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We develop a nonlinear force-free field (NLFFF) extrapolation code based onthe magnetohydrodynamic (MHD) relaxation method. We extend the classical MHDrelaxation method in two important ways. First, we introduce an algorithminitially proposed by cite{2002JCoPh.175..645D} to effectively clean thenumerical errors associated with $nabla cdot vec{B}$. Second, the multi-gridtype method is implemented in our NLFFF to perform direct analysis of thehigh-resolution magnetogram data. As a result of these two implementations, wesuccessfully extrapolated the high resolution force-free field introduced bycite{1990ApJ...352..343L} with better accuracy in a drastically shorter time.We also applied our extrapolation method to the MHD solution obtained from theflux-emergence simulation by cite{2012ApJ...748...53M}. We found that NLFFFextrapolation may be less effective for reproducing areas higher than ahalf-domain, where some magnetic loops are found in a state of continuousupward expansion. However, an inverse S shaped structure consisting of thesheared and twisted loops formed in the lower region can be captured wellthrough our NLFFF extrapolation method. We further discuss how well thesesheared and twisted fields are reconstructed by estimating the magnetictopology and twist quantitatively.

Nonlinear Force-Free Extrapolation of the Coronal Magnetic Field Based on the MHD Relaxation Method