A dual time-stepping method, combined with artificial compressibility, was developed for the solution of the 3-D Navier-Stokes equations of incompressible fluids. These concept of artificial compressibility allows the adoption of numerical techniques originally developed for compressible flows. Thus, an incompressible flow solver is described, in which upwind discretizations, implicit relaxation schemes with flux-matrix splitting and a Runge-Kutta time-stepping scheme with multigrid are used. The computation of a bursting vortex, called vortex breakdown, is presented as one important application of the implicit version of the present method of solution, which of course is not restricted to vortical flows. © 1993.