  We present STARFORGE (STAR FORmation in Gaseous Environments): a new
numerical framework for 3D radiation MHD simulations of star formation that
simultaneously follow the formation, accretion, evolution, and dynamics of
individual stars in massive giant molecular clouds (GMCs) while accounting for
stellar feedback, including jets, radiative heating and momentum, stellar
winds, and supernovae. We use the GIZMO code with the MFM mesh-free Lagrangian
MHD method, augmented with new algorithms for gravity, timestepping, sink
particle formation and accretion, stellar dynamics, and feedback coupling. We
survey a wide range of numerical parameters/prescriptions for sink formation
and accretion and find very small variations in star formation history and the
IMF (except for intentionally-unphysical variations). Modules for
mass-injecting feedback (winds, SNe, and jets) inject new gas elements
on-the-fly, eliminating the lack of resolution in diffuse feedback cavities
otherwise inherent in Lagrangian methods. The treatment of radiation uses
GIZMO's radiative transfer solver to track 5 frequency bands (IR, optical, NUV,
FUV, ionizing), coupling direct stellar emission and dust emission with gas
heating and radiation pressure terms. We demonstrate accurate solutions for
SNe, winds, and radiation in problems with known similarity solutions, and show
that our jet module is robust to resolution and numerical details, and agrees
well with previous AMR simulations. STARFORGE can scale up to massive ($>10^5
M_\odot $) GMCs on current supercomputers while predicting the stellar
($\gtrsim 0.1 M_\odot$) range of the IMF, permitting simulations of both high-
and low-mass cluster formation in a wide range of conditions.
