Entropy-Stable Schemes in the Low-Mach-Number Regime: Flux-Preconditioning, Entropy Breakdowns, and Entropy Transfers

Abstract

Entropy-Stable (ES) schemes, specifically those built from [Tadmor \textit{Math. Comput.} 49 (1987) 91], have been gaining interest over the past decade, especially in the context of under-resolved simulations of compressible turbulent flows using high-order methods. These schemes are attractive because they can provide stability in a global and nonlinear sense (consistency with thermodynamics). However, fully realizing the potential of ES schemes requires a better grasp of their local behavior. Entropy-stability itself does not imply good local behavior [Gouasmi \textit{et al.} \textit{J. Sci. Comp.} 78 (2019) 971, Gouasmi \textit{et al.} \textit{Comput. Methd. Appl. M.} 363 (2020) 112912]. In this spirit, we studied ES schemes in problems where \textit{global stability is not the core issue}. In the present work, we consider the accuracy degradation issues typically encountered by upwind-type schemes in the low-Mach-number regime [Turkel \textit{Annu. Rev. Fluid Mech.} 31 (1999) 285] and their treatment using \textit{Flux-Preconditioning} [Turkel \textit{J. Comput. Phys.} 72 (1987) 277, Miczek \textit{et al.} \textit{A \& A} 576 (2015) A50]. ES schemes suffer from the same issues and Flux-Preconditioning can improve their behavior without interfering with entropy-stability. This is first demonstrated analytically: using similarity and congruence transforms we were able to establish conditions for a preconditioned flux to be ES, and introduce the ES variants of the Miczek's and Turkel's preconditioned fluxes. This is then demonstrated numerically through first-order simulations of two simple test problems representative of the incompressible and acoustic limits, the Gresho Vortex and a right-moving acoustic wave. The results are overall consistent with previous studies [...]