Magnetically Controlled Spasmodic Accretion During Star Formation. I.   Formulation of the Problem and Method of Solution
  We formulate the problem of the late accretion phase of the evolution of an isothermal magnetic disk surrounding a forming star. The evolution is described by the six-fluid MHD equations, accounting for the presence of neutrals, atomic and molecular ions, electrons, and neutral, positively, and negatively charged grains. Only the electron fluid is assumed to be attached to the magnetic field, in order to investigate the effect of the detachment of the ions from the magnetic field lines that begins at densities as low as 10^8 cm^-3. The "central sink approximation" is used to circumvent the problem of describing the evolution inside the opaque central region for densities greater than 10^11 cm^-3. In this way, the structure and evolution of the isothermal disk surrounding the forming star can be studied at late times without having to implement the numerically costly radiative transfer required by the physics of the opaque core. The mass and magnetic flux accumulating in the forming star arecalculated, as are their effects on the structure & evolution of the surrounding disk. The numerical method of solution first uses an adaptive grid and later, after a central region a few AU in radius becomes opaque, switches to a stationary but nonuniform grid with a central sink cell. It also involves an implicit time integrator, an advective difference scheme that possesses the transportive property, a second-order difference approximation of forces inside a cell, an integral approximation of the gravitational and magnetic fields, and tensor artificial viscosity that permits an accurate investigation of the formation and evolution of shocks in the neutral fluid. 
Konstantinos Tassis and Telemachos Ch. Mouschovias (Departments of   Physics and Astronomy, University of Illinois at Urbana...