An Adaptive Finite Element Method for Magnetohydrodynamics

Abstract

A finite element discretization for two-dimensional MHD is described. The elements are triangles with piecewise linear basis functions. The main computational difficulty is the accurate calculation of the current. The most effective solution is to employ a current-vorticity advection formulation of the equations. Acceptable results can also be obtained with a two-step calculation of the current from the vector potential. Mesh operations are described to reconnect and refine the mesh adaptively in the vicinity of nearly singular currents to improve magnetic flux conservation. Example computations of the coalescence instability, tilt mode, and divertor tokamak equilibrium, validating and illustrating the method, are presented. The simulations show the formation of current sheets, with the current density increasing exponentially in time. During this increase, the grid of initially ∼10 4 points adapts to provide resolution comparable to a uniform grid of up to 1.6 × 10 8 grid points.

Interacting Tilt and Kink Instabilities in Repelling Current Channels

“…First, purely planar tilt evolutions, as well as translationally invariant extensions of the basic setup where two neighboring islands carry opposite currents, were studied in compressible MHD. As in the previous incompressible planar (Strauss & Longcope 1998;Lankalapalli et al 2007) or compressible forcefree (Richard et al 1990) scenarios, the ideal tilt manifests itself over a wide range of plasma beta (β ∈ [0.12, 12.7] have been explored, see table 1) and has Alfvénic growth rates which decrease with decreasing β. Using extreme resolution adaptive simulations, we showed that the near-singular current sheets that develop ahead of the tilt-displaced islands additionally become liable to tearing-type disruptions.…”

“…Both tilt and coalescence instability have been studied extensively in pure twodimensional (2D), planar configurations (see, e.g., Longcope & Strauss 1993;Strauss & Longcope 1998;Marliani & Strauss 1999;Ng et al 2008) where the poloidal field distribution forms islands that repel or attract, causing localized reconnection and strong hints of singular current layer development. The singular nature of the current concentrations has turned them into popular testbeds for adaptive mesh refinement (AMR) strategies in MHD simulations (Strauss & Longcope 1998;Lankalapalli et al 2007;Ng et al 2008). Here, we use high-resolution, fixed grid and AMR simulations to study how adjacent repelling current channels, without initial curvature or line-tying, evolve through combined tilt and kink evolutions in up to 3D configurations.…”

Additive Schwarz-based fully coupled implicit methods for resistive Hall magnetohydrodynamic problems

“…Solving the MHD equations numerically is a challenge because of the complex, not yet fully understood behavior of the solution. The system admits phenomena such as Alfvén waves and their instabilities, and one of the intrinsic features of the system is the formation of a singular current density sheet [29], which is linked to the reconnection of magnetic field lines.…”

A multiscale stabilization of the streamfunction form of the steady state Navier-Stokes equations

“…Second, C 1 finite elements must be employed, which are not trivial to develop or implement. Magnetohydrodynamics (MHD) has caused a renewed interest in the streamfunction form of the Navier-Stokes equations as (1) the geometries in consideration can be decomposed into twodimensional elements in the plane and spectral elements in the toroidal plane and (2) other operators representing MHD physics also benefit from C 1 continuity [5,6]. …”