  An implicit method for the ohmic dissipation is proposed. The proposed method
is based on the Crank-Nicolson method and exhibits second-order accuracy in
time and space. The proposed method has been implemented in the SFUMATO
adaptive mesh refinement (AMR) code. The multigrid method on the grids of the
AMR hierarchy converges the solution. The convergence is fast but depends on
the time step, resolution, and resistivity. Test problems demonstrated that
decent solutions are obtained even at the interface between fine and coarse
grids. Moreover, the solution obtained by the proposed method shows good
agreement with that obtained by the explicit method, which required many time
steps. The present method reduces the number of time steps, and hence the
computational costs, as compared with the explicit method.

Instrumentation and Methods for Astrophysics;Solar and Stellar Astrophysics