Oscillation Mitigation of Hyperbolicity-Preserving Intrusive Uncertainty Quantification Methods for Systems of Conservation Laws
Oscillation Mitigation of Hyperbolicity-Preserving Intrusive Uncertainty Quantification Methods for Systems of Conservation Laws
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DOI：
10.1016/j.cam.2021.113714
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发表时间：
2020-08
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影响因子：
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通讯作者：
J. Kusch;L. Schlachter
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文献类型：
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作者：
J. Kusch;L. Schlachter
本文研究了具有不确定性的守恒律系统的侵入式不确定性量化方案。虽然侵入式方法继承了某些优势，如适应性和改进的准确性，但它们受到两个关键问题的影响。首先，侵入性方法倾向于表现出振荡，尤其是在激波结构上；其次，标准侵入性方法可能会失去双曲性。这项工作的目的是通过两种不同的战略来应对这些挑战。首先，我们将滤波器与多元方法相结合来求解双曲性保持随机Galerkin(HSG)格式。虽然HSG格式中使用的限制器确保了双曲性，但滤波器以及多元素ANSAZ减少了振荡。其次，我们给出了一种求解多项式矩(IPM)方法的多元方法。尽管IPM方法本质上是双曲线的，但它在每个空间单元和每个时间步长都需要求解优化问题时，会受到振荡的影响。所提出的多单元IPM方法使得优化问题在每个多单元中解耦。因此，我们能够显著降低计算成本，同时提高并行化能力。这两种策略都被扩展到自适应，允许在每个多元素中调整基函数以适应解的光滑性。最后，我们在NACA翼型和喷管试验算例中对这两种方法进行了评估和比较。在我们的数值实验中，我们观察到伪影的减轻。此外，将多元素ANSAZ用于IPM显著降低了计算成本。
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.