Koordinationsantrag

非线性双曲平衡定律在流体力学过程的建模中是普遍存在的。它们能够开发强大的数值模拟方法，支持关键应用的决策制定，例如计算机空气和航天器设计或气候变化研究。然而，关于独特的双曲线特征的基本问题仍然是开放的，包括冲击波和剪切波的多尺度干扰，或双曲线运输和随机环境的相互作用。多维无粘流方程的适定性问题在很大程度上尚未解决，这与高雷诺数极限下的湍流运动规律有着密切的联系。进一步的进展需要流体力学和分析、数值和随机数学领域的共同努力。优先方案致力于开发新的数学模型和方法，以了解小尺度和机制的动态创建，这些小尺度和机制被双曲非线性增强或耗尽。它努力在一个新的分析和数值模式的双曲线运输，可以提供坚实的基础，即将到来的理论小尺度湍流在大雷诺数限制。优先计划将主要围绕三个主要研究方向发展：新的解决方案概念：这包括对流体力学中出现的双曲系统的分析（通过例如广义熵方法，耗散极限或概率和基于矩的解决方案），为这些解决方案概念设计高分辨率数值，并探索与现代统计湍流建模和扰动/过滤技术的联系。 多尺度模型和渐近状态：研究包括模型层次（例如Boltzmann-Euler或统计湍流）的开发和分析，以及解释渐近流态（例如极端马赫数）的闭合。熵和结构保持的数值方法需要设计，允许保存的渐近状态，而通过层次结构和政权的错误控制模型选择遍历。 概率模型：该领域包括流体力学中双曲系统随机模型的分析、数值和不确定性量化。它包括概率建模的概念，探索统计湍流，例如随机变分原理和随机/数据驱动的工具混合扰动/过滤技术的探索。不确定性量化的数值方法应考虑到基本模型的双曲线特征的保留。

Nonlinear hyperbolic balance laws are ubiquitous in the modelling of fluidmechanical processes. They enable the development of powerful numerical simulation methods that back decision-making for critical applications such as in-silico air- and spacecraft design or climate change research. However, fundamental questions about distinctive hyperbolic features remain open including the multi-scale interference of shock and shear waves, or the interplay of hyperbolic transport and random environments. The largely unsolved well-posedness problem for multi-dimensional inviscid flow equations is deeply connected to the laws of turbulent fluid motion in the high Reynolds-number limit. Further progress requires a concerted effort of both fluid mechanics and the mathematical fields of analysis, numerics, and stochastics. The Priority Programme is devoted to the development of new mathematical models and methods to understand the dynamic creation of small scales and mechanisms which are either enhanced or depleted by the hyperbolic nonlinearity. It strives at a novel analytical and numerical paradigm for hyperbolic transport that can provide firm grounds for the upcoming theory of small-scale turbulence in the large Reynolds number limit. The Priority Programme will mostly evolve around three major research directions: Novel solution concepts: This includes the analysis for hyperbolic systems arising in fluid mechanics (via e.g. generalized entropy methods, dissipative limits or probabilistic and moment-based solutions), the design of high-resolution numerics for these solution concepts, and exploring the connections to modern statistical turbulence modelling and perturbation/filtering techniques. Multi-scale models and asymptotic regimes: Research includes the development and analysis of model hierarchies (e.g. Boltzmann-Euler or in statistical turbulence) and their closures that account for asymptotic flow regimes (e.g. extreme Mach numbers). Entropy- and structure-preserving numerical methods need to be designed that allow the preservation of asymptotic states while traversing through hierarchies and regimes by error-controlled model selection. Probabilistic models: This area comprises the analysis, numerics and uncertainty quantification for stochastic models of hyperbolic systems arising in fluid mechanics. It includes probabilistic modelling concepts to explore statistical turbulence using e.g. stochastic variational principles and the exploration of stochastic/data-driven tools for hybrid perturbation/filtering techniques. Numerical methods of uncertainty quantification should account for the preservation of hyperbolic features of the underlying model.