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

TL;DR

This paper develops a unified high-order split form discontinuous Galerkin framework for the compressible Euler equations, systematically generating and analyzing split forms for improved robustness in turbulent flow simulations.

Contribution

It introduces a systematic method to generate and analyze split forms of Euler advection terms within a high-order DG framework, enhancing robustness and energy preservation.

Findings

Split forms improve robustness in turbulent flow simulations.

The framework can generate all common split forms systematically.

Certain split forms are shown to be kinetic energy preserving.

Abstract

Fisher and Carpenter (\textit{High-order entropy stable finite difference schemes for non-linear conservation laws: Finite domains, Journal of Computational Physics, 252:518--557, 2013}) found a remarkable equivalence of general diagonal norm high-order summation-by-parts operators to a subcell based high-order finite volume formulation. This equivalence enables the construction of provably entropy stable schemes by a specific choice of the subcell finite volume flux. We show that besides the construction of entropy stable high order schemes, a careful choice of subcell finite volume fluxes generates split formulations of quadratic or cubic terms. Thus, by changing the subcell finite volume flux to a specific choice, we are able to generate, in a systematic way, all common split forms of the compressible Euler advection terms, such as the Ducros splitting and the Kennedy and Gruber splitting. Although these split forms are not entropy stable, we present a systematic way to 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 inviscid Taylor-Green vortex and show that the new split forms enhance the robustness of high order simulations in comparison to the standard scheme when solving turbulent vortex dominated flows. In fact, we show that for certain test cases, the novel split form discontinuous Galerkin schemes are more robust than the discontinuous Galerkin scheme with over-integration.