In this work, we present a positivity-preserving entropy-based adaptive filtering method for shock capturing in discontinuous spectral element methods. By adapting the filter strength to enforce positivity and a local discrete minimum entropy principle, the resulting approach can robustly resolve strong discontinuities with sub-element resolution, does not require problem-dependent parameter tuning, and can be easily implemented on general unstructured meshes with relatively low computational cost. The efficacy of the approach is shown in numerical experiments on hyperbolic and mixed hyperbolic-parabolic conservation laws such as the Euler and Navier-Stokes equations for problems including extreme shocks, shock-vortex interactions, and complex compressible turbulent flows.

在这项工作中，我们提出了一种基于正性保持熵的自适应滤波方法，用于在不连续谱元方法中捕获冲击波。通过调整滤波器强度以强制执行正性和局部离散最小熵原理，所得到的方法可以稳健地解决具有子元素分辨率的强不连续性，不需要与问题相关的参数调整，并且可以很容易地在具有相对计算成本低。该方法的有效性在双曲线和混合双曲线-抛物线守恒定律的数值实验中得到证明，例如欧拉和纳维-斯托克斯方程，用于解决极端冲击、冲击-涡流相互作用和复杂可压缩湍流等问题。