A Parallel, Adaptive, First-Order System Least-Squares (FOSLS) Algorithm for Incompressible Resistive, Magnetohydrodynamics (MHD)

Magnetohydrodynamics (MHD) is a model of plasma physics that treats the plasma as a charged fluid. As a result, the set of partial differential equations that describe this model are a time-dependent, nonlinear system of equations. Thus, the equations can be difficult to solve and efficient numerical algorithms are needed. This work shows the use of such an efficient algorithm on the incompressible, resistive MHD equations. A first-order systems least-squares (FOSLS) [1, 2] finite element discretization is used along with nested iteration and algebraic multigrid (AMG) [3, 4, 5, 6, 7, 8]. The main focus of this work is to show that if a nested iteration algorithm along with an efficiency-based adaptive mesh refinement (AMR) scheme is used, then a nonlinear system of equations, such as the MHD equations, can be solved in only a handful of work units per time step. Here, a work unit is defined as the equivalent of one relaxation sweep on the finest grid. In other words, the accuracy-per-computational-cost for solving the MHD equations can be maximized by the use of nested iteration and AMR. An island coalescence instability was able to be resolved in less than 10 work units per time step. Further details of this work can be found in several companion papers. In [9], the FOSLS method applied to MHD is described. The nested iteration algorithm has also been develoeped in [10] and [11] and the efficiency-based AMR method known as ACE, is discussed in [12, 13, 14, 15].