  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.
