An E cient Pseudospectral Method for the Computation of the Self-force on a Charged Particle: Circular Geodesics around a Schwarzschild Black Hole

The description of the inspiral of a stellar-mass compact object into a massive black hole sitting at a galactic centre is a problem of major relevance for the future space-based gravitational-wave observatory LISA (Laser Interferometer Space Antenna), as the signals from these systems will be buried in the data stream and accurate gravitational-wave templates will be needed to extract them. The main di culty in describing these systems lies in the estimation of the gravitational e ects of the stellar-mass compact object on his own trajectory around the massive black hole, which can be modeled as the action of a local force, the self-force. In this paper, we present a new timedomain numerical method for the computation of the self-force in a simpli ed model consisting of a charged scalar particle orbiting a nonrotating black hole. We use a multi-domain framework in such a way that the particle is located at the interface between two domains so that the presence of the particle and its physical e ects appear only through appropriate boundary conditions. In this way we eliminate completely the presence of a small length scale associated with the need of resolving the particle. This technique also avoids the problems associated with the impact of a low di erentiability of the solution in the accuracy of the numerical computations. The spatial discretization of the eld equations is done by using the pseudospectral collocation method and the time evolution, based on the method of lines, uses a Runge-Kutta solver. We show how this special framework can provide very e cient and accurate computations in the time domain, which makes the technique amenable for the intensive computations required in the astrophysically-relevant scenarios for LISA.