A procedure to obtain the SF directly from the Weyl scalars through direct regularization of the latter has been suggested recently, but has not yet been implemented for orbiting particles [13].

One may consider an alternative puncture function, obtained by replacing $2(1-\cos \delta \phi )\to {sin}^2\delta \phi$ in Eq. (28). This has the advantage that the odd $m$-modes of ${\Phi }_{\mathrm{P}}$ and $S_{\mathrm{R}}$ can be written in terms of elementary functions. The disadvantage is that, to avoid the singularity of this alternative ${\Phi }_{\mathrm{P}}$ at $\rho =0$, $\delta \phi =\pi$, one has to introduce a cutoff on ${\Phi }_{\mathrm{P}}$ at some $|\delta \phi |<\pi$, which then generates distributional contributions to the source modes $S_{\mathrm{R}}^m$, complicating their form considerably.

This can be seen by considering the volume integral of $\square {\Phi }_{\mathrm{R}}$ over a small 3-ball (in $\Sigma$) surrounding the particle, at the limit where the radius of the ball tends to zero. Using the Gauss theorem, this can be converted to a surface integral of ${\Phi }_{\mathrm{R},\alpha }$ over the 2-sphere. By virtue of Eq. (35) we have that ${\Phi }_{\mathrm{R}}$ is bounded at the particle, and that the gradient ${\Phi }_{\mathrm{R},\alpha }$ can at most diverge as $\sim 1/{ϵ}_0$ there. Hence, the surface integral of ${\Phi }_{\mathrm{R},\alpha }$ vanishes as the radius of the 2-sphere is taken to zero, implying $\square {\Phi }_{\mathrm{R}}$ (and hence also $S_{\mathrm{R}}$) contains no Dirac deltas.

The relation between local and global errors in the scheme (61) can be explained as follows: At each grid point, the field accumulates local errors from $\propto h^{-2}$ points belonging to the same $\theta =\mathrm{const}$ slice. [Note that the leading-order contribution to ${\Psi }_1^m$ in Eq. (61) comes from points 2–4, which lie on the same $\theta =\mathrm{const}$ slice as point 1.] Assuming the local $\mathcal{O}(h^4)$ errors are not strongly correlated, they accumulate to give a global error of $\mathcal{O}(h^{-2}\times h^4)=\mathcal{O}(h^2)$.