A new numerical method for the solution of an internally mixed spatially homogeneous sectional model for aerosol nucleation and condensation is proposed. The characteristics method is used to predict droplet sizes within a discrete time step. The method is designed such that 1) a pre-specified number of moments of the droplet size distribution may be preserved, 2) there exists no time step stability restriction related to the condensation rate and section size, 3) highly skewed fixed sectional distributions may be used and 4) it is straightforward to extend to spatially inhomogeneous settings and to incorporate droplet coagulation and break-up. We derive, starting from mass conservation, a consistent internally mixed multi-species aerosol model. For certain condensational growth laws analytical solutions exist, against which the method is validated. Using two-moment and four-moment-preserving schemes, we find first order convergence of the numerical solution to the analytical result, as a function of the number of sections. As the four-moment-preserving scheme does not guarantee positivity of the solution, a hybrid scheme is proposed, which, when needed, locally reverts back to two-moment preservation, to prevent negativity. As an illustration, the method is applied to a complete multi-species homogeneous nucleation and condensation problem.