A Semi-Discrete Optimal Transport Scheme for the Semi-Geostrophic Slice Compressible Model

We develop a semi-discrete optimal transport scheme for the compressible semi-geostrophic equations, a system that plays an important role in modelling large-scale atmospheric dynamics and frontogenesis. Unlike the incompressible case, the compressible equations involve variable density and internal energy, but can be recast into a variational framework that naturally couples the dynamics with an optimal transport formulation. This is done by a change to the so-called geostrophic coordinates, via a transformation inspired by the incompressible case. The discrete version of this variational formulation provides the basis for a numerical particle scheme. The implementation of this scheme presents considerable challenges, due to a non-quadratic cost function and parabolic $c$-Laguerre cells. To address these challenges, we use $c$-exponential charts to construct $c$-Laguerre tessellations efficiently, ensuring conservation of mass and energy while preserving key geometric structures. We analyse the scheme and validate its convergence through numerical experiments, including a single-seed benchmark and error analysis. This work provides a significant new generalisation of existing semi-discrete optimal transport techniques, offering a robust and structure-preserving tool for simulating realistic atmospheric flows.