Computational Magnetohydrodynamics with Discrete Differential Forms
The equations of magnetohydrodynamics (MHD) model the interaction of conducting fluids with electromagnetic fields, and provide the mathematical description of problems arising in areas as diverse as plasma physics, astrophysics, and thermonuclear fusion. They comprise balance equations for mass, momentum and energy, the magneto quasistatic Maxwell's equations for the electromagnetic fields, and material laws. This thesis is devoted to the development and analysis of novel numerical methods for the MHD problem based on Galerkin schemes for the electromagnetic fields via finite element exterior calculus (FEEC) coupled with finite volume (FV) schemes for the conservation laws of fluid mechanics. Finite element exterior calculus relies on discrete differential forms which provide structure preserving discretizations by supporting a discrete de Rham cohomology. The magneto quasistatic model underlying resistive MHD yields a magnetic advection diffusion problem for the magnetic potential. We consider the singular perturbation limit and devise robust numerical discretizations of generalized transient advection problems for differential forms, through an Eulerian method of lines with explicit time stepping and stabilized Galerkin schemes. The spatial approximations include both conforming discrete differential forms and genuinely discontinuous finite elements, and are designed to accommodate discontinuous advection velocities which inevitably occur in MHD flows. A priori convergence estimates are established for Lipschitz continuous velocities, conforming meshes and polynomial finite element spaces of discrete differential forms. Additionally, we explore an alternative class of numerical schemes for the discretization of the Lie derivative, built on the duality between the contraction operator and the extrusion of manifolds. These methods incorporate an upwinding element and deliver discrete advection operators that commute with the exterior derivative, hence ensuring that closed forms are Lie advected into closed forms. Nonlinear flux limiters, in the form of residual based artificial viscosity, are designed to curb the spurious oscillations resulting from higher order polynomial discretizations. In the resistive MHD system, the eddy current model with non vanishing diffusion gives rise to a parabolic problem in H(curl ,O) Discretizations with Galerkin schemes, and implicit explicit (IMEX) Runge Kutta time stepping, entail solving, in each time step, discrete boundary value problems for the double curl operator. Spatial discretizations with discrete differential forms pave the way for applying fast iterative solvers, viz. multigrid. For discontinuous Galerkin discretizations, we develop a family of preconditioners based on an auxiliary space of H(curl ,O) conforming finite elements, together with a smoother. The resulting iterative solvers are shown to be asymptotically optimal in terms of independence from the mesh width. With particular regard to the case of locally dominant transport, robustness with respect to jumps in the zerothand second order parts of the operator is shown to hold in almost all configurations, except when the problem changes from being curl dominated to reaction dominated. The balance laws for the fluid variables can be considered as a system of conservation laws with the magnetic induction field as a space variable coefficient, supplied at every time step by one of the foregoing structure preserving discretizations. The design of finite volume schemes for this extended Euler problem relies on approximate Riemann solvers, adapted to accommodate the electromagnetic contributions to the momentum and energy directly entering the fluxes. High order spatial accuracy is achieved via non oscillatory reconstruction techniques, such as TVD limiters and (W)ENO type reconstructions. A full discretization of the MHD system results from coupling the FEEC based numerical schemes for the magnetic advection diffusion problem with finite volume approximations of the conservation laws for the fluid. The lowest order fully coupled scheme is tested on a set of benchmark tests for the two dimensional planar ideal MHD equations. The method based on extrusion contraction upwind schemes for the magnetic advection preserves the divergence constraint exactly, and proves first order accurate for smooth solutions, conservative, and stable. This research was partly supported by the Swiss NSF Grant No. 146355.