Peter Korn

Peter Korn

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 $ν\ge 0$, convergence to smooth solutions uniformly in $ν$, 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.

Peter Korn

We develop a rigorous theory for a structure-preserving discretisation of the incompressible Euler and Navier--Stokes equations, based on discrete exterior calculus on prismatic Delaunay--Voronoi meshes over closed Riemannian manifolds. The central result is a selection principle: exact algebraic conservation at the discrete level is not merely a fidelity property but rules out entire classes of weak solutions that other discretisations reach unconditionally. We establish this in four regimes. \emph{Smooth solutions}: convergence at rate $\mathcal{O}(h^{\min(r_{\rm rec},\,r_\star)}\,|\log h|^{β_d})$, uniformly in viscosity $ν\ge 0$, with $β_3 = 0$ and $β_2 = 1$; first order on general meshes and second order on meshes with centroid proximity and reconstruction symmetry. \emph{Leray--Hopf weak regime}: subsequential $L^2$ limits are weak solutions of the viscous system. \emph{Inviscid measure-valued regime}: limits are conservative measure-valued Euler solutions; their concentration defect vanishes above the Onsager threshold $α> 1/3$ \emph{provided the discrete solutions admit a uniform $C^{0,α}$ bound there}. \emph{Dissipative regime}: no subsequence converges to an energy-dissipating Euler solution at any regularity, a structural exclusion that follows from exact discrete energy conservation and distinguishes the scheme. % from all Galerkin and finite-volume methods. The gap $1/3 < α< 1$, where energy conservation and defect-free convergence hold but uniqueness remains open, isolates the central open problem of inviscid fluid dynamics.

Maximilian Witte, Fabricio Rodrigues Lapolli, Philip Freese et al.

Using the nonlinear shallow water equations as benchmark, we demonstrate that a simulation with the ICON-O ocean model with a 20km resolution that is frequently corrected by a U-net-type neural network can achieve discretization errors of a simulation with 10km resolution. The network, originally developed for image-based super-resolution in post-processing, is trained to compute the difference between solutions on both meshes and is used to correct the coarse mesh every 12h. Our setup is the Galewsky test case, modeling transition of a barotropic instability into turbulent flow. We show that the ML-corrected coarse resolution run correctly maintains a balance flow and captures the transition to turbulence in line with the higher resolution simulation. After 8 day of simulation, the $L_2$-error of the corrected run is similar to a simulation run on the finer mesh. While mass is conserved in the corrected runs, we observe some spurious generation of kinetic energy.