  We introduce TIGRESS, a novel framework for multi-physics numerical
simulations of the star-forming interstellar medium (ISM) implemented in the
Athena MHD code. The algorithms of TIGRESS are designed to spatially and
temporally resolve key physical features, including: (1) the gravitational
collapse and ongoing accretion of gas that leads to star formation in clusters,
(2) the explosions of supernovae (SNe) both near their progenitor birth sites
and from runaway OB stars, with time delays relative to star formation
determined by population synthesis, (3) explicit evolution of SN remnants prior
to the onset of cooling, which leads to the creation of the hot ISM, (4)
photoelectric heating of the warm and cold phases of the ISM that tracks the
time-dependent ambient FUV field from the young cluster population, (5)
large-scale galactic differential rotation, which leads to epicyclic motion and
shears out overdense structures, limiting large-scale gravitational collapse,
(6) accurate evolution of magnetic fields, which can be important for vertical
support of the ISM disk as well as angular momentum transport. We present tests
of the newly-implemented physics modules, and demonstrate application of
TIGRESS in a fiducial model representing the Solar neighborhood environment. We
use a resolution study to demonstrate convergence and evaluate the minimum
resolution dx required to correctly recover several ISM properties, including
the star formation rate, wind mass-loss rate, disk scale height, turbulent and
Alfvenic velocity dispersions, and volume fractions of warm and hot phases. For
the Solar neighborhood model, all these ISM properties are converged at dx <=
8pc.
