Numerical methods which can be used to calculate flows in a rapidly rotating fluid are presented. The flows considered are those within a closed cylindrical container rotating about an axis aligned with the sidewalls, with a slow motion superposed upon the rigid rotation of the fluid. The Rossby number in this case is small; the Eckman number is also considered to be small, so that viscous diffusion is confined to the thin layers of the fluid. The flows calculated are those in the geostrophic region of the fluid, where the motion is depth-independent and the governing equations are two-dimensional. Between the geostatic flow and the walls of the container, however, there are ageostrophic layers of two types. By analyzing these layers theoretically, appropriate boundary conditions on the geostrophic flow can be calculated.