On various treatments of potential equations at shocks

The potential equation that governs transonic inviscid flowfields is elliptic in subsonic regions and hyperbolic in supersonic regions. The transition from supersonic to subsonic flow may occur either continuously or across a surface of discontinuity known as a shock. Because of the changing type of governing equation and the presence of the surfaces of discontinuity, special treatments of the governing equations are needed at shocks to preserve mass conservation and to ensure satisfaction of the governing equation. In the so-called shock-capturing techniques, shocks are captured by smearing the discontinuity over several mesh cells in the computational domain. Most existing conservative schemes conserve mass flux throughout the flowfield, including shock boundaries; however, they also introduce zero-order errors in the approximation to the governing equation. It is shown that the zero-order errors, which do not diminish as the mesh spacing approaches zero, can cause discrepancies in the prediction of shock strength and location. These parameters are extremely sensitive features of transonic inviscid flowfield calculations. It is also shown that the zero-order errors can be avoided by developing higher-order schemes that properly model the governing equation at shocks.