EsCUT: Entropy-stable high-order CUT-cell discontinuous Galerkin methods

该项目的目的是开发新的稳定技术，以获得切割网格上非线性双曲守恒定律的鲁棒和有效的熵稳定数值方法，用于复杂几何形状中欠分解流动的苛刻模拟。这些新的高分辨率，结构保持方法将被设计用于新的解概念，如耗散弱解。非线性双曲平衡律在飞机设计和环境/气候研究等许多重要应用中起着至关重要的作用。这些应用的高分辨率数值模拟方法面临的两个主要挑战是复杂的几何形状和湍流中必然存在的低分辨率特征。为了解决这些问题，需要两种类型的稳定：(i)切割单元稳定，以应对小切割单元的时间步长限制；（ii）基线方案的稳定，以应对未分解的流动和湍流。为了避免过度耗散的格式，并为更好地理解复杂几何中的湍流铺平道路，我们将为不连续伽辽金方法开发具有鲁棒切割单元稳定性的新型熵稳定高分辨率格式。基本的想法是在第一步中重新制定切割细胞稳定性，使它们可以通过熵分析来接近。在此基础上，我们将详细分析稳定特性，并开发保证熵稳定特性的新方法。该项目的最终成果是切细胞网格上的熵稳定DG方法，该方法可以在合理的时间步长约束下与显式时间积分方法有效地用于多维可压缩欧拉方程。在即将到来的项目扩展中，我们计划将这些新算法扩展到可压缩的Navier-Stokes方程，并将其应用于复杂几何形状湍流行为的模拟，特别是在高雷诺数区域。

The aim of this project is the development of new stabilisation techniques to obtain robust and efficient entropy-stable numerical methods for non-linear hyperbolic conservation laws on cut-cell meshes for demanding simulations of under-resolved flows in complex geometries. These new high-resolution, structure-preserving methods will be designed for novel solution concepts such as dissipative weak solutions. Nonlinear hyperbolic balance laws play a crucial role in many important applications such as aircraft design and environmental/climate research. Two major challenges for high-resolution numerical simulation methods for these applications are complex geometries and under-resolved features, necessarily present in turbulent flows. To solve these issues, two types of stabilisation are required: (i) cut-cell stabilisations to cope with time step restrictions of small cut cells (ii) stabilisation of the baseline scheme for under-resolved flows and turbulence. To avoid overly dissipative schemes and pave the way to a better understanding of turbulent flows in complex geometries, we will develop novel entropy-stable high-resolution schemes with robust cut-cell stabilisations for discontinuous Galerkin methods. The basic idea is to reformulate cut-cell stabilisations in a first step to make them approachable by an entropy analysis. Based thereon, we will analyse the stability properties in detail and develop novel approaches guaranteeing entropy stability properties. The final deliverable of this project are entropy-stable DG methods on cut-cell meshes that can be used efficiently with explicit time integration methods under reasonable time step constraints for the multi-dimensional compressible Euler equations. In an upcoming extension of this project, we plan to extend these novel algorithms to the compressible Navier-Stokes equations and apply them to the simulation of turbulent behaviour in complex geometries, in particular in the high Reynolds number regime.