Discretisations and preconditioners for magnetohydrodynamics models
Keywords: Magnetohydrodynamics
Preconditioners

Magnetohydrodynamics (MHD) models describe the behaviour of electrically conducting fluids such as astrophysical and laboratory plasmas or liquid metals in the presence of magnetic fields. They are generally known to be difficult to solve numerically, due to their highly nonlinear structure and the strong coupling between the electromagnetic and hydrodynamic variables, especially for high Reynolds and coupling numbers.
In the first part of this work, we present a scalable augmented Lagrangian preconditioner for a finite element discretisation of the B-E formulation of the incompressible viscoresistive MHD equations. For stationary problems, our solver achieves robust performance with respect to the Reynolds and coupling numbers in two dimensions and good results in three dimensions. We extend our method to fully implicit methods for time-dependent problems which we solve robustly in both two and three dimensions. Our approach relies on specialised parameter-robust multigrid methods for the hydrodynamic and electromagnetic blocks. The scheme ensures exactly divergence-free approximations of both the velocity and the magnetic field up to solver tolerances.
We confirm the robustness of our solver by numerical experiments in which we consider fluid and magnetic Reynolds numbers and coupling numbers up to 10,000 for stationary problems and up to 100,000 for transient problems in two and three dimensions.
In the second part, we focus on incompressible, resistive Hall MHD models and derive structure-preserving finite element methods for these equations. These equations incorporate the Hall current term in Ohm's law and provide a more appropriate description of fully ionized plasmas than the standard MHD equations on length scales close to or smaller than the ion skin depth. In particular, we present a variational formulation of Hall MHD that enforces the magnetic Gauss's law precisely (up to solver tolerances) and prove the well-posedness of a Picard linearisation. For the transient problem, we present time discretisations that preserve the energy and magnetic and hybrid helicity precisely in the ideal limit for two types of boundary conditions. Additionally, we investigate an augmented Lagrangian preconditioning technique for both the stationary and transient cases. Finally, we confirm our findings by several numerical experiments.
In the third part, we investigate anisothermal MHD models. We start by performing a bifurcation analysis for a magnetic Rayleigh-Benard problem at a high coupling number S=1,000 by choosing the Rayleigh number in the range between 0 and 100,000 as the bifurcation parameter. We study the effect of the coupling number on the bifurcation diagram and outline how we create initial guesses to obtain complex solution patterns and disconnected branches for high coupling numbers. Moreover, we extend the parameter-robust augmented Lagrangian preconditioner for the standard MHD equations to the anisothermal case. Again, we obtain excellent robustness with respect to the Rayleigh number, Prandtl number, magnetic Prandtl number and coupling number in two dimensions and good robustness in three dimensions. We verify our finding by reporting iteration numbers for a magnetic double glazing problem and a magnetic cooling channel problem.