  We present a first numerical implementation of a new scheme by Pound et al.
that enables the calculation of the gravitational self-force in Kerr spacetime
from a reconstructed metric-perturbation in a radiation gauge. The numerical
task of the metric reconstruction essentially reduces to solving the fully
separable Teukolsky equation, rather than having to tackle the linearized
Einstein's equations themselves. The method offers significant computational
saving compared to existing methods in the Lorenz gauge, and we expect it to
become a main workhorse for precision self-force calculations in the future.
Here we implement the method for circular orbits on a Schwarzschild background,
in order to illustrate its efficacy and accuracy. We use two independent
methods for solving the Teukolsky equation, one based on a direct numerical
integration, and the other on the analytical approach of Mano, Suzuki, and
Takasugi. The relative accuracy of the output self-force is at least $10^{-7}$
using the first method, and at least $10^{-9}$ using the second; the two
methods agree to within the error bars of the first. We comment on the relation
to a related approach by Shah et al., and discuss foreseeable applications to
more generic orbits in Kerr spacetime.
