Cosmological N-body simulations are done on massively parallel computers. This necessitates the use of simple time integrators, and, additionally, of mesh-grid approximations of the potentials. Recently, Adamek et al. (2015); Barrera-Hinojosa et al. (2019) have developed general relativistic N-body simulations to capture relativistic effects mainly for cosmological purposes. We therefore ask whether, with the available technology, relativistic effects like perihelion advance can be detected numerically to a relevant precision. We first study the spurious perihelion shift in the Kepler problem, as a function of the integration method used, and then as a function of an additional interpolation of forces on a 2-dimensional lattice. This is done for several choices of eccentricities and semi-major axes. Using these results, we can predict which precisions and lattice constants allow for a detection of the relativistic perihelion advance in N-body simulation. We find that there are only small windows of parameters -- such as eccentricity, distance from the central object and the Schwarzschild radius -- for which the corrections can be detected in the numerics.

翻译：宇宙学N体模拟通常在超大规模并行计算机上进行，这要求使用简单的时间积分器以及网格逼近势函数。近年来，Adamek等人（2015）和Barrera-Hinojosa等人（2019）发展了广义相对论N体模拟，主要用于捕捉宇宙学相关的相对论效应。为此，我们探究在现有技术条件下，能否在数值上以相关精度探测到如近日点进动等相对论效应。首先，我们研究开普勒问题中虚假的近日点进动，作为所用积分方法的函数，随后将其作为二维格点上的附加力插值函数进行分析。这一研究针对多种偏心率和半长轴参数组合进行。基于这些结果，我们能够预测在N体模拟中探测相对论近日点进动所需的精度和格点常数。研究发现，仅在小范围参数窗口内（如偏心率、距中心天体的距离和史瓦西半径），相对论修正才能在数值模拟中被探测到。