A Semi-Discrete Optimal Transport Scheme for The 3D Incompressible Semi-Geostrophic Equations

We describe a mesh-free 3D numerical scheme for solving the incompressible semi-geostrophic equations based on semi-discrete optimal transport techniques. These results generalize previous 2D implementations. The optimal transport methods we adopt are known for their structural preservation and energy conservation qualities and achieve an excellent level of efficiency and numerical energy-conservation. We use this scheme to generate numerical simulations of an important cyclone benchmark problem. To our knowledge, this is the first fully 3D simulation of the semi-geostrophic equations, evidencing semi-discrete optimal transport as a novel, robust numerical tool for meteorological and oceanographic modelling.