| In this thesis we develop scalable parallel methods and software for solutions of problems in magnetohydrodynamics (MHD). The behavior of an MHD system is complex since it admits various wave phenomena and their instabilities. One of the intrinsic features of the MHD model is the formation of a singular current density sheet, which is linked to the reconnection of magnetic field lines. Reconnection of magnetic flux-tubes leads to a release of energy stored in the magnetic field. Numerical simulation of the reconnection problem plays an important role in our understanding of physical systems ranging from the solar corona to laboratory fusion devices.; Robust iterative solvers are needed to simulate this time-dependent, multi-scale, multi-physics phenomenon. We develop a fully-coupled, nonlinearly implicit technique, in which no operator splitting is applied, for solving the system of MHD equations. More precisely, we first apply a high-order implicit time integration scheme, and then, to guarantee the nonlinear consistency, we use a Newton-Krylov-Schwarz algorithm to solve the large sparse nonlinear system of algebraic equations containing all physical variables at every time step. We investigate the performance of the one- and two-level additive Schwarz as well as the inner-outer preconditioning techniques and focus on the scalability studies of the linear and nonlinear solvers on fine meshes and on machines with thousands of processors. |
