In this talk we discuss the performance of some two-level Schwarz preconditioning techniques applied to Newton-Krylov algorithms for the numerical solution of magnetohydrodynamics. Good parallel performance makes the one-level additive Schwarz preconditioner a sound choice for the large-scale simulation when the number of processors is not too large. However, the growth in the number of linear iterations substantially degrades the parallel efficiency of the one-level preconditioner as the number of subdomains increases beyond one thousand. Two-level preconditioners may provide more efficient means of the inter-subdomain communication than the overlap between subdomains by incorporating a coarse space. In this work we compare the computing time and linear iteration counts of one- and two-level preconditioning methods including the additive, the multiplicative, and the F-cycle versions of the method. We report experimental results obtained on an IBM BG/L with thousands of processors.