We present a Stochastic Galerkin (SG) scheme for Uncertainty Quantification (UQ) of the compressible Navier-Stokes and Euler equations. For spatial discretization, we rely on the high-order Discontinuous Galerkin Spectral Element Method (DGSEM) in combination with an explicit time-stepping scheme. We simulate complex flow problems in two- and three-dimensional domains using curved unstructured hexahedral meshes. The dimension of the stochastic space can be arbitrarily large. In order to treat discontinuities and to ensure hyperbolicity, we employ a multi-element approach in combination with a hyperbolicity-preserving limiter in the stochastic domain and a Finite Volume subcell shock capturing scheme in physical space. Special emphasis is put on code performance and massive parallel scalability. We demonstrate the versatility and broad applicability of our code with various numerical experiments including the supersonic flow around a spacecraft geometry. By this, we wish to contribute to the path of the Stochastic Galerkin method towards practical applicability in research and industrial engineering.

我们提出了可压缩 Navier-Stokes 和 Euler 方程的不确定性量化 (UQ) 的随机伽辽金 (SG) 方案。对于空间离散化，我们依靠高阶不连续伽辽金谱元方法 (DGSEM) 与显式时间步进方案相结合。我们使用弯曲的非结构化六面体网格模拟二维和三维域中的复杂流动问题。随机空间的维度可以任意大。为了处理不连续性并确保双曲线性，我们采用多元素方法结合随机域中的双曲线性限制器和物理空间中的有限体积子单元激波捕获方案。特别强调代码性能和大规模并行可扩展性。我们通过各种数值实验证明了我们的代码的多功能性和广泛的适用性，包括围绕航天器几何形状的超音速流动。通过这个，我们希望为随机伽辽金方法在研究和工业工程中的实际适用性做出贡献。