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

该项目的目的是开发新的稳定化技术，以获得强大的和有效的熵稳定的数值方法，非线性双曲守恒律的切割细胞网格要求模拟欠解析流在复杂的几何形状。这些新的高分辨率，结构保持方法将被设计用于新的解决方案的概念，如耗散弱解。非线性双曲平衡律在许多重要的应用中起着至关重要的作用，如飞机设计和环境/气候研究。这些应用的高分辨率数值模拟方法的两个主要挑战是复杂的几何形状和分辨率不足的功能，必然存在于湍流。为了解决这些问题，需要两种类型的稳定化：（i）切割单元稳定化，以科普小切割单元的时间步长限制（ii）稳定化的基线计划，为欠分辨率的流动和湍流。为了避免过度耗散的计划，并铺平道路，以更好地了解湍流在复杂的几何形状，我们将开发新的熵稳定的高分辨率计划与强大的切割细胞stabilisations的不连续Galerkin方法。其基本思想是在第一步重新制定削减细胞stabilisations，使他们接近的熵分析。在此基础上，我们将详细分析稳定性，并开发新的方法，保证熵稳定性。该项目的最终成果是切割网格上的熵稳定DG方法，该方法可以在合理的时间步长约束下有效地与显式时间积分方法一起用于多维可压缩欧拉方程。在即将到来的扩展这个项目中，我们计划将这些新的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的湍流行为的模拟，特别是在高雷诺数制度。

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.