  Motivated by the desire to understand the leading order nonlinear
gravitational wave interactions around arbitrarily rapidly rotating Kerr black
holes, we describe a numerical code designed to compute second order vacuum
perturbations on such spacetimes. A general discussion of the formalism we use
is presented in (arXiv:2008.11770); here we show how we numerically implement
that formalism with a particular choice of coordinates and tetrad conditions,
and give example results for black holes with dimensionless spin parameters
$a=0.7$ and $a=0.998$. We first solve the Teukolsky equation for the linearly
perturbed Weyl scalar $\Psi_4^{(1)}$, followed by direct reconstruction of the
spacetime metric from $\Psi_4^{(1)}$, and then solve for the dynamics of the
second order perturbed Weyl scalar $\Psi_4^{(2)}$. This code is a first step
toward a more general purpose second order code, and we outline how our basic
approach could be further developed to address current questions of interest,
including extending the analysis of ringdown in black hole mergers to before
the linear regime, exploring gravitational wave "turbulence" around
near-extremal Kerr black holes, and studying the physics of extreme mass ratio
inspiral.
