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Dust grains influence many aspects of star formation, including planet formation and the opacities for radiative transfer, chemistry, and the magnetic field via Ohmic, Hall, as well as ambipolar diffusion. The size distribution of the dust grains is the primary characteristic influencing all these aspects. Grain size increases by coagulation throughout the star formation process. In this work, we describe numerical simulations of protostellar collapse using methods described in earlier papers of this series. We compute the evolution of the grain size distribution from coagulation and the non-ideal magnetohydrodynamics effects self-consistently and at low numerical cost. We find that the coagulation efficiency is mostly affected by the time spent in high-density regions. Starting from sub-micron radii, grain sizes reach more than 100 µm in an inner protoplanetary disk that is only 1000 yr old. We also show that the growth of grains significantly affects the resistivities, while also having an indirect effect on the dynamics and angular momentum of the disk. 

Thermal ionization is a critical process at temperatures T  > 10 3 K, particularly during star formation. An increase in ionization leads to a decrease in nonideal magnetohydrodynamics (MHD) resistivities, which has a significant impact on protoplanetary disks and protostar formation. We developed an extension of the fast computational ionization method presented in our recent paper to include thermal ionization. The model can be used to inexpensively calculate the density of ions and electrons and the electric charge of each size of grains for an arbitrary size distribution. This tool should be particularly useful for the self-consistent calculation of nonideal MHD resistivities in multidimensional simulations, especially of protostellar collapse and protoplanetary disks. 

Dust grains play a major role in many astrophysical contexts. They affect the chemical, magnetic, dynamical, and optical properties of their environment, from galaxies down to the interstellar medium, star-forming regions, and protoplanetary disks. Their coagulation leads to shifts in their size distribution and ultimately to the formation of planets. However, although the coagulation process is reasonably uncomplicated to numerically implement by itself, it is difficult to couple it with multidimensional hydrodynamics numerical simulations because of its high computational cost. We propose here a simple method for tracking the coagulation of grains at far lower cost. Given an initial grain size distribution, the state of the distribution at time t is solely determined by the value of a single variable integrated along the trajectory, independently of the specific path taken by the grains. Although this method cannot account for processes other than coagulation, it is mathematically exact, fast, inexpensive, and can be used to evaluate the effect of grain coagulation in most astrophysical contexts. It is applicable to all coagulation kernels in which local physical conditions and grain properties can be separated. We also describe another method for calculating the average electric charge of grains and the density of ions and electrons in environments that are shielded from radiation fields, given the density and temperature of the gas, the cosmic-ray ionization rate, and the average mass of the ions. The equations we provide are fast to integrate numerically and can be used in multidimensional numerical simulations to self-consistently calculate on the fly the local resistivities that are required to model nonideal magnetohydrodynamics.