Large-Eddy Simulations (LES) of supercritical cold flow in a swirl injector with axi-symmetric configuration, and supercritical flames for LOX/methane stabilized after a splitter plate have been performed with various grid resolution. LES is used for turbulence closure. Three to four level of grid resolution is used to examine the grid sensitivity in both cold and reaction flows at supercritical conditions. One level of grid resolution corresponds to a 50% grid size decrease for all directions. In addition, the effect of ideal-gas and real-gas thermodynamic and transport property evaluations, sub-grid scale (SGS) models, and turbulent combustion models are also studied under reacting conditions. The results identify that the most important factor for accurate solutions is the grid resolution with the current numerical methodology. Turbulent combustion models are the second most important modeling factor. The effect of SGS models and property evaluation is much less important. To make high-fidelity modeling of supercritical mixing and combustion less expensive, so that they can be widely used in solving practical problems associated with supercritical mixing and combusting, appropriate models are needed to alleviate the grid resolution requirement.