Entropy stable discontinuous Galerkin methods for balance laws in non-conservative form: Applications to the Euler equations with gravity

The Flux‐Differencing Discontinuous Galerkin Method Applied to an Idealized Fully Compressible Nonhydrostatic Dry Atmosphere

“…In summary, the averaging prescription changes to "average along the coordinate direction" depending on which derivative is being taken. There are restrictions in the treatment of metric terms for curvilinear coordinates, see Winters et al (2021), Waruszewski et al (2022) for details.…”

“…A detailed explanation of the flux‐differencing method used in this work is in Chan (2018) and Waruszewski et al. (2022), but here we focus on a heuristic overview. Furthermore we concentrate on the aspect that makes flux‐differencing different from traditional DG methods, the volume term calculations.…”

“…Gassner et al (2016a), Winters et al (2021), G. J. Gassner and Winters (2021) for insights on why equations remain conservative for operators that satisfy a summation by parts property and Chan (2018), Waruszewski et al (2022) for more general operators. Deciding between which form of equations to use is not a matter of guesswork or experimentation; rather, they are derived based on which discrete properties one wants to preserve.…”

The Flux-Differencing Discontinuous Galerkin Method Applied to an Idealized Fully Compressible Nonhydrostatic Dry Atmosphere

“…To solve the compressible Euler equations in three-dimensional domains, we use the FDDG formulation of Chan (2018) and Waruszewski et al (2022) and construct metric terms as outlined by D. A. Kopriva (2006).…”

“…where I is the identity matrix. The modification to the gravity source term was motivated by combining the entropy stable scheme of Waruszewski et al (2022) with the Kennedy-Gruber flux.…”