A simple rigid plasma model is used to show that axisymmetric plasma instabilities (in two dimensions) will occur on a resistive time scale and do not depend on the plasma mass. This is the justification for ignoring the inertial term in two-dimensional studies of plasma shape control and vertical stability. In three dimensions, it is not normally possible to ignore the inertial terms when computing plasma instabilities. This results in a stiff system of equations (with multiple time scales) in which the driving terms causing plasma instabilities are small compared with the stable compressive terms. Techniques are described for implicit time integration and for representing the vector fields in a way to facilitate obtaining accurate solutions for plasma instabilities when a strong background magnetic field is present.