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.

翻译：本文针对可压缩Navier-Stokes方程中出现的直接与混合非线性二阶导数项，提出了一套基于中点的全新高阶离散格式。首先，我们详细阐述了一组传统的四阶与六阶基准格式，这些格式利用中心中点导数计算二阶导数项。为提升基准格式的频谱特性，本文提出了一种优化方法，在保持整体模板宽度与格式阶数不变的前提下，调整中点导数近似的阶数与截断误差。在混合导数离散化中，我们引入了一种新的滤波惩罚项至中点导数计算过程，以实现高波数精度与高频阻尼。对直接与混合二阶导数项进行的傅里叶分析表明，该格式具有高频谱效率与极小的数值耗散，且无奇偶失耦效应。通过多种基准测试案例，我们对所得优化格式进行了数值验证，评估了其理论精度阶数与解的分辨率。结果表明，当前优化格式能有效利用控制方程固有的粘性以提升模拟稳定性——这一特性归因于其在高波数范围内优异的频谱分辨率。该方法还在曲线坐标系下的非均匀结构网格上进行了测试与应用，采用了超声速冲击射流测试案例。