A new class of non-linear compact interpolation schemes is introduced in this dissertation that have a high spectral resolution and are non-oscillatory across discontinuities. The Compact-Reconstruction Weighted Essentially Non-Oscillatory (CRWENO) schemes use a solution-dependent combination of lower-order compact schemes to yield a high-order accurate, non-oscillatory scheme. Fifth-order accurate CRWENO schemes are constructed and their numerical properties are analyzed. These schemes have lower absolute errors and higher spectral resolution than the WENO scheme of the same order.

The schemes are applied to scalar conservation laws and the Euler equations of fluid dynamics. The order of convergence and the higher accuracy of the CRWENO schemes are verified for smooth solutions. Significant improvements are observed in the resolution of discontinuities and extrema as well as the preservation of flow features over large convection distances. The computational cost of the CRWENO schemes is assessed and the reduced error in the solution outweighs the additional expense of the implicit scheme, thus resulting in higher numerical efficiency. This conclusion extends to the reconstruction of conserved and primitive variables for the Euler equations, but not to the characteristic-based reconstruction. Further improvements are observed in the accuracy and resolution of the schemes with alternative formulations for the non-linear weights.

The CRWENO schemes are integrated into a structured, finite-volume Navier-Stokes solver and applied to problems of practical relevance. Steady and unsteady flows around airfoils are solved to validate the scheme for curvi-linear grids, as well as overset grids with relative motion. The steady flow around a three-dimensional wing and the unsteady flow around a full-scale rotor are solved. It is observed that though lower-order schemes suffice for the accurate prediction of aerodynamic forces, the CRWENO scheme yields improved resolution of near-blade and wake flow features, including boundary and shear layers, and shed vortices. The high spectral resolution, coupled with the non-oscillatory behavior, indicate their suitability for the direct numerical simulation of compressible turbulent flows. Canonical flow problems -- the decay of isotropic turbulence and the shock-turbulence interaction -- are solved. The CRWENO schemes show an improved resolution of the higher wavenumbers and the small-length-scale flow features that are characteristic of turbulent flows.

Overall, the CRWENO schemes show significant improvements in resolving and preserving flow features over a large range of length scales due to the higher spectral resolution and lower dissipation and dispersion errors, compared to the WENO schemes. Thus, these schemes are a viable alternative for the numerical simulation of compressible, turbulent flows.