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：在未来十年内，激光干涉空间天线（LISA）预计将发射，为研究恒星天体及系统（例如极端质量比旋进系统，EMRIs）的物理过程提供机会，这类系统是地基干涉仪和脉冲星计时阵列（PTA）无法探测的。与当前观测方法所检测到的先前源不同，这些源只能通过精确计算引力自力的方式来模拟。尽管频域研究已取得显著进展，但在时域中，度规重构和自力计算仍然是开放性的挑战。这类计算不仅能进一步验证频域计算与模型，还能实现轨道在自力影响下的完全自洽演化。鉴于我们预先掌握粒子处间断局部结构的信息，本文将通过操作不连续拉格朗日和埃尔米特插值公式，构建空间与时间的不连续离散化方案，从而恢复高阶精度。本研究展示了如何将该技术与合适的规范选择（双曲分层）及数值方法（非连续配点法与时间对称法）相结合，为问题提供一种相对简便的线法数值算法。这是系列论文中的第一篇，旨在研究施瓦西时空中点粒子沿圆形测地线运动的时域行为。本文描述了这些计算所需的数值机制，并表明我们的工作不仅能高精度测量辐射通量，还适用于评估粒子极限处所需场及其导数的计算。