  A variety of gravitational dynamics problems in asymptotically anti-de Sitter
(AdS) spacetime are amenable to efficient numerical solution using a common
approach involving a null slicing of spacetime based on infalling geodesics,
convenient exploitation of the residual diffeomorphism freedom, and use of
spectral methods for discretizing and solving the resulting differential
equations. Relevant issues and choices leading to this approach are discussed
in detail. Three examples, motivated by applications to non-equilibrium
dynamics in strongly coupled gauge theories, are discussed as instructive test
cases. These are gravitational descriptions of homogeneous isotropization,
collisions of planar shocks, and turbulent fluid flows in two spatial
dimensions.
