Kovasznay Mode Preserving and Entropy-Stable Numerical Methods

Entropy-stable, kinetic energy-stable, and nonlinearly stable numerical methods have gained popularity in computational fluid dynamics because they guarantee robustness on coarse meshes. Although nonlinear stability ensures that perturbations caused by aliasing errors remain bounded, as proven by Nordström 2022, under-resolved flow simulations can still suffer—Gassner and coauthors 2022 demonstrated that the bounded perturbations may introduce spurious oscillations that cause nonphysical results, such as negative density. In this presentation, we use the theory from Kovasznay's mode decomposition and Ribner's linear interaction analysis to present a set of weak conservation conditions termed Kovasznay mode preservation (KMP). Here, KMP is added to both entropy and kinetic energy conservation. We demonstrate how to satisfy KMP in skew-symmetric finite-volume and uncollocated discontinuous Galerkin methods. Numerically, for inviscid compressible flows, we demonstrate that KMP preserves the structure of acoustic, entropic, and vortical waves, and we present its impact on shock-vortex, shock-acoustic, and shock-entropic interactions on coarse, under-resolved meshes.