The use of a non-staggered computational grid for the numerical solutions of the incompressible flow equations has many advantages over the use of a staggered grid. A penalty, however, is inherent in the finite-difference approximations of the governing equations on non-staggered grids. In the primitive-variable solutions, the penalty is that the discrete continuity equation does not converge to machine accuracy. Rather it converges to a source term which is proportional to the fourth-order derivative of the pressure, the time increment, and the square of the grid spacing. An approach which minimizes the error in the discrete continuity equation is developed. Numerical results obtained for the driven cavity problem confirm the analytical developments.