Astrophysical and cosmological structure formation are challenging problems because they involve dynamical and hydrodynamical processes that can span a large range in scale, mass, and energy. Hydrodynamic and N-body simulations are powerful tools with which to solve the nonlinear physics, and their continuing development and application is the focus of this thesis.

I present a new approach to Eulerian computational fluid dynamics that is designed to work at high Mach numbers encountered in astrophysical simulations. The Eulerian conservation equations are solved in an adaptive frame moving with the fluid where Mach numbers are minimized. The Moving Frame code separately tracks local and bulk flow components, allowing thermodynamic variables to be accurately calculated in both subsonic and supersonic fluid.

An out-of-core hydrodynamic code has been developed for high resolution cosmological simulations. Out-of-core computation refers to the technique of using disk space as virtual memory and transferring data in and out of main memory at high I/O bandwidth. The code is based on a two-level mesh scheme where short-range physics is solved on a high-resolution, localized mesh while long-range physics is captured on a lower resolution, global mesh.

This thesis includes the first astrophysical application of Eulerian hydrodynamic simulations to model the formation of blue stragglers through stellar mergers. The off-axis collision of equal mass stars produces a single merger remnant. The merger of n = 3 polytropes results in substantial chemical mixing throughout the remnant, while the merger of realistic M = 0.8M_solar main sequence stars produces significant mixing only outside of the core.

The Out-of-core Hydro code is applied to running the largest Eulerian hydrodynamic simulation to date for studying the thermal history of the high redshift 3 ≤ z ≤ 7 intergalactic medium. The temperature-density and gas-dark matter density relations, as well as the scatter in these relations, are robustly quantified. Reionization and shock heating are observed to influence the temperature of the photoionized gas.

Furthermore, a parallel particle-mesh N-body code is applied to simulating the clustering of dark matter halos. The PMFAST simulations show that that several bias parameters are consistent with being scale-invariant, a useful property for doing cosmology with galaxy clustering.