We have developed a new Stokesian dynamics (SD) method for nondilute colloidal dispersions, which enables us to reduce drastically the computation time. To verify the validity of the present method, which is called the "cluster-based SD method," three-dimensional simulations of a ferromagnetic colloidal dispersion have been carried out for a simple shear flow.. The correlation function and viscosity have been evaluated to compare the results obtained by the present method with those obtained by the ordinary SD method and by the method of ignoring hydrodynamic interactions between particles. The results obtained here are summarized as follows. The transient properties from an initial state obtained by the present method agree well with those obtained by the ordinary method, even if a radius r(clstr), which defines the cluster formation, is taken as a small value such as r(clstr) = 1.2d (d is the particle diameter). Also, the equilibrium properties such as the pair correlation function and viscosity obtained by the present cluster-based method are in satisfactory agreement with those obtained by the ordinary SD method. Furthermore, the cluster-based method drastically reduces the computation time to about one-fourteenth to one-seventieth that of the ordinary method. It is clear from these results that the cluster-based SD method is significantly superior to the ordinary SD method for ferromagnetic colloidal dispersions for which a large model system such as N = 1000 or 10,000 is indispensable in simulations.