A high-order discontinuous Galerkin pressure robust splitting scheme for incompressible flows

The accurate numerical simulation of high Reynolds number incompressible flows is a challenging topic in computational fluid dynamics. Classical inf-sup stable methods like the Taylor-Hood element or only $L^{2$-conforming} discontinuous Galerkin (DG) methods relax the divergence constraint in the variational formulation. However, unlike divergence-free methods, this relaxation leads to a pressure-dependent contribution in the velocity error which is proportional to the inverse of the viscosity, thus resulting in methods that lack pressure robustness and have difficulties in preserving structures at high Reynolds numbers. The present paper addresses the discretization of the incompressible Navier-Stokes equations with high-order DG methods in the framework of projection methods. The major focus in this article is threefold: i) We present a novel postprocessing technique in the projection step of the splitting scheme that reconstructs the Helmholtz flux in $H(\text{div})$. In contrast to the previously introduced $H(\text{div})$ postprocessing technique, the resulting velocity field is pointwise divergence-free in addition to satisfying the discrete continuity equation. ii) Based on this Helmholtz flux $H(\text{div})$ reconstruction, we obtain a high order in space, pressure robust splitting scheme as numerical experiments in this paper demonstrate. iii) With this pressure robust splitting scheme, we demonstrate that a robust DG method for underresolved turbulent incompressible flows can be realized.