Summary: A method of solving the time accurate Navier-Stokes equations for three- dimensional incompressible flow or low Mach number flows has been developed and applied to the problem of the rotational spin-up of a liquid in cylinders. The method is of the operator splitting type and makes use of a direct banded solver to simultaneously correct both the velocity and pressure fields with the constraint of the continuity equation. The time splitting forms a consistent and time accurate solution of the Navier-Stokes equations in a finite volume form, and it is also consistent with the Poisson equation for the pressure field at the forward time level. The direct solver method for the Poisson equation has been used in a planar manner, and this has been combined with line iteration for the other coordinate direction. A major advantage of the present technique is that the LU decomposition of the resulting banded matrix has only to be formed once in a given plane, and then saved for future use. When combined with a time accurate predictor/corrector method for the momentum equation, the present solution technique represents an efficient method of solving both the time dependent and the steady three- dimensional Navier-Stokes equations.