Spatio-temporal Lie-Poisson discretization for incompressible magnetohydrodynamics on the sphere

TL;DR

This paper develops a structure-preserving discretization method for incompressible magnetohydrodynamics on the sphere, combining geometric quantization and Lie-Poisson integrators, ensuring accurate long-term simulation of magnetic fluid models.

Contribution

It introduces a novel spatio-temporal discretization framework for MHD on the sphere that preserves geometric structures and Casimir invariants, advancing numerical methods for magnetic fluid dynamics.

Findings

The discretization preserves a modified Lie--Poisson structure.

The method converges to continuous Casimir functions.

Demonstrated on MHD and Hazeltine's magnetic fluid models.

Abstract

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.