  We describe a grid-based numerical method for 3D hydrodynamic cosmological
simulations which is adaptive in space and time and combines the best features
of higher order--accurate Godunov schemes for Eulerian hydrodynamics with
adaptive particle--mesh methods for collisionless particles. The basis for our
method is the structured adaptive mesh refinement (AMR) algorithm of Berger &
Collela (1989), which we have extended to cosmological hydro + N-body
simulations. The resulting multiscale hybrid method is a powerful alternative
to particle-based methods in current use. The choices we have made in
constructing this algorithm are discussed, and its performance on the Zeldovich
pancake test problem is given. We present a sample application of our method to
the problem of first structure formation. We have achieved a spatial dynamic
range $L_{box}/\Delta x > 250,000$ in a 3D multispecies gas + dark matter
calculation, which is sufficient to resolve the formation of primordial
protostellar cloud cores starting from linear matter fluctuations in an
expanding FRW universe.
