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. The full method preserves the underlying geometry, namely the Lie--Poisson structure and all the Casimirs, and nearly preserves the Hamiltonian function in the sense of backward error analysis. To showcase the method, we apply it to two models for magnetic fluids: incompressible magnetohydrodynamics and Hazeltine's model. For the latter, our simulations reveal the formation of large scale vortex condensates, indicating a backward energy cascade analogous to two-dimensional turbulence.

翻译：我们提出了一种保结构的球面上不可压缩磁流体动力学（MHD）时空离散化方案。空间离散化基于几何量子化理论，在磁扩展李代数$\mathfrak{f}=\mathfrak{su}(N)\ltimes\mathfrak{su}(N)^{*}$的对偶空间上，将MHD方程离散化为有限维李-泊松系统。对于形如$\mathfrak{f}=\mathfrak{g}\ltimes\mathfrak{g^{*}}$的半直积李代数对偶空间上的李-泊松系统（其中$\mathfrak{g}$为$J$-二次李代数），我们还给出了相应的保结构时间离散化方法。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。完整方法能够保持底层几何结构（即李-泊松结构和所有卡西米尔不变量），并在后向误差分析意义上近似保持哈密顿函数。为展示该方法，我们将其应用于两种磁流体模型：不可压缩磁流体动力学模型和哈泽泰因模型。对于后者，我们的模拟揭示了大规模涡旋凝聚体的形成，表明存在类似二维湍流的逆向能量级串现象。