Fourth order indirect integration method for black hole perturbations: even modes

TL;DR

This paper introduces a fourth order finite element time domain algorithm for simulating even mode black hole perturbations caused by a plunging particle, improving accuracy without directly handling source terms.

Contribution

A novel fourth order integration scheme for black hole perturbations that avoids direct source term computation, enhancing numerical stability and accuracy.

Findings

Effective for non-rotating black hole perturbations

Applicable to particles in plunging, circular, and eccentric orbits

Achieves higher order accuracy in time domain simulations

Abstract

On the basis of a recently proposed strategy of finite element integration in time domain for partial differential equations with a singular source term, we present a fourth order algorithm for non-rotating black hole perturbations in the Regge-Wheeler gauge. Herein, we address even perturbations induced by a particle plunging in. The forward time value at the upper node of the $(r^*,t)$ grid cell is obtained by an algebraic sum of i) the preceding node values of the same cell, ii) analytic expressions, related to the jump conditions on the wave function and its derivatives, iii) the values of the wave function at adjacent cells. In this approach, the numerical integration does not deal with the source and potential terms directly, for cells crossed by the particle world line. This scheme has also been applied to circular and eccentric orbits and it will be object of a forthcoming publication.