Two transonic computational schemes which are based on the Integral Equation Formulation of the full potential equation were presented. The first scheme is a Shock Capturing-Shock Fitting (SCSF) scheme which uses the full potential equation throughout with the exception of the shock wave where the Rankine-Hugoniot relations are used to cross and fit the shock. The second scheme is an Integral Equation with Embedded Euler (IEEE) scheme which uses the full potential equation with an embedded region where the Euler equations are used. The two schemes are applied to several transonic airfoil flows and the results were compared with numerous computational results and experimental domains with fine grids. The SCSF-scheme is restricted to flows with weak shock, while the IEEE-scheme can handle strong shocks. Currently, the IEEE scheme is applied to other transonic flows with strong shocks as well as to unsteady pitching oscillations.
