Knowledge: Of particular interests within the NASA’s EDL activities are steady and unsteady high-speed flows over space vehicles. The environment of these vehicles generally contain high energetic discontinuities (e.g., shocks and rapid expansion), which lead to spurious numerical oscillations. State-of-the-art: Use of eigenvalue, flux, or a combination of the two limiters that are of first-order and likely to affect the formal order of accuracy of the scheme in the smooth region, possibly making the predicted solution less accurate. Proposal: To construct a compact high-order discontinuity capturing scheme for DG that is also convergent for energetic discontinuities. End Goal: 1) A prerequisite to ultimately produce high-order solution and solution gradients for Navier-Stokes using purely unstructured grids; 2) Greatly exceeds current capabilities by offering high-order compact schemes that are oscillation-free and positive for unstructured grids.