The Nonisothermal Stage of Magnetic Star Formation. I. Formulation of   the Problem and Method of Solution
  We formulate the problem of the formation and subsequent evolution of fragments (or cores) in magnetically-supported, self-gravitating molecular clouds in two spatial dimensions. The six-fluid (neutrals, electrons, molecular and atomic ions, positively-charged, negatively-charged, and neutral grains) physical system is governed by the radiative, nonideal magnetohydrodynamic (RMHD) equations. The magnetic flux is not assumed to be frozen in any of the charged species. Its evolution is determined by a newly-derived generalized Ohm's law, which accounts for the contributions of both elastic and inelastic collisions to ambipolar diffusion and Ohmic dissipation. The species abundances are calculated using an extensive chemical-equilibrium network. Both MRN and uniform grain size distributions are considered. The thermal evolution of the protostellar core and its effect on the dynamics are followed by employing the grey flux-limited diffusion approximation. Realistic temperature-dependent grain opacities are used that account for a variety of grain compositions. We have augmented the publicly-available Zeus-MP code to take into consideration all these effects and have modified several of its algorithms to improve convergence, accuracy and efficiency. Results of magnetic star formation simulations that accurately track the evolution of a protostellar fragment from a density ~10^3 cm^-3 to a density ~10^15 cm^-3, while rigorously accounting for both nonideal MHD processes and radiative transfer, are presented in a separate paper. 
Matthew W. Kunz (UIUC), Telemachos Ch. Mouschovias (UIUC)