E. C. Cyr, J. N. Shadid, R. S. Tuminaro, R. P. Pawlowski, L. Chacon
A New Approximate Block Factorization Preconditioner for Two-Dimensional Incompressible
(Reduced) Resistive MHD, SIAM J. SCI. COMPUT. 35:B701-730, 2013.
The one-fluid visco-resistive MHD model provides a description of the dynamics of a charged fluid under the influence of an electromagnetic field. This model is strongly coupled, highly nonlinear, and characterized by physical mechanisms that span a wide range of interacting time scales. Solutions of this system can include very fast component time scales to slowly varying dynamical time scales that are long relative to the normal modes of the model equations. Fully implicit time stepping is attractive for simulating this type of wide-ranging physical phenomena. However, it is essential that one has effective preconditioning strategies so that the overall fully implicit methodology is both efficient and scalable. In this paper, we propose and explore the performance of several candidate block preconditioners for this system. One of these preconditioners is based on an operator-split approximation. This method reduces the 3x3 system (momentum, continuity, and magnetics) into two 2x2 operators: a Navier-Stokes operator (momentum and continuity) and a magnetics-velocity operator (momentum and magnetics) which takes into account the critical Lorentz force coupling. Using previously developed preconditioners for Navier-Stokes, and an initial Schur-complement approximation for the magnetics-velocity system, we show that the split preconditioner is scalable and competitive with other preconditioners, including a fully coupled algebraic multigrid method.