### Abstract

The numerical simulaton of the dynamics of magnetized plasmas is among the most challenging problems in computational physics. Strongly magnetized plasmas are characterized widely separated space and time scales, and by extreme anisoptopy. All of these issues affect the design of algorithms. The fundamental mathematical description requires the simultaneous solution of the 6-dimensional kinetic equation along with Maxwell's equations. This is impossible in all but the very simplest cases, so reduced fluid models can be derived by taking velocity moments of the kinetic equation and assuming a closure condition. Different closure assumptions result in different fluid models. MHD is the simplest of these, although by no means universally applicable. MHD appears to be an excellent model for the dynamics of stellar interiors, where the problem reduces to computing large Reynolds' number turbulence. Memory and speed limitations of even the most powerful computer then dictate a further reduction by means of averaging and statistical closures to capture the effect of the sub-grid scale dynamics on the long wave length motions. There is no concensus on the form of these closures. For the case of low density, high temperature, strongly magnetized plasmas, as occur in laboratory fusion experiments, MHD is clearly not a good model on the smallest scales, and the closure problem becomes one of characterizing non-local kinetic effects in a local transport formalism. This is also an unsolved problem, so in both cases it can be said that there is no agreement on what equations to solve. Because of the differing plasma parameters in these 2 cases, different algorithms must be applied. The audience is already familiar with techniques for computing MHD turbulence. Here we will primarily be concerned with methods developed to simulate ther long time scale dynamics of highly magnetized plasmas, as occur in fusion plasmas and the solar corona. Methods of spatial and temporal differencing will be discussed, and examples of the computed dynamics of laboratory and coronal plasmas will be given. Limitations on the scope of simulations for the foreseeable future will be given. Perhaps some of the issues discussed here will also prove to be useful for stellar interiors.