  If a small "particle" of mass $\mu M$ (with $\mu \ll 1$) orbits a
Schwarzschild or Kerr black hole of mass $M$, the particle is subject to an
$\O(\mu)$ radiation-reaction "self-force". Here I argue that it's valuable to
compute this self-force highly accurately (relative error of $\ltsim 10^{-6}$)
and efficiently, and I describe techniques for doing this and for obtaining and
validating error estimates for the computation. I use an
adaptive-mesh-refinement (AMR) time-domain numerical integration of the
perturbation equations in the Barack-Ori mode-sum regularization formalism;
this is efficient, yet allows easy generalization to arbitrary particle orbits.
I focus on the model problem of a scalar particle in a circular geodesic orbit
in Schwarzschild spacetime.
  The mode-sum formalism gives the self-force as an infinite sum of regularized
spherical-harmonic modes $\sum_{\ell=0}^\infty F_{\ell,\reg}$, with
$F_{\ell,\reg}$ (and an "internal" error estimate) computed numerically for
$\ell \ltsim 30$ and estimated for larger~$\ell$ by fitting an asymptotic
"tail" series. Here I validate the internal error estimates for the individual
$F_{\ell,\reg}$ using a large set of numerical self-force computations of
widely-varying accuracies. I present numerical evidence that the actual
numerical errors in $F_{\ell,\reg}$ for different~$\ell$ are at most weakly
correlated, so the usual statistical error estimates are valid for computing
the self-force. I show that the tail fit is numerically ill-conditioned, but
this can be mostly alleviated by renormalizing the basis functions to have
similar magnitudes.
  Using AMR, fixed mesh refinement, and extended-precision floating-point
arithmetic, I obtain the (contravariant) radial component of the self-force for
a particle in a circular geodesic orbit of areal radius $r = 10M$ to within
$1$~ppm relative error.
