A no-go theorem and its resolution for the discrete compressible barotropic Navier--Stokes equations

TL;DR

This paper proves a no-go theorem for discrete compressible Navier--Stokes equations, introduces a density-weighted remedy that restores energy conservation, and establishes stability and convergence results for the proposed scheme.

Contribution

It identifies a fundamental limitation in discrete PDE formulations and proposes a density-weighted approach to overcome it, ensuring energy conservation and stability.

Findings

The no-go theorem shows energy residuals cannot be eliminated by any operator choice.

The density-weighted scheme restores exact total energy and preserves key physical laws.

The scheme is globally well-posed, convergent, and asymptotically preserves low-Mach limits.

Abstract

The compressible barotropic Navier--Stokes equations in vector-invariant form preserve the vorticity structure of the system and underlie modern atmospheric and ocean dynamical cores, yet no PDE theory has been developed for the compressible discrete system in this form. On a Delaunay--Voronoi mesh we prove via discrete exterior calculus, that every density-independent mass matrix with integration-by-parts-consistent divergence carries a sharp $\OO(h^2)$ energy residual of indeterminate sign that no operator choice can eliminate. This no-go theorem covers A-, B-, C-, D-, and quasi-B-grid staggerings. The density-weighted mass matrix is the unique algebraic remedy: it restores exact total energy while preserving the vector-invariant momentum equation, Lamb antisymmetry, and the topological conservation laws, at the cost of an $\OO(h^{r_\star})$ Kelvin defect matching the convergence rate. The residual is the cause of the Hollingsworth instability that has shaped vector-invariant dynamical-core design; the density-weighted construction removes it structurally. For the density-weighted~(DW) scheme on closed oriented Riemannian manifolds in $d = 2, 3$ we establish global well-posedness for $\nu \ge 0$, convergence to smooth solutions uniformly in $\nu$, and asymptotic preservation in the low-Mach limit; the density-free residual diverges as $\OO(M^{-1})$. Via a discrete Arnold energy-Casimir construction, exact discrete conservation forces Lyapunov stability around three classes of equilibria, excluding Hollingsworth instability: unconditional stability around hydrostatic and constant-flow stratified states, and conditional stability around sheared baroclinic states under a discrete Charney--Stern criterion. The DW scheme admits genuine baroclinic instability only when the continuum equations themselves do.