A fourth-order indirect integration method for black hole perturbations: even modes

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 wavefunction and its derivatives and (iii) the values of the wavefunction 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 the object of a forthcoming publication.