The interchange instability induced by gravity in a compressible plasma is investigated by numerical methods based on the equations of ideal magnetohydrodynamics. Eulerian computer codes have been developed to integrate in time both the linear and nonlinear forms of these equations. In the linear case the results are compared with a generalization of a normal mode analysis performed by Newcomb. Predictions regarding the growth rate and spatial distribution of these modes and their stabilization by sheared magnetic fields are confirmed in the computer runs. In the nonlinear case inclusion of diffusion and damping terms in the numerical code permits the unstable mode to be followed to a final stable state. The numerical results indicate that for a sheared magnetic field there is a displaced equilibrium state involving a strongly distorted plasma distribution.