  In this paper, we propose a new set of midpoint-based high-order
discretization schemes for computing straight and mixed nonlinear second
derivative terms that appear in the compressible Navier-Stokes equations.
Firstly, we detail a set of conventional fourth and sixth-order baseline
schemes that utilize central midpoint derivatives for the calculation of second
derivatives terms. To enhance the spectral properties of the baseline schemes,
an optimization procedure is proposed that adjusts the order and truncation
error of the midpoint derivative approximation while still constraining the
same overall stencil width and scheme order. A new filter penalty term is
introduced into the midpoint derivative calculation to help achieve high
wavenumber accuracy and high-frequency damping in the mixed derivative
discretization. Fourier analysis performed on the both straight and mixed
second derivative terms show high spectral efficiency and minimal numerical
viscosity with no odd-even decoupling effect. Numerical validation of the
resulting optimized schemes is performed through various benchmark test cases
assessing their theoretical order of accuracy and solution resolution. The
results highlight that the present optimized schemes efficiently utilize the
inherent viscosity of the governing equations to achieve improved simulation
stability - a feature attributed to their superior spectral resolution in the
high wavenumber range. The method is also tested and applied to non-uniform
structured meshes in curvilinear coordinates, employing a supersonic impinging
jet test case.
