The computation of the self-force constitutes one of the main challenges for the construction of precise theoretical waveform templates in order to detect and analyze extreme-mass-ratio inspirals with the future space-based gravitational-wave observatory LISA. Since the number of templates required is quite high, it is important to develop fast algorithms both for the computation of the self-force and the production of waveforms. In this paper, we show how to tune a recent time-domain technique for the computation of the self-force, what we call the particle without particle scheme, in order to make it very precise and at the same time very efficient. We also extend this technique in order to allow for highly eccentric orbits.
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