The inertial confinement approach to controlled fusion requires that small thin-walled spherical shells of fuel and other materials be imploded, compressed, and heated by laser or charged particle beams. In most cases of interest, the implosion of such thin shells is unstable to the growth of spherical asymmetries. We have developed and used two numerical simulation techniques to study these instabilities. The first technique is used to study the small amplitude growth of the instabilities by employing a perturbation method. The derivation of the Hamiltonian model on which the technique is based is developed here. The second technique is a fully nonlinear two-dimensional hydrodynamics and heat flow technique that we have used to follow the large-amplitude development and saturation of the instabilities. The examples of calculations shown demonstrate the utility of the method and the range of different saturation phenomena that may be expected.