Ambipolar diffusion, interstellar dust, and the formation of cloud cores and protostars. 2: Numerical method of solution

The problem of the self-initiated (due to ambipolar diffusion) formation and contraction of protostellar fragments in model molecular clouds was formulated in a previous paper in axisymmetric geometry, taking advantage of a recent result by Fiedler and Mouschovias; namely, that balance of forces along magnetic field lines is rapidly established and maintained even during dynamic contraction (i.e., collapse) perpendicular to the field lines, at least up to central densities approximately 3 x 109/cc. We describe a method for solving the reduced, multifluid, nonlinear, magnetohydrodynamic equations that govern the time evolution of such model clouds; algebraic equations for the equilibrium abundances of charged species are solved at each time step. The method consists of 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 gravitaional and magnetic fields; and an adaptive mesh capable of reliably following the formation and evolution of cloud cores and of resolving length scales smaller than approximately 10 AU at densities approximately 109/cc with only about 100 points to cover the radial extent (typically approximately 4.3 pc) of the entire cloud. The accuracy of the method is discussed.