A mapping approach is proposed for the numerical simulation of distributive mixing in three-dimensional laminar flows. The method is based on a spatial discretization of the locally averaged concentration of fluid components in the mixture (the so-called ''coarse grain density''). A distribution matrix, that describes the changes in component concentration, is composed. The proposed method makes it possible to rapidly predict the (short- and long-term) mixing performances, and to compare a large number of different mixing protocols in an efficient way. The consistency and accuracy of this algorithm is validated by comparing the results obtained on grids with different spatial resolution, and by comparison with front tracking results. The technique is evaluated in a prototype mixing flow in a cubic cavity, generated by sliding opposite walls. Different mixing protocols are compared quantitatively, and result in optimal mixing protocol parameters.