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Fisher and Carpenter (\textit{High-order entropy stable finite differenceschemes for non-linear conservation laws: Finite domains, Journal ofComputational Physics, 252:518--557, 2013}) found a remarkable equivalence ofgeneral diagonal norm high-order summation-by-parts operators to a subcellbased high-order finite volume formulation. This equivalence enables theconstruction of provably entropy stable schemes by a specific choice of thesubcell finite volume flux. We show that besides the construction of entropystable high order schemes, a careful choice of subcell finite volume fluxesgenerates split formulations of quadratic or cubic terms. Thus, by changing thesubcell finite volume flux to a specific choice, we are able to generate, in asystematic way, all common split forms of the compressible Euler advectionterms, such as the Ducros splitting and the Kennedy and Gruber splitting.Although these split forms are not entropy stable, we present a systematic wayto prove which of those split forms are at least kinetic energy preserving.With this, we show we construct a unified high-order split form DG framework.We investigate with three dimensional numerical simulations of the inviscidTaylor-Green vortex and show that the new split forms enhance the robustness ofhigh order simulations in comparison to the standard scheme when solvingturbulent vortex dominated flows. In fact, we show that for certain test cases,the novel split form discontinuous Galerkin schemes are more robust than thediscontinuous Galerkin scheme with over-integration.

Split Form Nodal Discontinuous Galerkin Schemes with Summation-By-Parts Property for the Compressible Euler Equations