In this manuscript we propose and test a strategy 'Brownian dynamics and distribution function storing' (BDDFS) for performing numerical calculations of viscoelastic complex flows based on the unapproximated CONNFFESSIT type approach. Hardware limits this established approach for the 'calculation of non-newtonian flows using finite elements and stochastic simulation techniques' for highly complex flows due to fluctuations which come together with the stochastic determination of the macroscopic extra stress tensor. As soon as the number of cells in the flow domain becomes large an even much larger number of freedom degrees must be used to extract accurate results. Usually, variance reduction techniques are used to suppress noise, lower the memory requirements, produce correlated dynamics, and obtain approximate, and 'good' results. BDDFS is a numerical method for the still approximate, but 'uncorrelated' solution of the same problem with limited memory needs. It relies on a discrete storage of the configurational distribution function (D-CDF) for dumbbells, or polymers. Configurational variables subject to standard BD are sampled consistently with the D-CDF. Compared with the original approach, the memory requirement is reduced by the ratio between the number of D-CDF grid points and the number of molecules. The hybrid method has similarities with both spectral methods and BD. but remains conceptually simple and computationally feasible also for short chains. The strategy has yet been tested against a homogeneous shear flow of dumbbells where the advantages should be reduced to a minimum. Results reveal that the BDDFS concept may offer advantages upon alternative approaches, which must become larger with the complexity of the system under study and, whenever molecular correlations on length scales larger than the grid size contain information relevant to interpret experiments. (C) 2004 Elsevier B.V. All rights reserved.