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Within the next decade the Laser Interferometer Space Antenna (LISA) is dueto be launched, providing the opportunity to extract physics from stellarobjects and systems, such as \textit{Extreme Mass Ratio Inspirals}, (EMRIs)otherwise undetectable to ground based interferometers and Pulsar Timing Arrays(PTA). Unlike previous sources detected by the currently availableobservational methods, these sources can \textit{only} be simulated using anaccurate computation of the gravitational self-force. Whereas the field hasseen outstanding progress in the frequency domain, metric reconstruction andself-force calculations are still an open challenge in the time domain. Suchcomputations would not only further corroborate frequency domain calculationsand models, but also allow for full self-consistent evolution of the orbitunder the effect of the self-force. Given we have \textit{a priori} informationabout the local structure of the discontinuity at the particle, we will showhow to construct discontinuous spatial and temporal discretisations byoperating on discontinuous Lagrange and Hermite interpolation formulae andhence recover higher order accuracy. In this work we demonstrate how thistechnique in conjunction with well-suited gauge choice (hyperboloidal slicing)and numerical (discontinuous collocation with time symmetric) methods canprovide a relatively simple method of lines numerical algorithm to the problem.This is the first of a series of papers studying the behaviour of apoint-particle prescribing circular geodesic motion in Schwarzschild in the\textit{time domain}. In this work we describe the numerical machinerynecessary for these computations and show not only our work is capable ofhighly accurate flux radiation measurements but it also shows suitability forevaluation of the necessary field and it's derivatives at the particle limit.

Hyperboloidal discontinuous time-symmetric numerical algorithm with higher order jumps for gravitational self-force computations in the time domain