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 semidirect product Lie algebras of the form $\mathfrak{f}=\mathfrak{g}\ltimes\mathfrak{g^{*}}$, where $\mathfrak{g}$ is a $J$-quadratic Lie algebra. Critically, the time integration method is free of computationally costly matrix exponentials. We prove that the full method preserves the underlying geometry, namely the Lie--Poisson structure and all the Casimirs. To showcase the method, we apply it to two models for magnetic fluids: incompressible magnetohydrodynamics and Hazeltine's model.

翻译：我们提出了一种球面上不可压缩磁流体动力学（MHD）的保结构时空离散化方法。空间离散化基于几何量子化理论，通过磁扩展李代数$\mathfrak{f}=\mathfrak{su}(N)\ltimes\mathfrak{su}(N)^{*}$的对偶空间，得到MHD方程的空间离散化模拟作为有限维李-泊松系统。针对形如$\mathfrak{f}=\mathfrak{g}\ltimes\mathfrak{g^{*}}$的半直积李代数对偶空间上的李-泊松系统，我们还给出了相应的保结构时间离散化方法，其中$\mathfrak{g}$为$J$-二次李代数。关键之处在于，该时间积分方法避免了计算代价高昂的矩阵指数运算。我们证明了该完整方法能够保持底层几何结构，即李-泊松结构及所有卡西米尔函数。为展示该方法，我们将其应用于两个磁流体模型：不可压缩磁流体动力学和哈泽泰因模型。