The dynamics of a single-species aerosol composed of droplets in air is described in terms of nucleation, evaporation, condensation, and coagulation processes. We present a comprehensive overview of the Euler-Euler formulation, which gives rise to a model in which fast nucleation that initiates aerosol droplets co-exists with comparably slow condensation. The latter process is responsible for the subsequent growth of the droplets. To accurately represent the dynamical consequences of the fast nucleation process, while retaining numerical efficiency, a new second-order time-integration method for the nucleation, evaporation, and condensation processes is proposed and analyzed. The new time-integration method takes the form of a 'corrected Euler forward' method. It includes rapid nucleation bursts and their possible cessation within a time step Δt. If the current nucleation burst persists for longer than the next time step, it is included fully, whereas cessation of the nucleation burst within the next Δt implies corrections to the effective rates in the algorithm. The identification of these two situations corresponds to the physical mechanism by which nucleation of a supersaturated vapor is halted because of the progressing condensation onto the already formed droplets. The resulting time-integration method is shown to be second-order accurate in time, whereas the computational costs per time step were found to be increased by less than 25% compared with the Euler forward method. The new method is also applied in combination with advective transport of the aerosol forming vapor to investigate a front of rapid nucleation. Adopting robust first-order upwinding for the spatial discretization, we arrive at a flexible method that shows an overall first-order convergence in Δt. For the full, spatially dependent system motivated by an aerosol of water droplets in air, the computational benefits of the new time-integration method over the Euler forward scheme, are a factor of about 10 improvements in accuracy at a given Δt and a similar factor in computing time when keeping the same level of accuracy.