Mathematics and Applications Seminar

Michael Roop, Chalmers tekniska högskola is giving a talk with the title: ”Structure preserving discretization and long-time behaviour of ideal magnetohydrodynamics on the sphere”

Michael Roop is a PhD student at Chalmers tekniska högskola. His PhD project ”Geometric Hydrodynamics” is concerned with numerical investigations of 2D incompressible Euler’s equations, using tools from Lie algebra theory and quantum theory.

Abstract:

An important class of models arising in fluid mechanics represents PDEs formulated as geodesic flows on the group of diffeomorphisms of some manifold. They are usually referred to as Euler-Arnold equations. Their Hamiltonian nature suggests that they have a lot of symmetries and conservation laws, such as energy and Casimir functions. A natural approach to discretization of such equations is to preserve those conservation laws. Preservation of Casimir functions is known to be essential for long time simulations of fluids. In this talk, we will address one example of such equations, the system of incompressible magnetohydrodynamics (MHD) equations on the sphere, and will present its fully discrete analogue that completely preserves the underlying Lie-Poisson structure. In particular, the numerical method exactly preserves Casimirs and nearly preserves the Hamiltonian function. The method is applied to two models describing the motion of magnetized ideal fluids, reduced MHD (RMHD) model, and Hazeltine’s model (also called Alfvén wave turbulence equations). Numerical simulations reveal formation of large scale vortex blob structures in the long time behaviour.

Welcome!