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

TL;DR

This paper introduces a mesh-free, three-dimensional numerical scheme based on semi-discrete optimal transport for solving the incompressible semi-geostrophic equations, enabling the first fully 3D simulation of such models with high efficiency and energy conservation.

Contribution

It extends semi-discrete optimal transport methods from 2D to 3D, providing a robust, energy-conserving numerical tool for meteorological and oceanographic simulations.

Findings

Successful 3D cyclone benchmark simulation

Scheme preserves structural properties and energy efficiently

First fully 3D semi-geostrophic simulation achieved

Abstract

We describe a mesh-free three-dimensional numerical scheme for solving the incompressible semi-geostrophic equations based on semi-discrete optimal transport techniques. These results generalise previous two-dimensional 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 three-dimensional simulation of the semi-geostrophic equations, evidencing semi-discrete optimal transport as a novel, robust numerical tool for meteorological and oceanographic modelling.