A Semi-Discrete Optimal Transport Scheme for the Semi-Geostrophic Slice Compressible Model
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910046202363904 |
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| author | Lavier, Théo Pelloni, Beatrice |
| author_facet | Lavier, Théo Pelloni, Beatrice |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08105 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Semi-Discrete Optimal Transport Scheme for the Semi-Geostrophic Slice Compressible Model Lavier, Théo Pelloni, Beatrice Numerical Analysis 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. |
| title | A Semi-Discrete Optimal Transport Scheme for the Semi-Geostrophic Slice Compressible Model |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2603.08105 |