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：本文提出了一类基于中点的高阶离散格式，用于计算可压缩Navier-Stokes方程中的非线性和混合二阶导数项。首先，详细阐述了一组传统的四阶和六阶基准格式，这些格式利用中心中点导数计算二阶导数项。为改善基准格式的谱特性，提出了一种优化方法，在保持相同模板宽度和格式阶数的条件下，调整中点导数近似的阶数和截断误差。在中点导数计算中引入新的滤波惩罚项，以帮助在混合导数离散中实现高波数精度和高频阻尼。对直项和混合二阶导数项的傅里叶分析表明，该格式具有高谱效率、极小数值粘性且无奇偶解耦效应。通过多个基准算例对所得到的优化格式进行数值验证，评估其理论精度阶数和解的分辨率。结果表明，当前优化格式有效利用了控制方程固有的粘性特性以实现改进的模拟稳定性——这一特性归因于其在波数范围内的优异谱分辨率。该方法还通过超声速冲击射流算例，在曲线坐标系下的非均匀结构网格上进行了测试与应用。