Optimal transport and the compressible semi-geostrophic equations
Date:
The semi-geostrophic equations for a compressible fluid, first analyzed by Cullen and Maroofi [2003], provide a simplified model of the formation and evolution of atmospheric fronts. I will describe the use of semi-discrete optimal transport theory to construct a numerical particle method. This method is structure preserving in the sense that numerical solutions conserve energy. I will then present numerical results and discuss the challenges we faced in implementing this numerical method. Using this approach, we give a constructive proof of the existence of global-in-time weak solutions as the limit of spatially discrete approximations. This work directly extends the work of Bourne et al. [2022] from the incompressible to the compressible setting.
This is joint work with David Bourne, Charlie Egan, and Beatrice Pelloni at Heriot-Watt University.