Semi-implicit methods for 3-D MHD computations

Nonlinear MHD systems often evolve on time scales that are long compared to those associated with the fastest normal modes of the system. Stability restrictions placed on explicit temporal approximations may result in uneconomically small time steps, while the fully implicit treatment of nonlinear terms requires iteration, or results in unacceptably large matrices. Recently, a new class of methods for solving the time-dependent MHD equations, based on an algorithm developed for long-range weather simulation, has been introduced. These semi-implicit methods allow very large time steps, yet avoid the complexity and large memory requirements associated with implicit methods. In the semi-implicit methods for MHD new terms are added to the time-discretized equations that do not affect the consistency of the solution, yet provide a simple and efficient means of enhancing stability. This method is unconditionally stable with respect to all Alfven modes, and consequently permits such large time steps that accuracy becomes the most important consideration in the choice of step size.