Nested Iteration and First-Order System Least Squares for Incompressible, Resistive Magnetohydrodynamics

Abstract

Abstract. This paper develops a nested iteration algorithm to solve time-dependent nonlinear systems of partial differential equations. For each time step, Newton's method is used to form approximate solutions from a sequence of nested spaces, where the resolution of the approximations increases as the algorithm progresses. Nested iteration results in most of the iterations being performed on coarser grids, where minimal work is needed to reduce error to the level of discretization error. The approximate solution on a given coarse grid is interpolated to a refined grid and used as an initial guess for the problem posed there. The approximation is then already close enough to the solution on the current grid that a minimal amount of work is needed to solve the refined problem, due to the rapid convergence of Newton's method near a solution. The paper develops an algorithm that attempts to optimize accuracy-percomputational-cost on each grid, so that essentially no unnecessary work is done on any grid. The nested iteration algorithm is then applied to a reduced 2D model of the incompressible, resistive magnetohydrodynamic (MHD) equations. Using this algorithm on the MHD equations in the context of a first-order system least-squares (FOSLS) finite element discretization and algebraic multigrid (AMG) to solve the linearized systems, instabilities in a model tokamak fusion reactor are simulated. Numerical results show that this highly complex nonlinear problem is solved in an equivalent of 30-80 fine-grid relaxation sweeps per time step.Key words. magnetohydrodynamics, algebraic multigrid, FOSLS, nested iteration AMS subject classifications. 65F10, 65M55, 76W05Acknowledgments. This work was sponsored by the Department of Energy under grant numbers DE-FG02-03ER25574 and DE-FC02-06ER25784, Lawrence Livermore National Laboratory under contract numbers B568677, and the National Science Foundation under grant numbers DMS-0621199, DMS-0749317, and DMS-0811275.1. Introduction. This paper applies a nested iteration algorithm to a set of incompressible resistive Magnetohydrodynamic (MHD) equations. A first-order systems least-squares (FOSLS) [11,12] finite element discretization is used. The main focus of this paper is to show that nested iteration is a crucial component to solving complicated systems of nonlinear partial differential equations. As is shown for the MHD equations, nested iteration allows for a complicated system of nonlinear equations to be solved to within discretization accuracy in only a handful of work units (WUs). A WU is defined as the amount of computation required to perform one fine-grid relaxation sweep. The main idea is that most of the work is done on coarse grids where computation is much cheaper. The approximation to the solution on these coarse grids is interpolated up to successively finer grids, where linearizations and algebraic solves are applied. The process is then continued to finer grids until a desired error tolerance or resolution is reached. The result is that coarse grids give fairly...

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

“…In this way, standard Newton steps can be used in conjunction with classical AMG iterations on the resulting discrete linear systems. This approach is described in more detail in [4,27]. In addition, other approaches to solving sets of nonlinear transient equations can lead to unconditionally stable methods that can reduce the cost of the linearizations and increase the accuracy of the time discretization (e.g., in [40,51]).…”

“…These initial conditions are perturbed away from equilibrium as follows: [32,33], and, thus, the linearized version of the FOSLS formulation, (3.8)-(3.13), is elliptic in a subspace of a product H 1 -norm [5]. No higher regularity for the variables is needed to achieve this, since the weak form comes from a minimization of a quadratic functional.…”

“…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.…”

“…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.…”

A Combined Preconditioning Strategy for Nonsymmetric Systems

“…Many efforts are usually devoted to design proper preconditioning strategies that allow for efficient and fast solution of the resulting algebraic systems [27,28]. The development of preconditioners is very often guided by the properties of the underlying problem and it sometimes can even dictate particular aspects that should be accounted for, when devising the numerical discretization of the continuous problem (as for instance in [18,1]). …”