Island Coalescence Using Parallel First-Order System Least Squares on Incompressible Resistive Magnetohydrodynamics

Abstract

This paper investigates the performance of a parallel Newton, first-order system least-squares (FOSLS) finite-element method with local adaptive refinement and algebraic multigrid (AMG) applied to incompressible, resistive magnetohydrodynamics. In particular, an island coalescence test problem is studied that models magnetic reconnection using a reduced two-dimensional (2D) model of a tokamak fusion reactor. The results show that, using an appropriate temporal and spatial resolution, these methods are capable of resolving the physical instabilities accurately at small computational cost. The time-dependent, nonlinear system of PDEs is solved using work equivalent to about 50-60 simple relaxation sweeps (Gauss-Seidel iterations) per time step. Experiments show that, unless the time step is sufficiently small, nonphysical numerical instabilities may occur. Further, decreasing the time step size does not proportionally increase the cost of the computation, because AMG convergence is improved. In addition, an effective implementation of the methods in parallel keeps load balancing issues to a minimum. Various quantities, such as the reconnection rate and the "sloshing" effect of the plasma instability, are measured to confirm that the correct physics is reproduced. Introduction.The island coalescence problem for studying fast magnetic reconnection in a plasma has been studied extensively (e.g., in [9,11,34,41,42,45,52]). Many numerical algorithms have been implemented to simulate this problem using various types of physical and mathematical models [1,15,22,24,25,35,37,38,41,43,44,46,47,49,50]. The aim of this paper is to extend the results of [2, 3, 4] and show that by using the first-order system least-squares (FOSLS) finite-element method along with nested iteration (NI), algebraic multigrid (AMG), and an efficiency-based adaptive local refinement scheme (ACE) in parallel, the relevant physics of the reconnection is modeled with a low amount of computational cost. Preliminary results are obtained in [2,3,4] using an incompressible resistive magnetohydrodynamics (MHD) model with the above methods. However, limitations with the computational resources used exposed several deficiencies in the method. Namely, at high Lundquist numbers (low resistivity), time integration was not resolving the magnetic instabilities, resulting in numerical oscillations in the current density peak. These, in turn, affected the performance of the solvers and the convergence of the discrete methods. With an efficient implementation of the method on computers with a distributed

An adaptive scalable fully implicit algorithm based on stabilized finite element for reduced visco-resistive MHD

“…For instance, a large "plasmoid" may form at the reconnection site for small Lundquist numbers. We observe one giant plasmoid forms for Lundquist number of 5 × 10 4 when it is under-resolved, and similar unphysical solutions were observed previously in [55,56]. Note the current sheet thickness scales as √ η…”

“…In all these tests, a smaller time step of 0.025 is used. When a large time step of 0.1 is used, we found the reconnection rate may not be accurate in the large Lundquist number runs, which has been also pointed out in the study of [55]. Our results match very well with [4,57].…”

“…Many of the previous studies applied various numerical schemes with some preconditioning techniques to this problem; see [4,9,13,25,26,55,[57][58][59] for instance. Most of those studies however focus on the moderately high Lundquist number cases (the maximal Lundquist number considered is typically under 10 5 ).…”

“…Most of those studies however focus on the moderately high Lundquist number cases (the maximal Lundquist number considered is typically under 10 5 ). Among them, [26,55] also develop an implicit AMR FEM solver, but owing to low Lundquist numbers in their tests the plasmoid instability is not observed. One of the highest Lundquist number considered in previous studies is [59] in which S L up to 10 9 is studied for this example using a resistive MHD model, while [13] considers a simulation of S L = 10 7 for the same example.…”

Monolithic multigrid for implicit Runge-Kutta discretizations of incompressible fluid flow

“…When a large external magnetic field is imposed in the "toroidal" direction of the tokamak, essentially two-dimensional dynamics result. This model geometry is then mapped and rescaled to a square domain, Ω = (−1, 1) 2 , with periodic boundary conditions on the left and right edges (see [22,63,64] for more details). 1 Using the same stopping tolerance led to even more inconsistent data.…”