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[abridged] The inspiral of a stellar compact object into a massive black holeis one of the main sources of gravitational waves for the future space-basedLaser Interferometer Space Antenna. We expect to be able to detect and analyzemany cycles of these slowly inspiraling systems. To that end, the use of veryprecise theoretical waveform templates in the data analysis is required. Tobuild them we need to have a deep understanding of the gravitationalbackreaction mechanism responsible for the inspiral. The self-force approachdescribes the inspiral as the action of a local force that can be obtained fromthe regularization of the perturbations created by the stellar compact objecton the massive black hole geometry. In this paper we extend a new time-domaintechnique for the computation of the self-force from the circular case to thecase of eccentric orbits around a non-rotating black hole. The main idea behindour 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 convergenceproperties for the numerical computations. This formulation is implemented byusing a pseudospectral collocation method for the spatial discretization,combined with a Runge Kutta algorithm for the time evolution. We presentresults from several simulations of eccentric orbits in the case of a scalarcharged particle around a Schwarzschild black hole. In particular, we show theconvergence of the method and its ability to resolve the field and itsderivatives across the particle location. Finally, we provide numerical valuesof the self-force for different orbital parameters.

Pseudospectral Collocation Methods for the Computation of the Self-Force on a Charged Particle: Generic Orbits around a Schwarzschild Black Hole