  The numerical evolution of Einstein's field equations in a generic background
has the potential to answer a variety of important questions in physics: from
applications to the gauge-gravity duality, to modelling black hole production
in TeV gravity scenarios, analysis of the stability of exact solutions and
tests of Cosmic Censorship. In order to investigate these questions, we extend
numerical relativity to more general space-times than those investigated
hitherto, by developing a framework to study the numerical evolution of D
dimensional vacuum space-times with an SO(D-2) isometry group for D\ge 5, or
SO(D-3) for D\ge 6.
  Performing a dimensional reduction on a (D-4)-sphere, the D dimensional
vacuum Einstein equations are rewritten as a 3+1 dimensional system with source
terms, and presented in the Baumgarte, Shapiro, Shibata and Nakamura (BSSN)
formulation. This allows the use of existing 3+1 dimensional numerical codes
with small adaptations. Brill-Lindquist initial data are constructed in D
dimensions and a procedure to match them to our 3+1 dimensional evolution
equations is given. We have implemented our framework by adapting the LEAN code
and perform a variety of simulations of non-spinning black hole space-times.
Specifically, we present a modified moving puncture gauge which facilitates
long term stable simulations in D=5. We further demonstrate the internal
consistency of the code by studying convergence and comparing numerical versus
analytic results in the case of geodesic slicing for D=5,6.
