Geometric discretization for incompressible magnetohydrodynamics on the sphere

Licentiate thesis, 2023

Many physical processes are modelled by partial differential equations (PDE), and their efficient discretization is still a challenging problem and an actively developing field. An important class of models arising in mathematical physics represents PDEs formulated in terms of a Lie-Poisson structure on the dual of infinite-dimensional Lie algebras, such as the Lie algebra of vector fields. They are usually referred to as Euler-Arnold systems. A natural approach to discretizing such PDEs is to develop numerical schemes that preserve the underlying Lie-Poisson structure. In the present thesis, an important class of such equations is addressed, namely equations of incompressible magnetohydrodynamics (MHD) on the sphere. The thesis comprises two papers.

In the first paper, a spatio-temporal discretization of MHD on the sphere is developed. This numerical scheme fully preserves the underlying Lie-Poisson structure. The discretization is performed into two steps. First, space discretization based on geometric quantization provides a finite-dimensional Lie-Poisson system on the dual of a semidirect product Lie algebra. Second, structure preserving time integrator is developed. It exactly preserves all the Casimirs and nearly preserves the Hamiltonian function in the sense of backward error analysis.

In the second paper, the developed structure preserving integrator is applied to Hazeltine's model of 2D turbulence in magnetized plasma. Simulations reveal formation of large-scale coherent structures in the long time behaviour, which indicates the presence of an inverse energy cascade.