A filtering operation, based on B-spline discretizations, is introduced to target weakly growing mesh-scale oscillations that can arise in high-fidelity turbulence simulations. This is a spectral regularization that can be described using the singular values of a banded matrix operator, with the filtering strength set by a scalar- or vector-valued penalty parameter. The penalty parameter can be specified though it can also be advantageously selected to minimize the generalized cross validation (GCV) measure of distance between the pre- and post-filtered solutions. Efficient algorithms are developed to compute both the scalar and vector penalty parameters. The B-spline filter has a sharper localization to high-wavenumber than compact or explicit filters of the same stencil width and is demonstrated for solutions of the Burgers' equation, decaying Burgers' turbulence, and compressible Navier–Stokes turbulent channel flow. These simulations confirm the scheme's numerical stability and ability to narrowly target the high wavenumber components of numerical solutions. An advantage over finite-difference filters is that these B-spline filters are stable on bounded domains and even preserve formal order of accuracy.

引入了基于B样条离散化的滤波操作，以针对可能在高保真湍流模拟中出现的微弱增长的网格规模振荡。这是可以使用带状矩阵算子的奇异值描述的频谱正则化，其滤波强度由标量或矢量值的罚分参数设置。惩罚参数可以被指定，尽管也可以被有利地选择以最小化过滤前后溶液之间距离的广义交叉验证（GCV）量度。开发了有效的算法来计算标量和矢量惩罚参数。与相同模板宽度的紧凑型滤波器或显式滤波器相比，B样条滤波器对高波数的定位更加清晰，并已证明可用于Burgers方程的解，降低了Burgers的湍流，以及可压缩的Navier–Stokes湍流通道。这些仿真证实了该方案的数值稳定性以及将数值解的高波数分量作为目标的能力。相对于有限差分滤波器的一个优势在于，这些B样条滤波器在有界域上是稳定的，甚至可以保留形式上的精度顺序。