There is a wide variety of interesting and challenging problems to be
solved in this field which require a correspondingly wide variety of
computational techniques and machine capabilities. A partial list
of possible applications includes multidimensional stellar evolution-from
star formation to the death of the star in a supernova explosion,
cosmological simulations, galaxy formation, accretion disk structure
and evolution, and the formation of supersonic jets. There are
several reasons why these problems will require (at least) a PetaFLOPS
computer in order to obtain reliable solutions. One is the need to
simulate enormous ranges in length and time scales, requiring very
large computational grids (possibly using adaptive mesh techniques)
and huge numbers of time steps. Very large grids also are required
to reduce the numerical dissipation in order to represent accurately
the extremely low viscosity of most astrophysical gases. This is
particularly important in regimes where turbulence and convective
motions are important. Most problems will require fully three-dimensional
simulations. Calculations containing grid points or more will
have to be done routinely. Many problems will require complex physics
calculations at each grid point, such as nuclear reaction networks
involving perhaps hundreds of species and thousands of reactions,
detailed equation of state calculations, complex radiative transfer,
and solution of elliptic equations to calculate, for example, self
gravity. Calculations of galaxy formation will require coupling
of a gas dynamics code to an N-body code to follow the motions of
individual stars. Finally, it will be necessary to run each simulation
many times to fully understand the physics of
the object(s) being studied. These additional simulations will involve
parameter studies, modifying the initial conditions, using different
input physics, and so on.