Random compressible Euler equations: Numerics and its Analysis

在科学和工程中出现的数学模型继承了几个不确定性来源，如模型参数、初始条件和边界条件。为了预测可靠的结果，确定性模型是不够的，需要更复杂的方法来分析不确定性对数值解的影响。然而，椭圆型和抛物型方程的解析结果和相应的数值格式设计方面的进展尚未完全扩展到双曲型问题。一个主要的障碍是非线性输运，它会导致在随机空间中传播的解的正则性丧失，并且在侵入伽辽金方法中可能会失去双曲性。欧拉方程的不确定性量化与统计流体力学有着内在的联系。考虑可压缩流体流动模型的随机或统计解的想法是很自然的，以便描述湍流的流体行为。我们在这一建议的目的是从理论和数值的角度加深对可压缩欧拉方程的随机/统计解的理解。本研究主要有三个目标：首先，我们引入并分析了一个新的集合平均解的概念，即随机耗散解。在多项式混沌展开的基础上，通过适当的不确定性量化方法的收敛性证明了它们的存在性。为此，我们使用固有的随机紧性参数。为了量化数值近似的误差，将推导出随机相对能量不等式。应用集值紧性框架，k收敛，我们近似紊流雷诺应力和能量耗散。其次，利用随机耗散解的公式，提出并分析了新的数值格式。在这里，将使用矩近似来推导统计矩演化的有效方程。第三，利用渐近保持数值格式研究了随机弱可压缩欧拉方程的低马赫数极限。因此，我们解决了双曲问题中的多尺度现象，研究了随机性和双曲输运之间的微妙相互作用，并为SPP 2410的总体目标做出了贡献，即设计熵稳定和结构保持的数值方案。

Mathematical models arising in science and engineering inherit several sources of uncertainties, such as model parameters, initial and boundary conditions. In order to predict reliable results, deterministic models are insufficient and more sophisticated methods are needed to analyse the influence of uncertainties on numerical solutions. Progress in analytical results and corresponding design of numerical schemes for elliptic and parabolic equations, have, however, not yet been fully expanded towards the hyperbolic problems. A main obstacle is posed by the nonlinear transport that causes the loss of regularity of a solution that propagates also in the random space and the possible loss of hyperbolicity in intrusive Galerkin methods. Uncertainty quantification of the Euler equations is intrinsically connected to statistical hydrodynamics. The idea of considering random or statistical solutions of compressible fluid flow models is natural in order to describe turbulent fluid behaviour. Our aim in this proposal is to deepen the understanding of random/statistical solutions of compressible Euler equations both from the theoretical as well as numerical point of view. The proposed project aims to achieve three goals: Firstly, we introduce and analyse a new concept of ensemble-averaged solutions, the random dissipative solutions. Their existence will be proved via convergence of suitable uncertainty quantification methods, based on polynomial chaos expansion. To this end, we work with inherently stochastic compactness arguments. In order to quantify errors of numerical approximations the random relative energy inequality will be derived. Applying a set-valued compactness framework, K-convergence, we approximate turbulent Reynolds stress and energy dissipation. Secondly, using the formulation of random dissipative solutions we propose and analyse novel numerical schemes. Here, moment approximations will be used to derive effective equations for the evolution of statistical moments. Thirdly, we study the low Mach number limit of the random weakly-compressible Euler equations by means of asymptotic preserving numerical schemes. Consequently, we address multiscale phenomena in hyperbolic problems, investigate a delicate interplay between randomness and hyperbolic transport and contribute to overarching goals of SPP 2410 to design entropy stable and structure-preserving numerical schemes.