  We present an improved high-order weighted compact high resolution (WCHR)
scheme that extends the idea of weighted compact nonlinear schemes (WCNS's)
using nonlinear interpolations in conjunction with compact finite difference
schemes for shock-capturing in compressible turbulent flows. The proposed
scheme has better resolution property than previous WCNS's. This is achieved by
using a compact (or spatially implicit) form instead of the traditional fully
explicit form for the nonlinear interpolation. Since compact interpolation
schemes tend to have lower dispersion errors compared to explicit interpolation
schemes, the proposed scheme has the ability to resolve more fine-scale
features while still having the ability to provide sufficiently localized
dissipation to capture shocks and discontinuities robustly. Approximate
dispersion relation characteristics of this scheme are analyzed to show the
superior resolution properties of the scheme compared to other WCNS's of
similar orders of accuracy. Conservative and high-order accurate boundary
schemes are also proposed for non-periodic problems. Further, a new
conservative flux-difference form for compact finite difference schemes is
derived and allows for the use of positivity-preserving limiters for improved
robustness. Different test cases demonstrate the ability of this scheme to
capture discontinuities in a robust and stable manner while also localizing the
required numerical dissipation only to regions containing discontinuities and
very high wavenumber features and hence preserving smooth flow features better
in comparison to WCNS's.
