Nodal discontinuous Galerkin methods for non-ideal equations of state: pressure equilibrium preservation and entropy correction
2026-08-14 • Computational Engineering, Finance, and Science
Computational Engineering, Finance, and Science
AI summaryⓘ
The authors studied special mathematical methods called discontinuous Galerkin (DG) methods used to simulate fluids more accurately and safely. They focused on how to preserve important properties like pressure balance and entropy in these simulations, especially when using complex fluid models. They developed new ways to analyze and design DG methods that keep pressure stable and added corrections to maintain entropy, which helps make the simulations more reliable over long times. Their work shows that certain adjustments do not cause big errors and can improve simulation stability for challenging cases.
Discontinuous Galerkin methodsPressure equilibriumEntropy stabilityFlux differencingNon-ideal equations of stateTadmor's shuffle conditionDissipative correctionsNumerical fluxConservative discretizationHigh order simulations
Authors
Jesse CHan, Hendrik Ranocha, Raymond Park, Joshua Lampert, Eric Ching, Ayaboe Edoh
Abstract
Structure-preserving discontinuous Galerkin (DG) methods typically improve the robustness of high order simulations of real fluids. In addition to conservation, key structures include the preservation of pressure equilibrium and satisfaction of at least one entropy inequality. In this work, we investigate conservative discretizations using exactly pressure equilibrium conserving (EPEC) and approximately pressure equilibrium conserving (APEC) flux differencing DG formulations, as well as entropy stable formulations through the use of minimally dissipative corrections for non-ideal equations of state (EOS). We introduce an analysis of EPEC schemes and a new procedure for designing such fluxes based on a generalization of Tadmor's shuffle condition. We also analyze APEC DG schemes and show that the incorporation of dissipative interface penalization terms does not significantly increase pressure equilibrium errors, especially at higher orders of approximation. Finally, we observe that when combined with APEC flux differencing formulations, entropy correction improves robustness for under-resolved solutions and long-time simulations.