Bernstein spectral method for quasinormal modes and other eigenvalue problems

Abstract

Spectral methods are now common in the solution of ordinary differential eigenvalue problems in a wide variety of fields, such as in the computation of black hole quasinormal modes. Most of these spectral codes are based on standard Chebyshev, Fourier, or some other orthogonal basis functions. In this work we highlight the usefulness of a relatively unknown set of non-orthogonal basis functions, known as Bernstein polynomials, and their advantages for handling boundary conditions in ordinary differential eigenvalue problems. We also report on a new user-friendly package, called SpectralBP, that implements Berstein-polynomial-based spectral routines for eigenvalue problems. We demonstrate the functionalities of the package by applying it to a number of model problems in quantum mechanics and to the problem of computing scalar and gravitational quasinormal modes in a Schwarzschild background. We validate our code against some known results and achieve excellent agreement. Compared to continued-fraction or series methods, global approximation methods are particularly well-suited for computing the algebraically special modes for gravitational perturbations of the Schwarzschild geometry. We demonstrate this by reporting the most accurate numerical calculation of these modes to date, achieved with only modest resources.

Bernstein spectral method for quasinormal modes of a generic black hole spacetime and application to instability of dilaton-de Sitter solution

“…First, we will discuss the Bernstein spectral method for asymptotically de Sitter spacetimes, when the purely outgoing wave is imposed at the de Sitter horizon. Following [14], we introduce the compact coordinate, which is defined as follows:…”

“…The method which is efficient for finding purely imaginary, that is, non-oscillatory, quasinormal modes is a spectral method based on the Bernstein polynomial nonorthogonal basis functions [14]. As was shown in [14] while reproducing only several first overtones during a short computing time, it perfectly detects the algebraically special mode of the Schwarzschild black hole with high accuracy, which occurs at n = 8, where n is the overtone number. Another spectral method based on the Chebyshev polynomials was used to obtain a family of the purely imaginary modes of the Schwarzschild-de Sitter black holes [15].…”

“…Here, using the initial code of [14] as a basis, we develop it to be even more user-friendly and solve two main problems related to it:…”

Electromagnetic quasinormal modes of Schwarzschild-anti-de Sitter black holes: bifurcations, spectral similarity, and exact solutions in the large black hole limit

“…The code we have used is implemented in a Mathematica package we call SpectralBP. This Bernstein spectral implementation and all its unappreciated advantages are fully described in [28]. It is publicly available and may be found at https://github.com/slashdotfield/SpectralBP.…”

“…Where the numerical results are concerned, a novel pseudospectral method was used using the Bernstein polynomial basis. The code we used is distributed as a Mathematica package we call SpectralBP [28].…”