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 (1987) [54], 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 et al., 2019 [17], Gouasmi et al., 2020 [18]). In this spirit, we studied ES schemes in problems where 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, 1999 [59]) and their treatment using Flux-Preconditioning (Turkel, 1987 [58], Miczek et al., 2015 [41]). 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. For instance, we observe that Turkel's preconditioner improves accuracy in the incompressible limit with the downside of overly damping acoustic waves (Bruel et al., 2019 [4]). For Miczek's matrix however, we came across unexpected spurious transients in both problems (a small left-moving acoustic wave in the latter), motivating further analysis. We revisited the pressure fluctuation argument of Guillard and Viozat (1999) [26] in terms of entropy, showing how the standard ES dissipation operator (Ismail and Roe, 2009 [30]) can introduce inconsistent discrete entropy fluctuations in space. These analytical results are achieved by introducing mode-by-mode decompositions of the dissipation operator, similar to Roe and Pike (1984) [51], Tadmor (2003) [55], leading to what we call discrete Entropy Production Breakdowns (EPBs) in space. These EPBs outline the contributions of convective and acoustic modes to the discrete entropy production, both locally and globally. Ultimately, these EPBs enable us to single-out in the ES Miczek flux a skew-symmetric matrix component which we believe causes entropy transfers between acoustic waves. Removing this contribution eliminates the spurious transients without interfering with entropy-stability. This conjecture is explored numerically and analytically.