On fully implicit methods for extended magnetohydrodynamics

Luis Chacon


Magnetohydrodynamics (MHD) describes the behavior of charged hot gases (plasmas) in the presence of electromagnetic fields, and is a crucial tool for the understanding of solar, space, and laboratory (e.g. thermonuclear fusion) plasmas.

Mathematically, MHD is a hyperbolic PDE system which supports various fast waves. In its basic form (single fluid), these waves are linear ($\omega \sim k$). However, in extended (or two-fluid) MHD models (XMHD), these waves become dispersive ($\omega \sim k^2$). The numerical stiffness associated with such waves present severe challenges for the efficient and accurate numerical integration of MHD phenomena over long frequencies.

In this talk, we describe our discretization/solver strategy to address such challenges. Spatially, we employ a novel second-order finite-volume strategy which is cell-centered, conservative, stable linearly and nonlinearly, and suitable for general curvilinear geometries. Temporally, we employ Newton-Krylov-based fully implicit methods for robustness, efficiency, and accuracy. A crucial aspect of our approach is a novel preconditioning strategy, which we term "physics-based". It is based on a Schur complement treatment of the semi-discrete MHD equations, which in turn renders the linearized MHD system well conditioned for the use of classical multilevel techniques, thereby resulting in an optimal, scalable MHD solver.