Random compressible Euler equations: Numerics and its Analysis

随机可压缩欧拉方程：数值及其分析

• 批准号: 525853336
• 负责人: Professor Dr. Michael Herty
• 金额: --
• 依托单位: Institute of Mathematics
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/525853336?language=en
• 关键词: Random; compressible; Euler; equations; Numerics

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.

科学和工程中出现的数学模型继承了几种不确定性来源，例如模型参数，初始和边界条件。为了预测可靠的结果，确定性模型不足，需要更复杂的方法来分析不确定性对数值解决方案的影响。然而，椭圆方程和抛物线方程的数值方案的分析结果和相应设计的进展尚未完全扩展到双曲线问题。非线性转运构成了主要障碍，这会导致解决方案的规律性丧失，该解决方案在随机空间中也传播，并且在侵入性的Galerkin方法中可能丧失双曲线。 EULER方程的不确定性定量与统计流体动力学本质上连接。为了描述湍流行为，考虑可压缩流体流量模型的随机或统计解的想法是自然的。我们在该建议中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。拟议的项目旨在实现三个目标：首先，我们介绍并分析了一个新的合奏平均解决方案（随机耗散解决方案）的新概念。它们的存在将通过基于多项式混沌扩展的合适不确定性定量方法的收敛来证明。为此，我们使用固有的随机紧凑性参数。为了量化数值近似误差，将得出随机相对能量不等式。应用设定值的紧凑型框架，k连接，我们近似湍流的雷诺应力和能量耗散。其次，使用随机耗散溶液的制定，我们提出并分析了新型的数值方案。在这里，力矩近似将用于得出有效的方程，以实现统计矩的演变。第三，我们通过渐近保留数值方案研究了随机弱压缩欧拉方程的低马赫数极限。因此，我们解决了双曲线问题中的多尺度现象，研究随机性和双曲线运输之间的微妙相互作用，并促进了SPP 2410的总体目标，以设计熵稳定且具有结构的数值方案。