  This is the third in a series of papers aimed at developing a practical
time-domain method for self-force calculations in Kerr spacetime. The key
elements of the method are (i) removal of a singular part of the perturbation
field with a suitable analytic "puncture", (ii) decomposition of the
perturbation equations in azimuthal ($m$-)modes, taking advantage of the axial
symmetry of the Kerr background, (iii) numerical evolution of the individual
$m$-modes in 2+1-dimensions with a finite difference scheme, and (iv)
reconstruction of the local self-force from the mode sum. Here we report a
first implementation of the method to compute the gravitational self-force. We
work in the Lorenz gauge, solving directly for the metric perturbation in
2+1-dimensions. The modes $m=0,1$ contain nonradiative pieces, whose
time-domain evolution is hampered by certain gauge instabilities. We study this
problem in detail and propose ways around it. In the current work we use the
Schwarzschild geometry as a platform for development; in a forthcoming
paper---the fourth in the series---we apply our method to the gravitational
self-force in Kerr geometry.
