In this work we develop the Spectral Ewald Accelerated Stokesian Dynamics (SEASD), a novel computational method for dynamic simulations of polydisperse colloidal suspensions with full hydrodynamic interactions. SEASD is based on the framework of Stokesian Dynamics (SD) with extension to compressible solvents, and uses the Spectral Ewald (SE) method [Lindbo & Tornberg, J. Comput. Phys. 229 (2010) 8994] for the wave-space mobility computation. To meet the performance requirement of dynamic simulations, we use Graphic Processing Units (GPU) to evaluate the suspension mobility, and achieve an order of magnitude speedup compared to a CPU implementation. For further speedup, we develop a novel far-field block-diagonal preconditioner to reduce the far-field evaluations in the iterative solver, and SEASD-nf, a polydisperse extension of the mean-field Brownian approximation of Banchio & Brady [J. Chem. Phys. 118 (2003) 10323]. We extensively discuss implementation and parameter selection strategies in SEASD, and demonstrate the spectral accuracy in the mobility evaluation and the overall $mathcal{O}(Nlog N)$ computation scaling. We present three computational examples to further validate SEASD and SEASD-nf in monodisperse and bidisperse suspensions: the short-time transport properties, the equilibrium osmotic pressure and viscoelastic moduli, and the steady shear Brownian rheology. Our validation results show that the agreement between SEASD and SEASD-nf is satisfactory over a wide range of parameters, and also provide significant insight into the dynamics of polydisperse colloidal suspensions.
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