Entropy-Stable and Well-Balanced Discontinuous Galerkin Methods for the Compressible Euler Equations in Vector-Invariant Form
Abstract
We develop structure-preserving methods for the compressible Euler equations with gravity in vector-invariant form and potential temperature as a prognostic variable within the flux-differencing discontinuous Galerkin spectral element method (DGSEM) framework. By discretizing the nonconservative terms as symmetric and antisymmetric products, we derive two-point numerical fluxes that conserve both the thermodynamic entropy and the total energy. Moreover, we design an entropy-stable numerical flux that is well-balanced for both isothermal and isentropic background states. All properties are shown to carry over to the high-order DGSEM on general curvilinear meshes. Several numerical examples confirm the theoretical findings and show the robustness and accuracy of the scheme for use in modern dynamical cores for atmospheric flows.
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BibTeX
@misc{artiano2026entropystable,
title = {{Entropy-Stable and Well-Balanced Discontinuous Galerkin Methods for the Compressible Euler Equations in Vector-Invariant Form}},
author = {Marco Artiano and Kieran Ricardo and Oswald Knoth and Peter Spichtinger and Hendrik Ranocha},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05820},
eprint = {2610.05820},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05820},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Entropy-Stable and Well-Balanced Discontinuous Galerkin Methods for the Compressible Euler Equations in Vector-Invariant Form
AU - Marco Artiano
AU - Kieran Ricardo
AU - Oswald Knoth
AU - Peter Spichtinger
AU - Hendrik Ranocha
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05820
UR - https://arxiv.org/abs/2610.05820
AB - We develop structure-preserving methods for the compressible Euler equations with gravity in vector-invariant form and potential temperature as a prognostic variable within the flux-differencing discontinuous Galerkin spectral element method (DGSEM) framework. By discretizing the nonconservative terms as symmetric and antisymmetric products, we derive two-point numerical fluxes that conserve both the thermodynamic entropy and the total energy. Moreover, we design an entropy-stable numerical flux that is well-balanced for both isothermal and isentropic background states. All properties are shown to carry over to the high-order DGSEM on general curvilinear meshes. Several numerical examples confirm the theoretical findings and show the robustness and accuracy of the scheme for use in modern dynamical cores for atmospheric flows.
ER -