High resolution optimized high-order schemes for discretization of non-linear straight and mixed second derivative terms

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.