Axisymmetric fully spectral code for hyperbolic equations

Abstract

We present a fully pseudo-spectral scheme to solve axisymmetric hyperbolic equations of second order. With the Chebyshev polynomials as basis functions, the numerical grid is based on the Lobbato (for two spatial directions) and Radau (for the time direction) collocation points. The method solves two issues of previous algorithms which were restricted to one spatial dimension, namely, (i) the inversion of a dense matrix and (ii) the acquisition of a sufficiently good initial-guess for non-linear systems of equations. For the first issue, we use the iterative bi-conjugate gradient stabilized method, which we equip with a pre-conditioner based on a singly diagonally implicit Runge-Kutta ("SDIRK"-) method. In this paper, the SDIRK-method is also used to solve issue (ii). The numerical solutions are correct up to machine precision and we do not observe any restriction concerning the time step in comparison with the spatial resolution. As an application, we solve general-relativistic wave equations on a black-hole space-time in so-called hyperboloidal slices and reproduce some recent results available in the literature.

Hyperboloidal slicing approach to quasinormal mode expansions: The Reissner-Nordström case

“…The straight red line is obtained for κ = 0.5 in the "areal radius fixing" gauge, whereas the dotted blue line is for κ = 0.9 in the "Cauchy horizon fixing" gauge. The black straight line corresponds to the explicit time evolution with the code [44]. The inset focus on the match between both methods around τ = 0.…”

“…In fact, the minimal gauge was employed beyond spherical symmetry in [44] in the numerical evolution of the Teukolsky equation in the Kerr spacetime. Even though the qualitative results do not differ from other studies (e.g.…”

Holographic quenches and anomalous transport

“…It would be interesting to make an analogous study with the holographic model [47]. development in the application of spectral methods along the time direction in dynamical scenarios [34].…”

“…Here, we discuss the main features of the method with emphasis on its application for eigenvalue problems and for systems of differential equations with extra parameters. Then, we review the algorithm for the fully spectral code introduced in [34].…”

“…The most natural way to solve (2.25) is to numerically integrate it in time. In this work, the time evolution is performed with the fully spectral code 2 introduced in [34] (the numerical details are described in appendix A). We use this highly accurate numerical method to obtain the full response of the current operator.…”