The static and dynamic stability of Rayleigh-Bénard convection in a rectangular flow domain is computationally investigated. Sinusoidal vertical oscillations are applied to the system to provide dynamic flow stabilization. Stability maps are produced for a range of flow and heating conditions, and are compared to experimental measurements and linear stability analysis predictions from existing literature. Density variation is introduced through: 1) the Boussinesq approximation, 2) a linearly varying temperature dependent equation of state (EOS) and 3) the perfect gas EOS. Significant effects of choice of EOS on dynamic stability are observed. These weakly compressible flows are solved efficiently using an implicit numerical method that has been developed to solve the momentum, continuity, enthalpy and state equations simultaneously in fully coupled fashion. This block coupled system of equations is linearized with Newton’s method, and quadratic convergence is achieved. The details of these numerics are presented.