Within the next decade the Laser Interferometer Space Antenna (LISA) is due to be launched, providing the opportunity to extract physics from stellar objects 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 available observational methods, these sources can \textit{only} be simulated using an accurate computation of the gravitational self-force. Whereas the field has seen outstanding progress in the frequency domain, metric reconstruction and self-force calculations are still an open challenge in the time domain. Such computations would not only further corroborate frequency domain calculations and models, but also allow for full self-consistent evolution of the orbit under the effect of the self-force. Given we have \textit{a priori} information about the local structure of the discontinuity at the particle, we will show how to construct discontinuous spatial and temporal discretisations by operating on discontinuous Lagrange and Hermite interpolation formulae and hence recover higher order accuracy. In this work we demonstrate how this technique in conjunction with well-suited gauge choice (hyperboloidal slicing) and numerical (discontinuous collocation with time symmetric) methods can provide a relatively simple method of lines numerical algorithm to the problem. This is the first of a series of papers studying the behaviour of a point-particle prescribing circular geodesic motion in Schwarzschild in the \textit{time domain}. In this work we describe the numerical machinery necessary for these computations and show not only our work is capable of highly accurate flux radiation measurements but it also shows suitability for evaluation of the necessary field and it's derivatives at the particle limit.

翻译：摘要：未来十年内，激光干涉空间天线（LISA）计划发射，这将为从诸如《极端质量比旋近》（EMRIs）等恒星天体与系统中提取物理信息提供机遇——这些信号是地面干涉仪和脉冲星计时阵列（PTA）无法探测的。不同于当前观测方法所能捕获的先前源，这些源《只能》通过引力自力的精确计算进行模拟。尽管该领域在频域方面取得了显著进展，但度规重构与自力计算在时域中仍是开放性挑战。此类计算不仅能进一步验证频域计算结果与模型，还可实现轨道在自力作用下的完全自洽演化。鉴于我们掌握粒子处间断局部结构的《先验》信息，本文将展示如何通过操作不连续拉格朗日和埃尔米特插值公式来构造空间与时间的不连续离散化，进而恢复高阶精度。本研究证明，该技术与恰当的规范选择（双曲层切片）及数值方法（时间对称型不连续配置法）相结合，可为该问题提供一种相对简单的直线法数值算法。本文是系列论文的首篇，旨在研究施瓦西时域中沿圆测地线运动的点粒子行为。文中描述了此类计算所需的数值机制，并表明我们的工作不仅能实现高精度的辐射通量测量，还适用于评估粒子极限处的必要场量及其导数。