  In this work numerical methods for solving Einstein's equations are developed
and applied to the study of inhomogeneous cosmological models.
  A two-dimensional computer code is described which implements two advanced
numerical methods: LeVeque's multi-dimensional high-resolution integration
scheme which allows accurate evolution of solutions containing discontinuities
or steep gradients, and an adaptive mesh refinement (AMR) algorithm which
enables the local resolution of a simulation to vary dynamically in response to
the behaviour of the evolved solution.
  A family of hyperbolic formulations of the Einstein equations is derived by
generalization of an evolution system proposed by Frittelli and Reula, and
numerical solutions produced using these formulations are compared to solutions
produced using alternative reductions of the evolutions equations. Properties
of the harmonic time slicing condition are also investigated, and analytic and
numerical results concerning the formation of coordinate singularities are
presented.
  Numerical simulations are performed of planar cosmologies, described using
Gowdy's reduction of the Einstein equations, and U(1)-symmetric cosmologies,
described using Moncrief's reduction of the Einstein equations, with the
spacetimes in both cases being vacuum. Numerical studies follow up on the work
of Berger, Moncrief and co-workers, with attention being focused on the
small-scale features that develop in the models and the behaviour of linear and
nonlinear gravitational waves.
