Nodal discontinuous Galerkin methods for non-ideal equations of state: pressure equilibrium preservation and entropy correction

2026-08-14

AI Summary

This study addresses the challenges of pressure equilibrium and entropy stability in high-order simulations of non-ideal fluids by proposing a pressure-equilibrium-preserving and entropy-stable nodal discontinuous Galerkin scheme. A novel numerical framework is constructed by designing EPEC fluxes through the generalization of Tadmor’s condition, combined with dissipation corrections and interface penalty terms. Numerical investigations demonstrate that this method maintains high-order accuracy without significantly increasing pressure equilibrium errors. Furthermore, it effectively overcomes instabilities arising from under-resolution and long-time integration, thereby substantially enhancing the robustness and reliability of non-ideal fluid dynamics simulations.

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.

Problem

• non-ideal equations of state
• pressure equilibrium preservation
• entropy correction
• discontinuous Galerkin methods
• structure-preserving

Innovation

• Discontinuous Galerkin
• Pressure equilibrium preservation
• Entropy correction
• Non-ideal equations of state
• Flux differencing