Computational Methods for the Self-Force in Black Hole Spacetimes 349
convergence) and Lousto [76] (fourth-order convergence). This simple but powerful
idea is at the core of many of the 1C1D finite-difference implementations presented
in the last few years [24,59,79], including the work discussed in Section 4.
Despite such advances, 1C1D particles remain numerically expensive to handle,
because the nonsmoothness associated with them introduces a large scale variance
in the solutions: The l-mode field gradients grow sharply near the particle, and,
moreover, become increasingly more difficult to resolve with larger l (recall the
l-mode gradient is / ` at large l). The mode-sum formula, recall, converges rather
slowly (like 1= l), and so requires one to compute a considerably large number of
modes (typically 20 with even a moderate accuracy goal). This proves to stretch
the limit of what can be achieved today using finite differentiation on a fixed mesh.
Several methods have been proposed to address this problem in the current con-
text. Sopuerta and collaborators [35,36,110,111] explored the use of finite-element
discretization. This technique is particularly powerful in dealing with multi-scale
problems, and, being quasi-spectral, it benefits from an exponential convergence
rate. So far it has been applied successfully for generic orbits in Schwarzschild, and
higher-dimensional implementations (for Kerr studies) are currently being consid-
ered. A related quasi-spectral scheme was recently suggested by Field et al. [51].
Finally, Thornburg [115] very recently developed an adaptive mess refinement al-
gorithm for Lousto–Price’s finite differences scheme (with a global fourth-order
convergence). This was successfully implemented for a scalar charge in a circular
orbit in Schwarzschild, and generalizations are being considered.
Puncture Methods In Kerr spacetime, one no longer benefits from a 1C1D sepa-
rability. The Lorenz-gauge perturbation equations are only separable into azimuthal
m-modes, each a function of t; r; in a 2C1D space. The m modes are not finite
on the worldline, but rather they diverge there logarithmically (see the discussion in
Section II.C of Ref. [11]). Since the 2C1D numerical solutions are truly divergent, a
direct finite-difference treatment becomes problematic. However, since the singular
behavior of the perturbation can be approximated analytically, a simple remedy to
this problem suggests itself.
The idea, which has recently been studied independently by several groups
[11, 77, 116], is to utilize a new perturbation variable for the numerical time evo-
lution (the “residual” field), constructed from the full (retarded) field by subtracting
a suitable function (the “puncture” field), given analytically, which approximates
the singular part of the perturbation well enough that the residual field is (at least)
bounded at the particle. The perturbation equations are then recast with the residual
field as their independent variable, and with a new source term (depending on the
puncture field and its derivatives) which now extends off the worldline but contains
no delta function. The equations are then solved for the residual field in the time
domain, using, for example, standard finite differentiation.
Several variants of this method have been studied and tested with scalar-field
codes in 1C1D [116]and2C1D [11, 77], and also proposed for use in full 3C1D
[69]. The various schemes differ primarily in the way they handle the puncture func-
tion far from the particle: Barack and Golbourn [11] introduce a puncture with a