Hardbound. This book size article is dedicated to the numerical simulation of unsteady incompressible viscous flow modelled by the Navier - Stokes equations, or by non-Newtonian variants of them. In order to achieve this goal we have developed a methodology based on the following tools:

(i) Time discretization by operator splitting schemes such as Peaceman - Rachford's, Douglas - Rachford's ,Marchuk - Yanenko's, Strang's symmetrized, and the so-called theta - scheme introduced by the author in the mid - eighties.

(ii) Projection methods (in L2 or H1) for the treatment of the incompressibility condition div u = 0.

(iii) Treatment of the advection by: either a centered scheme leading to linear or nonlinear advection - diffusion problems solved by least squares / conjugate gradient algorithms, or to a linear wave - like equation well suited to finite element based solution methods.

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