Mon, 12 Dec 2022 16:00

online

In this talk, we present the class of split form discontinuous Galerkin methods. The notion of split form refers to different interpretations of the non-linear terms in the fluid dynamics equations. For instance the advective part of the momentum flux can be cast in analytically equivalent forms such as the advective form, or the conservative form, or convex combinations of both forms. While these forms are analytically equivalent for smooth solutions, it is interesting to understand that their discrete forms might have strongly different properties. It turns out that specific underlying split forms of the fluid equations give discontinuous Galerkin (DG) approximations with special favourable properties such as, kinetic energy preservation, energy consistency, pressure equilibrium preservation and even entropy conservation/stability. The most important improvement that can be observed is drastically increased non-linear robustness of the DG discretization, in particular when simulating under-resolved turbulence. A necessary ingredient to retain fully discrete conservation when using split formulations is the so-called summation-by-parts (SBP) property of the discrete derivative and integral operator. It turns out hat specific DG discretizations, such as the Legendre-Gauss-Lobatto (LGL) spectral element method, satisfy the SBP property. We will further discuss in this talk that it is possible to construct compatible low-order finite volume discretizations on the LGL subcell grid, such that a convex blending of the high-order DG method with the robust low-order method is feasible. This allows to construct provably positive hybrid discretizations that enable simulations of problems with strong shock waves, such as e.g. an astrophysical jet with Ma=2000.