  We present a fully relativistic numerical method for the study of
cosmological problems using the Baumgarte-Shapiro-Shibata-Nakamura formalism on
a dynamical Friedmann-Lema\^itre-Robertson-Walker background. This has many
potential applications including the study of the growth of structures beyond
the linear regime. We present one such application by reproducing the
Lema\^itre-Tolman-Bondi solution for the collapse of pressureless matter with
arbitrary lapse function. The regular and smooth numerical solution at the
center of coordinates proceeds in a natural way by relying on the Partially
Implicit Runge-Kutta algorithm described in Montero and Cordero-Carri\'on
[arXiv:1211.5930]. We generalize the usual radiative outer boundary condition
to the case of a dynamical background and show the stability and convergence
properties of the method in the study of pure gauge dynamics on a de Sitter
background.
