  Although the basic physics of star formation is classical, numerical
simulations have yielded essential insights into how stars form. They show that
star formation is a highly nonuniform runaway process characterized by the
emergence of nearly singular peaks in density, followed by the accretional
growth of embryo stars that form at these density peaks. Circumstellar disks
often form from the gas being accreted by the forming stars, and accretion from
these disks may be episodic, driven by gravitational instabilities or by
protostellar interactions. Star-forming clouds typically develop filamentary
structures, which may, along with the thermal physics, play an important role
in the origin of stellar masses because of the sensitivity of filament
fragmentation to temperature variations. Simulations of the formation of star
clusters show that the most massive stars form by continuing accretion in the
dense cluster cores, and this again is a runaway process that couples star
formation and cluster formation. Star-forming clouds also tend to develop
hierarchical structures, and smaller groups of forming objects tend to merge
into progressively larger ones, a generic feature of self-gravitating systems
that is common to star formation and galaxy formation. Because of the large
range of scales and the complex dynamics involved, analytic models cannot
adequately describe many processes of star formation, and detailed numerical
simulations are needed to advance our understanding of the subject.
