Orateur
Description
Abstract: Arnold identified ideal fluid motion with geodesics on the group of volume-preserving diffeomorphisms, whose curvature controls hydrodynamic stability. We develop this programme on finite meshes, establishing an approximate finite-dimensional Lie algebra supported by a discrete de Rham complex. This yields well-posed discrete Euler equations that converge to the continuum Euler equations. We derive Arnold's curvature formula and Lorenz's predictability barrier. Finite-dimensional Hopf–Rinow restores geodesic completeness, resolving Shnirelman's obstruction. We connect this to Brenier's relaxed least-action principle, and give a finite-dimensional realisation of the De Lellis–Székélyhidi convex integration scheme, making the construction of wild weak solutions explicit on the mesh. A phase transition emerges as $h\to 0$: local geometry converges while completeness and uniqueness change qualitatively, with curvature providing the geometric link between instability and non-uniqueness.