  We study rapidly accreting, gravitationally unstable disks with a series of
global, three dimensional, numerical experiments using the code ORION. In this
paper we conduct a numerical parameter study focused on protostellar disks, and
show that one can predict disk behavior and the multiplicity of the accreting
star system as a function of two dimensionless parameters which compare the
disk's accretion rate to its sound speed and orbital period. Although
gravitational instabilities become strong, we find that fragmentation into
binary or multiple systems occurs only when material falls in several times
more rapidly than the canonical isothermal limit. The disk-to-star accretion
rate is proportional to the infall rate, and governed by gravitational torques
generated by low-m spiral modes. We also confirm the existence of a maximum
stable disk mass: disks that exceed ~50% of the total system mass are subject
to fragmentation and the subsequent formation of binary companions.
