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

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.