Abstract
The long-time evolution of extreme mass-ratio inspiral systems requires
minimal phase and dispersion errors to accurately compute far-field waveforms,
while high accuracy is essential near the smaller black hole (modeled as a
Dirac delta distribution) for self-force computations. Spectrally accurate
methods, such as nodal discontinuous Galerkin (DG) methods, are well suited for
these tasks. Their numerical errors typically decrease as $\propto (\Delta
x)^{N+1}$, where $\Delta x$ is the subdomain size and $N$ is the polynomial
degree of the approximation. However, certain DG schemes exhibit
superconvergence, where truncation, phase, and dispersion errors can decrease
as fast as $\propto (\Delta x)^{2N+1}$. Superconvergent numerical solvers are,
by construction, extremely efficient and accurate. We theoretically demonstrate
that our DG scheme for the scalar Teukolsky equation with a distributional
source is superconvergent, and this property is retained when combined with the
hyperboloidal layer compactification technique. This ensures that waveforms,
total energy and angular-momentum fluxes, and self-force computations benefit
from superconvergence. We empirically verify this behavior across a family of
hyperboloidal layer compactifications with varying degrees of smoothness.
Additionally, we show that self-force quantities for circular orbits, computed
at the point particle's location, also exhibit a certain degree of
superconvergence. Our results underscore the potential benefits of numerical
superconvergence for efficient and accurate gravitational waveform simulations
based on DG methods.