Semi-discrete optimal transport methods for the semi-geostrophic equations

TL;DR

This paper introduces a new constructive proof for the existence of global weak solutions to the 3D semi-geostrophic equations using semi-discrete optimal transport, leading to improved regularity and explicit solutions.

Contribution

It presents a novel semi-discrete optimal transport approach to prove existence, regularity, and explicit solutions for semi-geostrophic equations, with an efficient numerical implementation.

Findings

Established existence of global weak solutions in 3D

Achieved improved time-regularity for discrete initial measures

Demonstrated numerical simulations of semi-geostrophic flows

Abstract

We give a new and constructive proof of the existence of global-in-time weak solutions of the 3-dimensional incompressible semi-geostrophic equations (SG) in geostrophic coordinates, for arbitrary initial measures with compact support. This new proof, based on semi-discrete optimal transport techniques, works by characterising discrete solutions of SG in geostrophic coordinates in terms of trajectories satisfying an ordinary differential equation. It is advantageous in its simplicity and its explicit relation to Eulerian coordinates through the use of Laguerre tessellations. Using our method, we obtain improved time-regularity for a large class of discrete initial measures, and we compute explicitly two discrete solutions. The method naturally gives rise to an efficient numerical method, which we illustrate by presenting simulations of a 2-dimensional semi-geostrophic flow in geostrophic coordinates generated using a numerical solver for the semi-discrete optimal transport problem coupled with an ordinary differential equation solver.