for estimating the growth in the width of a turbulent mixing zone at an interface between fluids of different density due to Richtmyer-Meshkov and Rayleigh-Taylor instabilities. The model is calibrated to give the required self-similar behaviour for mixing in simple situations. However, the early stages of the mixing process are very dependent on the initial conditions and modifications to the Buoyancy-Drag model are then needed to obtain correct results. In a recent paper, Thornber et al. [Phys. Fluids. 29 (2017) 105107], a range of three-dimensional simulation techniques were used to calculate the evolution of the mixing zone integral width due to single-shock Richtmyer-Meshkov mixing from narrowband initial random perturbations. Further analysis of the results of these simulations gives greater insight into the transition from the initial linear behaviour to late-time self-similar mixing and provides a way of modifying the Buoyancy-Drag model to treat the initial conditions accurately. Higher resolution simulations are used to calculate the early time behavior more accurately and compare with a multi-mode model based on the impulsive linear theory. The analysis of the iLES data also gives a new method for estimating the growth exponent , theta (mixing zone width ~ t^theta), which is suitable for simulations which do not fully reach the self-state. The estimates of theta are consistent with the theoretical model of Elbaz & Shvarts [Physics of Plasmas (2018), 25, 062126].