Spatio-temporal Lie-Poisson discretization for incompressible magnetohydrodynamics on the sphere
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| Format: | Preprint |
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2023
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| author | Modin, Klas Roop, Michael |
| author_facet | Modin, Klas Roop, Michael |
| contents | We give a structure preserving spatio-temporal discretization for incompressible magnetohydrodynamics (MHD) on the sphere. Discretization in space is based on the theory of geometric quantization, which yields a spatially discretized analogue of the MHD equations as a finite-dimensional Lie--Poisson system on the dual of the magnetic extension Lie algebra $\mathfrak{f}=\mathfrak{su}(N)\ltimes\mathfrak{su}(N)^{*}$. We also give accompanying structure preserving time discretizations for Lie--Poisson systems on the dual of semi-direct product Lie algebras of the form $\mathfrak{f}=\mathfrak{g}\ltimes\mathfrak{g^{*}}$, where $\mathfrak{g}$ is a $J$-quadratic Lie algebra. The time integration method is free of computationally costly matrix exponentials. We prove that the full method preserves a modified Lie--Poisson structure and corresponding Casimir functions, and that the modified structure and Casimirs converge to the continuous ones. The method is demonstrated for two models of magnetic fluids: incompressible magnetohydrodynamics and Hazeltine's model. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_16045 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Spatio-temporal Lie-Poisson discretization for incompressible magnetohydrodynamics on the sphere Modin, Klas Roop, Michael Numerical Analysis Mathematical Physics Differential Geometry 37M15, 65P10, 53D20, 76W05 We give a structure preserving spatio-temporal discretization for incompressible magnetohydrodynamics (MHD) on the sphere. Discretization in space is based on the theory of geometric quantization, which yields a spatially discretized analogue of the MHD equations as a finite-dimensional Lie--Poisson system on the dual of the magnetic extension Lie algebra $\mathfrak{f}=\mathfrak{su}(N)\ltimes\mathfrak{su}(N)^{*}$. We also give accompanying structure preserving time discretizations for Lie--Poisson systems on the dual of semi-direct product Lie algebras of the form $\mathfrak{f}=\mathfrak{g}\ltimes\mathfrak{g^{*}}$, where $\mathfrak{g}$ is a $J$-quadratic Lie algebra. The time integration method is free of computationally costly matrix exponentials. We prove that the full method preserves a modified Lie--Poisson structure and corresponding Casimir functions, and that the modified structure and Casimirs converge to the continuous ones. The method is demonstrated for two models of magnetic fluids: incompressible magnetohydrodynamics and Hazeltine's model. |
| title | Spatio-temporal Lie-Poisson discretization for incompressible magnetohydrodynamics on the sphere |
| topic | Numerical Analysis Mathematical Physics Differential Geometry 37M15, 65P10, 53D20, 76W05 |
| url | https://arxiv.org/abs/2311.16045 |