We present a stable spectral vanishing viscosity for discontinuous Galerkin schemes, with applications to turbulent and supersonic flows. The idea behind the SVV is to spatially filter the dissipative fluxes, such that it concentrates in higher wavenumbers, where the flow is typically under-resolved, leaving low wavenumbers dissipation-free. Moreover, we derive a stable approximation of the Guermond-Popov fluxes with the Bassi-Rebay 1 scheme, used to introduce density regularization in shock capturing simulations. This filtering uses a Cholesky decomposition of the fluxes that ensures the entropy stability of the scheme, which also includes a stable approximation of boundary conditions for adiabatic walls. For turbulent flows, we test the method with the three-dimensional Taylor-Green vortex and show that energy is correctly dissipated, and the scheme is stable when a kinetic energy preserving split-form is used in combination with a low dissipation Riemann solver. Finally, we test the shock capturing capabilities of our method with the Shu-Osher and the supersonic forward facing step cases, obtaining good results without spurious oscillations even with coarse meshes.

中文翻译：

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不连续伽辽金方案的熵稳定谱消失粘度：应用于冲击捕获和 LES 模型

我们为不连续伽辽金方案提供了稳定的谱消失粘度，适用于湍流和超音速流动。SVV 背后的想法是在空间上过滤耗散通量，使其集中在较高波数中，其中流动通常未得到充分解析，从而使低波数无耗散。此外，我们使用 Bassi-Rebay 1 方案推导出 Guermond-Popov 通量的稳定近似值，用于在冲击捕获模拟中引入密度正则化。此过滤使用通量的 Cholesky 分解，以确保方案的熵稳定性，其中还包括绝热壁边界条件的稳定近似。对于湍流，我们使用三维 Taylor-Green 涡旋测试该方法并表明能量正确耗散，当保持动能分裂形式与低耗散黎曼求解器结合使用时，该方案是稳定的。最后，我们使用 Shu-Osher 和超音速前向台阶情况测试了我们方法的冲击捕获能力，即使使用粗网格也能获得良好的结果而没有虚假振荡。