A computational scheme has been constructed for solving the equations that describe strong thermal convection in a two-dimensional gas layer that is heated from below and is stratified across many scaleheights by a uniform gravitational field. The purpose of this scheme is to mimic the physical conditions that may have existed in a section of the proto-solar cloud from which the planetary system formed. The vertical temperature gradient of the initial quiescent layer of diatomic gas is strongly superadiabatic and matches that of a polytrope of index m=1. The temperature at the upper boundary is kept fixed during the computation. Because of the highly compressible nature of the gas and the steep spatial gradients, a modified version of a flux-corrected transport scheme due to Zalesak is devised. The computations show that after the convection adopts a steady-state configuration, the flow consists of horizontal pairs of giant convective cells of opposing circulation. At the cell boundaries, the downflows are rapid and spatially concentrated while the upflows are broad and sluggish. Supersonic speeds are easily achieved in the downflows. Contrary to the expectations of the mixing length theory of convection, there is a net downward flux of kinetic energy at each level in the layer. The convecting layer is cooler on average compared with the initial temperature profile, and there is a net shift of mass towards the lower boundary. The implications of these results for the modern Laplacian theory of Solar system origin are briefly discussed.