  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 difficulty in describing these systems lies in the
estimation of the gravitational effects 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
time-domain numerical method for the computation of the self-force in a
simplified 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 effects 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 differentiability of
the solution in the accuracy of the numerical computations. The spatial
discretization of the field 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
efficient and accurate computations in the time domain, which makes the
technique amenable for the intensive computations required in the
astrophysically-relevant scenarios for LISA.

General Relativity and Quantum Cosmology;High Energy Astrophysical Phenomena