In this article we study intrusive uncertainty quantification schemes for systems of conservation laws with uncertainty. While intrusive methods inherit certain advantages such as adaptivity and an improved accuracy, they suffer from two key issues. First, intrusive methods tend to show oscillations, especially at shock structures and second, standard intrusive methods can lose hyperbolicity. The aim of this work is to tackle these challenges with the help of two different strategies. First, we combine filters with the multi-element approach for the hyperbolicity-preserving stochastic Galerkin (hSG) scheme. While the limiter used in the hSG scheme ensures hyperbolicity, the filter as well as the multi-element ansatz mitigate oscillations. Second, we derive a multi-element approach for the intrusive polynomial moment (IPM) method. Even though the IPM method is inherently hyperbolic, it suffers from oscillations while requiring the solution of an optimization problem in every spatial cell and every time step. The proposed multi-element IPM method leads to a decoupling of the optimization problem in every multi-element. Thus, we are able to significantly decrease computational costs while improving parallelizability. Both proposed strategies are extended to adaptivity, allowing to adapt the number of basis functions in each multi-element to the smoothness of the solution. We finally evaluate and compare both approaches on various numerical examples such as a NACA airfoil and a nozzle test case for the two-dimensional Euler equations. In our numerical experiments, we observe the mitigation of spurious artifacts. Furthermore, using the multi-element ansatz for IPM significantly reduces computational costs.

在本文中，我们研究具有不确定性的守恒定律系统的侵入性不确定性量化方案。虽然侵入式方法继承了某些优点，例如适应性和提高的准确性，但它们存在两个关键问题。首先，侵入式方法倾向于显示振荡，尤其是在冲击结构中，其次，标准侵入式方法可能会失去双曲线性。这项工作的目的是在两种不同策略的帮助下应对这些挑战。首先，我们将滤波器与多元素方法相结合，用于双曲线保留随机伽辽金 (hSG) 方案。虽然 hSG 方案中使用的限制器确保了双曲线性，但滤波器以及多元素 ansatz 可以减轻振荡。其次，我们为侵入式多项式矩 (IPM) 方法推导了一种多元素方法。尽管 IPM 方法本质上是双曲线的，但它会受到振荡的影响，同时需要在每个空间单元和每个时间步中解决优化问题。所提出的多元素 IPM 方法导致每个多元素中优化问题的解耦。因此，我们能够在提高并行性的同时显着降低计算成本。所提出的两种策略都扩展到自适应性，允许使每个多元素中的基函数数量适应解决方案的平滑度。我们最终在各种数值示例（例如 NACA 翼型和用于二维欧拉方程的喷嘴测试案例）上评估和比较这两种方法。在我们的数值实验中，我们观察到虚假伪像的减轻。此外，