The moving discontinuous Galerkin finite element method with interface condition enforcement (MDG-ICE) is applied to the case of viscous flows. This method uses a weak formulation that separately enforces the conservation law, constitutive law, and the corresponding interface conditions in order to provide the means to detect interfaces or under-resolved flow features. To satisfy the resulting overdetermined weak formulation, the discrete domain geometry is introduced as a variable, so that the method implicitly fits a priori unknown interfaces and moves the grid to resolve sharp, but smooth, gradients, achieving a form of anisotropic curvilinear $r$-adaptivity. This approach avoids introducing low-order errors that arise using shock capturing, artificial dissipation, or limiting. The utility of this approach is demonstrated with its application to a series of test problems culminating with the compressible Navier-Stokes solution to a Mach 5 viscous bow shock for a Reynolds number of $10^{5}$ in two-dimensional space. Time accurate solutions of unsteady problems are obtained via a space-time formulation, in which the unsteady problem is formulated as a higher dimensional steady space-time problem. The method is shown to accurately resolve and transport viscous structures without relying on numerical dissipation for stabilization.

中文翻译：

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可压缩粘性流界面条件强制的移动不连续伽辽金有限元方法

具有界面条件强制的移动不连续 Galerkin 有限元方法 (MDG-ICE) 应用于粘性流的情况。该方法使用弱公式，分别强制执行守恒定律、本构律和相应的界面条件，以提供检测界面或未解析流动特征的手段。为了满足由此产生的超定弱公式，将离散域几何作为变量引入，以便该方法隐式拟合先验未知界面并移动网格以解决尖锐但平滑的梯度，实现各向异性曲线 $r$ - 适应性。这种方法避免了引入因使用冲击捕获、人工耗散或限制而产生的低阶误差。这种方法的实用性通过它在一系列测试问题中的应用得到了证明，这些测试问题最终是对二维空间中雷诺数为 $10^{5}$ 的 Mach 5 粘性弓形激波的可压缩 Navier-Stokes 解。非定常问题的时间精确解是通过时空公式获得的，其中非定常问题被公式化为一个更高维的稳定时空问题。该方法被证明可以准确地解析和传输粘性结构，而不依赖于稳定的数值耗散。其中非定常问题被表述为一个更高维的稳定时空问题。该方法被证明可以准确地解析和传输粘性结构，而不依赖于稳定的数值耗散。其中非定常问题被表述为一个更高维的稳定时空问题。该方法被证明可以准确地解析和传输粘性结构，而不依赖于稳定的数值耗散。