Ambipolar diffusion, interstellar dust, and the formation of cloud cores and protostars. II. 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 ≃3 × 10⁹ cm^-3. 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 gravitational 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 ≃10 AU at densities ≃10⁹ cm^-3 with only about 100 points to cover the radial extent (typically ≃4.3 pc) of the entire cloud. The accuracy of the method is discussed.