Matrix method and the suppression of Runge's phenomenon

Dear Editor,
We are very grateful for the referee’s report on our manuscript. The questions, comments, and suggestions have
been constructive in improving our work. We have revised the manuscript accordingly. In the revised manuscript, we
adequately cite the relevant references regarding the manuscript as suggested and elaborate further on the numerical
scheme. Our answers and the corresponding modifications to the manuscript are summarized below.
REVIEWER REPORT:
Strengths
The article successfully addressed an important technical aspect of the matrix method related to the Runge’s phenomenon.
Weaknesses
There is an extensive discussion about the method, the QNMs and the somehow of academic only interesting
“pseudospectra”. But even thought the article cited dozens of publications it missed some of the very first articles
using the matrix method. Actually, in computational physics this is typically referred as “Numerov’s method”.
Report
This article is part of a series written by some of the authors over an 8-year time period. They were successful
in promoting the use of the matrix method for studying quasi-normal modes. In the article, they demonstrated the
potential of this method in addressing the issue of Runge’s phenomenon. This result alone is sufficient to make the
paper interesting to readers. However, the authors acknowledge that there is currently no solution for cases where
the effective potential is discontinuous. I consider this to be a minor weakness since these types of potentials have
limited importance for astrophysics and the instabilities reported are mostly of academic interest. Overall, the article
presents original research and, in my opinion, meets the criteria for publication in the journal.
Requested changes
While reading the review article by Kokkotas & Schmidt in Living Reviews (1999), I came across the first ever uses
of the matrix method in calculating QNMs:
** of black holes:
– Bhabani Majumdar and N. Panchapakesan PRD, 40, 2568 (1989)
– E.W. Leaver PRD 41, 2986 (1990)
** of neutron stars
– K.D. Kokkotas and J. Ruoff A&A 365, 565 (2001)
– S. Boutloukos and H-P. Nollert PRD 75, 043007 (2007)
I recommend to include these characteristic references in the introduction.
We sincerely thank the referee for the recognition and calling our attention to the relevant studies. Indeed, as the
referee pointed out, the approach, essentially based on the idea of expanding the waveform and its derivatives on all
the grid points, was pioneered by other authors earlier. Accordingly, the manuscript has been updated to include
proper references and additional discussions.
By studying the relevant references, we observe a minor subtlety: If one expands the waveform around a given
point, the master equation will give rise to an iteration relation between the expansion coefficients. The latter can
readily be written down as a matrix equation, as in [15, 37, 38]. Although similar, this is not entirely equivalent to a
similar approach, where the expansions are carried out at every grid point. The resulting discretized master equation
and its boundary condition can also be reformulated in a matrix form, which was first performed in Refs. [44, 45]
for the star QNMs. The later is referred to as matrix method in the present study. For both cases, the rank of
the matrix in question is more or less the number of grid points. However, for the former, typically, the matrix is only populated on its diagonal and sub-diagonal elements, while the matrix is mostly dense for the latter. This indicates
that the second matrix potentially carries more information owing to the expansion on the remaining grid points.
The Introduction of the manuscript has been modified to read: “As mentioned above, one of the most accurate
numerical schemes to date is the continued fraction method. While appropriately taken account the asymptotical
waveform, such an approach expands the waveform at a given coordinate [15, 37, 38]. As a result, the master equation
typically gives rise to a three or four-term iterative relation between the expansion coefficients, which can be
expressed in a mostly diagonal matrix form. The problem for the QNMs is thus effectively solved by using the Hill’s
determinant [39, 40] or Numerov’s method [41, 42]. Following this line of thought, instead of a given position, one may
discretize the entire spatial domain and perform the expansions of the waveform on the entire grid [43]. Subsequently,
the master equation can also be formulated into a mostly dense matrix equation, and the QNM problem is reiterated
as an algebraic nonlinear equation for the complex frequencies. Such an approach was employed to evaluate QNMs of
compact stars [44, 45] pioneered by Kokkotas, Ruoff, Boutloukos, and Nollert and recently promoted by Jansen [46].
Some of us have further pursued the idea and dubbed the approach as matrix method [43, 47–52]. Besides the metrics
with spherical symmetry [47], the method can be applied to black hole spacetimes with axial symmetry [48] and
system composed of coupled degrees of freedom [53].”
We also added Kokkotas and Schmidt’s review paper on QNM to the reference. The revised sentence reads: “The
black hole quasinormal modes (QNMs) carry essential information on one of the most crucial predictions of Einstein’s
general relativity. These distinctive dissipative oscillations bear intrinsic properties of the peculiar spacetime
region [1–4]. ”
Again, we appreciate the referees’ valuable comments and suggestions, based on which the manuscript has improved.
We hope that the revised version meets the standards for publication in SciPost Physics Core.
With best regards,
Shui-Fa Shen et al.