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

David P. Bourne, Charles P. Egan, Beatrice Pelloni, Mark Wilkinson

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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.
Original languageEnglish
Article number39
JournalCalculus of Variations and Partial Differential Equations
Volume61
Issue number1
Early online date16 Jan 2022
DOIs
Publication statusPublished - Feb 2022

Keywords

  • semigeostrophic system
  • atmospheric fluid dynamics

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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