  [abridged] The inspiral of a stellar compact object into a massive black hole
is one of the main sources of gravitational waves for the future space-based
Laser Interferometer Space Antenna. We expect to be able to detect and analyze
many cycles of these slowly inspiraling systems. To that end, the use of very
precise theoretical waveform templates in the data analysis is required. To
build them we need to have a deep understanding of the gravitational
backreaction mechanism responsible for the inspiral. The self-force approach
describes the inspiral as the action of a local force that can be obtained from
the regularization of the perturbations created by the stellar compact object
on the massive black hole geometry. In this paper we extend a new time-domain
technique for the computation of the self-force from the circular case to the
case of eccentric orbits around a non-rotating black hole. The main idea behind
our scheme is to use a multidomain framework in which the small compact object,
described as a particle, is located at the interface between two subdomains.
Then, the equations at each subdomain are homogeneous wave-type equations,
without distributional sources. In this particle-without-particle formulation,
the solution of the equations is smooth enough to provide good convergence
properties for the numerical computations. This formulation is implemented by
using a pseudospectral collocation method for the spatial discretization,
combined with a Runge Kutta algorithm for the time evolution. We present
results from several simulations of eccentric orbits in the case of a scalar
charged particle around a Schwarzschild black hole. In particular, we show the
convergence of the method and its ability to resolve the field and its
derivatives across the particle location. Finally, we provide numerical values
of the self-force for different orbital parameters.
