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In this paper, we propose a new set of midpoint-based high-orderdiscretization schemes for computing straight and mixed nonlinear secondderivative terms that appear in the compressible Navier-Stokes equations.Firstly, we detail a set of conventional fourth and sixth-order baselineschemes that utilize central midpoint derivatives for the calculation of secondderivatives terms. To enhance the spectral properties of the baseline schemes,an optimization procedure is proposed that adjusts the order and truncationerror of the midpoint derivative approximation while still constraining thesame overall stencil width and scheme order. A new filter penalty term isintroduced into the midpoint derivative calculation to help achieve highwavenumber accuracy and high-frequency damping in the mixed derivativediscretization. Fourier analysis performed on the both straight and mixedsecond derivative terms show high spectral efficiency and minimal numericalviscosity with no odd-even decoupling effect. Numerical validation of theresulting optimized schemes is performed through various benchmark test casesassessing their theoretical order of accuracy and solution resolution. Theresults highlight that the present optimized schemes efficiently utilize theinherent viscosity of the governing equations to achieve improved simulationstability - a feature attributed to their superior spectral resolution in thehigh wavenumber range. The method is also tested and applied to non-uniformstructured meshes in curvilinear coordinates, employing a supersonic impingingjet test case.

Low Dispersion Optimized High-Order Schemes for Discretization of Non-Linear Straight and Mixed Second Derivative Terms