New method to integrate 2+1 wave equations with Dirac's delta functions as sources

TL;DR

This paper introduces a novel numerical method for solving (2+1)-dimensional wave equations with Dirac delta function sources, relevant for modeling scalar perturbations around Kerr black holes, by analytically removing singularities for improved computation.

Contribution

It presents a new approach to handle singular sources in wave equations by analytically removing singularities, enabling more accurate numerical simulations in black hole perturbation theory.

Findings

Successfully derived a global effective source for wave equations with delta functions.

Demonstrated improved numerical stability and accuracy in simulations.

Applicable to scalar perturbations in Kerr black hole backgrounds.

Abstract

Gravitational perturbations in a Kerr black hole background can not be decomposed into simple tensor harmonics in the time domain. Here, we make the mode decomposition only in the azimuthal direction and discuss the resulting (2+1)-dimensional Klein-Gordon differential equation for scalar perturbations with a two dimensional Dirac's $\delta$-function as a source representing a point particle orbiting a much larger black hole. To make this equation amenable for numerical integrations we explicitly remove analytically the singular behavior of the source and compute a global effective source for the corresponding waveform.